The product of r−k and ∇δ on ℝm
In the theory of distributions, there is a general lack of definitions for products and powers of distributions. In physics (Gasiorowicz (1967), page 141), one finds the need to evaluate δ2 when calculating the transition rates of certain particle interactions and using some products such as (1/x)⋅...
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| Format: | Article |
| Language: | English |
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Wiley
2000-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171200004233 |
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| _version_ | 1832566773468102656 |
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| author | C. K. Li |
| author_facet | C. K. Li |
| author_sort | C. K. Li |
| collection | DOAJ |
| description | In the theory of distributions, there is a general lack of
definitions for products and powers of distributions. In physics
(Gasiorowicz (1967), page 141), one finds the need to evaluate
δ2 when calculating the transition rates of certain
particle interactions and using some products such as
(1/x)⋅δ. In 1990, Li and Fisher introduced a
computable delta sequence in an m-dimensional space to obtain
a noncommutative neutrix product of r−k and Δδ (Δ denotes the Laplacian) for any
positive integer k between 1 and m−1 inclusive. Cheng and
Li (1991) utilized a net δϵ(x) (similar to the
δn(x)) and the normalization procedure of μ(x)x+λ to deduce a commutative neutrix product of
r−k and δ for any positive real number k. The object
of this paper is to apply Pizetti's formula and the normalization
procedure to derive the product of r−k and ∇δ (∇ is the gradient operator) on ℝm. The nice properties
of the δ-sequence are fully shown and used in the proof of
our theorem. |
| format | Article |
| id | doaj-art-9d9b9ce8c9a649e3a1bb3dadf0875995 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2000-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-9d9b9ce8c9a649e3a1bb3dadf08759952025-02-03T01:03:21ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252000-01-0124636136910.1155/S0161171200004233The product of r−k and ∇δ on ℝmC. K. Li0University College of Cape Breton, Mathematics and Computer Science, Nova Scotia, Sydney B1P 6L2, CanadaIn the theory of distributions, there is a general lack of definitions for products and powers of distributions. In physics (Gasiorowicz (1967), page 141), one finds the need to evaluate δ2 when calculating the transition rates of certain particle interactions and using some products such as (1/x)⋅δ. In 1990, Li and Fisher introduced a computable delta sequence in an m-dimensional space to obtain a noncommutative neutrix product of r−k and Δδ (Δ denotes the Laplacian) for any positive integer k between 1 and m−1 inclusive. Cheng and Li (1991) utilized a net δϵ(x) (similar to the δn(x)) and the normalization procedure of μ(x)x+λ to deduce a commutative neutrix product of r−k and δ for any positive real number k. The object of this paper is to apply Pizetti's formula and the normalization procedure to derive the product of r−k and ∇δ (∇ is the gradient operator) on ℝm. The nice properties of the δ-sequence are fully shown and used in the proof of our theorem.http://dx.doi.org/10.1155/S0161171200004233Pizetti's formuladelta sequenceneutrix limit and distribution. |
| spellingShingle | C. K. Li The product of r−k and ∇δ on ℝm International Journal of Mathematics and Mathematical Sciences Pizetti's formula delta sequence neutrix limit and distribution. |
| title | The product of r−k and ∇δ on
ℝm |
| title_full | The product of r−k and ∇δ on
ℝm |
| title_fullStr | The product of r−k and ∇δ on
ℝm |
| title_full_unstemmed | The product of r−k and ∇δ on
ℝm |
| title_short | The product of r−k and ∇δ on
ℝm |
| title_sort | product of r k and ∇δ on rm |
| topic | Pizetti's formula delta sequence neutrix limit and distribution. |
| url | http://dx.doi.org/10.1155/S0161171200004233 |
| work_keys_str_mv | AT ckli theproductofrkanddonrm AT ckli productofrkanddonrm |