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The product of r − k and ∇ δ on ℝ m

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  • C. K. Li

Abstract

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 δ 2 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 Δ δ ( Δ denotes the Laplacian) for any positive integer k between 1 and m − 1 inclusive. Cheng and Li (1991) utilized a net δ ϵ ( x ) (similar to the δ n ( 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 ℝ m . The nice properties of the δ -sequence are fully shown and used in the proof of our theorem.

Suggested Citation

  • C. K. Li, 2000. "The product of r − k and ∇ δ on ℝ m," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 24, pages 1-9, January.
  • Handle: RePEc:hin:jijmms:569894
    DOI: 10.1155/S0161171200004233
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    Cited by:

    1. Shih, Han M., 2014. "Migrating product structure bill of materials Excel files to STEP PDM implementation," International Journal of Information Management, Elsevier, vol. 34(4), pages 489-516.
    2. Zaumanis, Martins & Mallick, Rajib B. & Frank, Robert, 2014. "100% recycled hot mix asphalt: A review and analysis," Resources, Conservation & Recycling, Elsevier, vol. 92(C), pages 230-245.

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