1,134 research outputs found
Characterization of the critical Sobolev space on the optimal singularity at the origin
Get PDFIn the present paper, we investigate the optimal singularity at the origin for the functions belonging to the critical Sobolev space H^[n/p, p] (ℝ^[n]), 1 < p < ∞. With this purpose, we shall show the weighted Gagliardo-Nirenberg type inequality: ||u|| _[Lq(ℝ^[n] ; dx/|x|s)] ≤ C(1/[n-s])^[1/q + 1/p'] q^[1/p'] ||u|| ^[(n-s)p/nq] _[Lp(ℝ^[n]] ||(-Δ)^[n/2p] u||^[1-[(n-s)p]/nq] _[Lp(ℝ^[n]], (GN) where C depends only on n and p. Here, 0 ≤ s < n and p~ ≤ q < ∞ with some p~ ∈ (p, ∞) determined only by n and p. Additionally, in the case n ≥ 2 and n/[n-1] ≤ p < ∞, we can prove the growth orders for s as s ↑ n and for q as q → ∞ are both optimal. (GN) allows us to prove the Trudinger type estimate with the homogeneous weight. Furthermore, it is obvious that (GN) can not hold with the weight |x|^[n] itself. However, with a help of the logarithmic weight of the type (log 1/|x|)^[r] |x|^[n] at the origin, we cover this critical weight. Simultaneously, we shall give the minimal exponent r = [q+p']/p' so that the continuous embedding can hold
Software Requirements SEI Curriculum Module SEI-CM-19-1.2
No full textSEI educational materials are being made available to educators throughout the academic, industrial, and governmentcommunities. The use of these materials in a course does not in any way constitute an endorsement of the course by the SEI, by Carnegie Mellon University, or by the United States government. SEI curriculum modules may be copied or incorporated into other materials, but not for profit, provided that appropriatecredit is given to the SEI and to the original author of the materials
Sei arie con istromenti, composte dal Sig. Gioseppe Manfredini.
No full textBound with: F. Palma, Sei arie con istromenti ... Opera terza, London 1749
The Latest Work from the SEI
No full textAs part of an ongoing effort to keep you informed about our latest work, this blog post summarizes some recent publications from the SEI in the areas of supply chain risk management, technical debt, large language models, quantum computing, acquisition, and trustworthiness in AI systems. These publications highlight the latest work of SEI technologists in these areas. This post includes a listing of each publication, author(s), and links where they can be accessed on the SEI website. </p
- …
