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For example, the real projective space of dimension , where is a power of two, requires = for an embedding. However, this does not apply to immersions; for instance, R P 2 {\displaystyle \mathbb {R} \mathrm {P} ^{2}} can be immersed in R 3 {\displaystyle \mathbb {R} ^{3}} as is explicitly shown by Boy's surface —which has self-intersections.
An infinite-dimensional example of a continuous embedding is given by the Rellich–Kondrachov theorem: let Ω ⊆ R n be an open, bounded, Lipschitz domain, and let 1 ≤ p < n. Set =. Then the Sobolev space W 1,p (Ω; R) is continuously embedded in the L p space L p ∗ (Ω; R).
The technical statement appearing in Nash's original paper is as follows: if M is a given m-dimensional Riemannian manifold (analytic or of class C k, 3 ≤ k ≤ ∞), then there exists a number n (with n ≤ m(3m+11)/2 if M is a compact manifold, and with n ≤ m(m+1)(3m+11)/2 if M is a non-compact manifold) and an isometric embedding ƒ: M → R n (also analytic or of class C k). [15]
The alignment of two decimal numbers (for example 1.3 + 27.08) is a simple, exact shift. Conversion to a character form or for display (e.g., to a text-based format such as XML , or to drive signals for a seven-segment display ) is a simple per-digit mapping, and can be done in linear ( O ( n )) time.
Sometimes, it is desired to order text with embedded numbers using proper numerical order. For example, "Figure 7b" goes before "Figure 11a", even though '7' comes after '1' in Unicode. This can be extended to Roman numerals. This behavior is not particularly difficult to produce as long as only integers are to be sorted, although it can slow ...
An embedded graph uniquely defines cyclic orders of edges incident to the same vertex. The set of all these cyclic orders is called a rotation system.Embeddings with the same rotation system are considered to be equivalent and the corresponding equivalence class of embeddings is called combinatorial embedding (as opposed to the term topological embedding, which refers to the previous ...
An embedded system on a plug-in card with processor, memory, power supply, and external interfaces. An embedded system is a specialized computer system—a combination of a computer processor, computer memory, and input/output peripheral devices—that has a dedicated function within a larger mechanical or electronic system.
Unums (universal numbers [1]) are a family of number formats and arithmetic for implementing real numbers on a computer, proposed by John L. Gustafson in 2015. [2] They are designed as an alternative to the ubiquitous IEEE 754 floating-point standard. The latest version is known as posits. [3]