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Near Extensions and Alignment of Data in R(superscript)n
book

Near Extensions and Alignment of Data in R(superscript)n

by Steven B. Damelin
November 2023
Intermediate to advanced content levelIntermediate to advanced
192 pages
3h 29m
English
Wiley
Content preview from Near Extensions and Alignment of Data in R(superscript)n

8 Algebraic Geometry: Approximation-varieties, Lojasiewicz, Quantification: (ε, δ)-Theorem 2.2 (part (2))

In this chapter, we are now going to formulate and prove a variant of Theorem 2.2 ((part (2)), namely Theorem 8.12 where we are able to now give quantifications of relations between epsilon and delta 1 in Theorem 2.2 ((part (2)).

Theorem 8.12 consists of two parts. Part (1) deals with the case when we force our distinct points to lie on an ellipse and in part (2), we assume that our distinct points have not too large diameter and are not too close to each other.

Here is our result.

Theorem 8.12. The following holds [40]:

  • (1) Let delta greater-than 0 be small enough depending on d. There exist c comma c prime small enough depending on d such the following holds. Let StartSet y 1 comma ellipsis comma y Subscript k Baseline EndSet and be two collections of distinct points in with

    (8.1)

    and with ...

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