Nested intervals

Nested intervals

Numerical Methods for Use in Preparing High-Quality Statistical Tables

Published in Donald B. Owen, Subir Ghosh, William R. Schucany, William B. Smith, Statistics of Quality, 2020

In practice, one would find a k* > j* using scalar computations. Then, beginning with k*, compute an interval enclosure Pk* = Rk* + Ek* of pρ,n(r > cα,n. Do this for k* + 1, k* + 2,… . and maintain the intersection of successive enclosing intervals Pk* ∩ Pk*+1 ∩ Pk*+2 …. The process terminates when the width of the intersection of enclosures is small enough to satisfy accuracy requirements or decreases in width cease to occur. The reason that intersection is used for what is theoretically a sequence of nested intervals is that in rounded interval arithmetic the nesting may not be complete but intersection enforces the nested effect.

Connected Sets and the Intermediate Value Theorem

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Published in John D. Ross, Kendall C. Richards, Introductory Analysis, 2020

John D. Ross, Kendall C. Richards

The Nested Interval Property guarantees that the infinite intersection of these nested intervals is nonempty. We define the point c to be the point that’s in this infinite intersection. We want to show that f(c) = 0, which we will do in a similar manner as the previous proof (by showing f(c) ≥ 0 and f(c) ≤ 0.

Shadowing property for the free group acting in the circle

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Journal Information

Published in Dynamical Systems, 2020

Jorge Iglesias, Aldo Portela

Now we construct inductively a sequence of nested intervals . Since , let We will prove that iff .

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Numerical Methods for Use in Preparing High-Quality Statistical Tables

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Shadowing property for the free group acting in the circle