Sentence examples for expansion maps from inspiring English sources

Foam expansion and pore shape were studied by the construction of expansion maps and image analysis on foam cross-sections.

As confirmation of the latter, an eye position map for each stimulus has been plotted (Fig. 5E,F), where the downward fixation has been subtracted from the upward fixation in the clockwise and expansion maps.

From book to book, the Histories lets you track Persia's expansion, mapped by its conflicts with whomever it is trying to subjugate at the time.

After Fukushima disaster in Japan, China NPPs' expansion map changed as shown in Figure 5.

Then for the expansion mapping T, there exists a unique fixed point in X. Proof Firstly, T is injective.

For ω = ω 1 ω 2 ⋯, the "binary decimal expansion map" π : = ∑ k = 1 ∞ ω k 2 − k is a coding map.

Here, 0 and 1 are two fixed points of T. Remark 3.3 As in the previous example, the expansion mapping theorem is not applicable in this case either.

In this paper, we prove expansion mapping theorems using the concept of compatible maps, weakly reciprocal continuity, R-weakly commuting mappings, R-weakly commuting of type ( A f ), ( A g ) and ( P ) in metric spaces and in G-metric spaces.

We call a mapping T is a C ∗ -algebra-valued expansion mapping on X, if T : X → X satisfies: (1) T ( X ) = X ;   (2) d ( T x, T y ) ⪰ A ∗ d ( x, y ) A, ∀ x, y ∈ X,  .

In this example, 0 and 3/2 are two fixed points of T. Remark 3.2 The expansion mapping theorem proved by Wang et al.[5] cannot be applied in the above example since we have d ( T ( 1 2 ), T ( 0 ) ) = 1 4 < 1 2 = d ( 1 / 2, 0 ).

We say that T is an -expansive mapping if there exist two functions ξ ∈ χ and α : X × X → [ 0, + ∞ ) such that ξ ( d ( T x, T y ) ) ≥ α ( x, y ) d ( x, y ) (1). for all x, y ∈ X. Remark 3.1 If T : X → X is an expansion mapping, then T is an -expansive mapping, where α ( x, y ) = 1 for all x, y ∈ X and ξ ( a ) = k a for all a ≥ 0 and some k ∈ [ 0, 1 ). We now prove our main results.