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In practice, the quantizer consists of two mappings: an encoder mapping and a decoder mapping.
Let be a set, as in Theorem 2.6, a surjective mapping, and a family of arbitrary mappings.
Let, be two set-valued mappings with for every, a single-valued mapping, and a subset of.
Some claims need extensive mapping and a lengthy verification of claimants.
Let be a single-valued mapping and a multivalued mapping.
Let be a single-valued mapping and a multivalued contraction.
Let be a semimonotone mapping and a closed convex set.
By Theorem 2.5, there are a unique quadratic mapping and a unique quartic mapping satisfying (2.26).
Then there exist a unique quadratic mapping and a unique quadratic mapping such that (2.17).
Let be a continuous single-valued mapping and a lower semicontinuous multivalued mapping.
Let f : I ∘ ⊆ R → R be a differentiable mapping and a, b ∈ I ∘ with a < b.
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