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Suppose { r k } is a bounded sequence of real numbers and { d k, l } is a doubly index sequence of real numbers which satisfy lim sup k → ∞ lim sup l → ∞ d k, l ≤ 0, and r k + l ≤ r k + d k, l. for each k, l ∈ N. Then lim k → ∞ r k = a for some a ∈ R. Lemma 2.7[30, 31].
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Suppose { r k } is a bounded sequence of real numbers and { d k, n } is a doubly-index sequence of real numbers which satisfy lim sup k → ∞ lim sup n → ∞ d k, n ≤ 0 and r k + n ≤ r k + d k, n. for each k, n ≥ 1.
where and throughout the paper we understand the EINSTEIN summation convention over doubly repeated indices.
The ProShares UltraShort QQQQ exchange-traded fund is designed not only to move opposite the Nasdaq index, but to move doubly opposite.
In practice, this probably means using an index structure with direct, indirect, and doubly indirect blocks.
A doubly resolvable packing design with block size k, index λ, replication number r, and v elements is called a generalized Kirkman square and denoted by GKSk v 1,λ r).
Table 2 summarizes model fit indices for confirmatory factor analyses and doubly latent models.
We write our real-valued series of observations as …X−2, X−1, X−0, X1, X2, …, a doubly infinite sequence of real-valued random variables indexed by integer numbers.
Existence of GKS3(4u;1,1;2(u−1))s and GKS3 6u 1,1 3 u−1))s is implied by existence of doubly resolvable group divisible designs with block size 3, index 1, and types 4u and 6u (i.e., (3,1 -DRGDDs of types 4u and 6u).
A doubly clad optical fiber with an inner cladding made of negative refractive index material was proposed and the modes distribution, dispersion and time delay were investigated by fully vectorial numerical method.
Any rebound in domestic investing would now doubly help boosting both the firms' own profits and the returns of the index in general.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com