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Some of these systems provide the end user not only with such a personalized item list but also with an explanation which describes why a specific item is recommended and why the system supposes that the user will like it.
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For a dynamic stochastic system, suppose that the random process y ∈ [ a, b ] is the output of the stochastic system, its output PDFs are defined by γ ( z, u ( t ) ), where u ( t ) ∈ R m is control input.
For the above system, suppose that there exists a function S: ℝ2 n × Π → ℝ which is sufficiently smooth and satisfies (13) Then (14) if the system is released from an initial state (x 0), p) ∈ D × Π where x 0) is in a level set of S entirely contained in D, || x1 0)|| = || x2(0)|| = β ≤ α.
But there's no system — I suppose that is a system.
Proof of Lemma 3. Suppose that system (2) is positive.
Suppose that system (2.1) satisfies the assumptions (H1)–(H4) and (A),, where,,,,,,,.
Necessity: Suppose that system (4) is asymptotically stable for any delays satisfying (5).
Proof Suppose that system (1.1) is not relatively disconjugate on Z [ a, b + 1 ].
Suppose that system (4.28) admits at least one stable equilibrium ((A_{1infty}, A_{2infty})).
Suppose that system (1) satisfies Assumption 1 and the observer has the form of (5).
Suppose that system (3) is Ω-controllable to X in time (t_{1}inmathbb{T}) from (x_{0}).
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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