Florian Cajori's A History of Mathematical Notations: Vol. I, Notations in PDF

By Florian Cajori

ISBN-10: 2102182312

ISBN-13: 9782102182310

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10) that φ(t) ≤ C0 (1 + t)−η with η = min1≤i≤m {( i − θi )/(1 − i )}. 11. Let φ(t) be a non-negative function on R+ ≡ [0, +∞) satisfying sup φ(s)1+γ ≤ K 0 (1 + t)γ {φ(t) − φ(t + 1)} t ≤s≤t +1 for some constants K 0 > 0, γ > 0, β < 1. Then φ(t) has the decay property: φ(t) ≤ C0 (1 + t) − (1−β) γ ; and if γ = 0, then φ(t) ≤ C0 exp{−λt 1−β } where C0 > 0, λ > 0 are constants. 3. 12. Let φ(t) be a non-negative function on R+ ≡ [0, +∞) satisfying sup t ≤s≤t +T φ(s)1+γ ≤ g(t)[φ(t) − φ(t + T )] with constants T > 0, γ > 0 and g(t) is a non-decreasing function.

A linear operator A : X ⊃ D(A) −→ R(A) ⊆ X is dissipative if and only if (λI − A)x ≤ λ x f or all x ∈ D(A), λ > 0. 6. The motivation for the use of the word “dissipative” comes from the case where X is a Hilbert space. ) denotes the scalar inner product on X. 11. A linear operator A : X ⊃ D(A) −→ R(A) ⊆ X is m-dissipative if A is dissipative and R(λI − A) = X f or all λ > 0 that is, for any given g ∈ X, there is f ∈ D(A) such that (λI − A) f = g. 1. Every m-dissipative operator is a dissipative operator.

Preliminary where h(t) ≥ 0 with +∞ 0 h(t)dt ≤ C2 < +∞ and f is a nondecreasing function from R+ into R+ . Then lim y(t) = 0. 6. 5. Then lim y(t) = 0. t →+∞ From the above context of this subsection, we only know that the non-negative function (y(t), say) goes to zero as time tends to infinity. We have no information on the decay rate of y(t). In fact, the decay rate of y(t) depends on some factors which include some terms in the inequality. , Mu˜noz Rivera [275]). 7. Suppose that y(t) ∈ C 1 (R+ ), y(t) ≥ 0, ∀t > 0 and satisfies y (t) ≤ −C0 y(t) + C1 e−γ t where C0 , C1 and γ are positive constants.

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A History of Mathematical Notations: Vol. I, Notations in Elementary Mathematics by Florian Cajori


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