Throughout the paper, we say that the expressions A and B are equivalent, and write A ≈ B whenever there exist positive constants C 1 and C 2 such that C 1 A ≤ B ≤ C 2 A. As usual, the letter C denotes a positive constant, possibly different on each occurrence.
This paper addresses the makespan minimization problem in scheduling flexible job shops whenever there exist separable sequence-dependent setup times.
This shows that AMD is able to quickly disseminate messages whenever there exist end-to-end connectivity to one of the fixed vehicles responsible for gathering data messages.
The selfmap T on X is called a generalized convex contraction whenever there exist a mapping α : X × X → [ 0, ∞ ) and a, b ∈ [ 0, 1 ), with a + b < 1, such that α ( x, y ) d ( T 2 x, T 2 y ) ≤ a d ( T x, T y ) + b d ( x, y ). for all x, y ∈ X. We say that α is the based mapping.
To this end, let us recall, according to [6], that an extended quasi-metric space ((X,d)) is said to be balanced if, given (r,s>0) and (x,yin X) we have (d x,y leq r+s) whenever there exist sequences ((x_{n})_{ninmathbb{N}}), and ((y_{n})_{ninmathbb{N}}) with (lim_{n,mrightarrowinfty }d y_{m},x_{n})=0) and such that (d x,x_{n})leq r) and (d y_{n},y leq s) for all (ninmathbb{N}).
Also, we say that the selfmap T on X is a generalized convex contraction of order 2 whenever there exist a mapping α : X × X → [ 0, ∞ ) and a 1, a 2, b 1, b 2 ∈ [ 0, 1 ) with a 1 + a 2 + b 1 + b 2 < 1 such that α ( x, y ) d ( T 2 x, T 2 y ) ≤ a 1 d ( x, T x ) + a 2 d ( T x, T 2 x ) + b 1 d ( y, T y ) + b 2 d ( T y, T 2 y ). for all x, y ∈ X. Other useful references: [12 15].
The conservativity of a minimal quantum dynamical semigroup is proved whenever there exists a "generalized" subharmonic operator bounded from below by the dissipative part of the infinitesimal generator.
In this paper, we investigate whenever there exists Λ such that {e−2πiλx}λ∈Λ is an orthonormal basis for L2(μρ,{ak,bk}).
A curve (omega :[0,1]rightarrow X) is said to be absolutely continuous whenever there exists (gin L^1 [0,1])) such that (d omega (t_0),omega (t_1))le int _{t_0}^{t_1}g(s),mathrm{d}s) for every (t_0
For any nonempty subset of a metric space, the diameter of is denoted and defined by = sup A mapping has diminishing orbital diameters [19, 20] if for each and whenever there exists such that.
A point x ∈ X is called an interior point of a set A ⊆ X whenever there exists 0 ≺ r ∈ C such that B ( x, r ) : = { y ∈ X : d ( x, y ) ≺ r } ⊆ A, where B ( x, r ) is an open ball.
