Sentence examples for and left limits from inspiring English sources

denotes the jump of at that is, where and represent the right and left limits of at, respectively.

where is a positive parameter,, is right-dense continuous,, and for each and represent the right and left limits of at.

where is a positive constant,, denotes the Caputo fractional derivative of order,,, and satisfy that,,, and represent the right and left limits of at.

For each interval of, we denote that by are the impulses at moments, and represent the right and left limits of and in the sense of time scales; is a strictly increasing sequence.

Furthermore, the fixed moments of times t k satisfy 0 < t 1 < ⋯ < t m < lim k → ∞ t k = ∞, x ( t k + ) and x ( t k − ) represent the right and left limits of x ( t ) at t = t k, respectively.

Similarly, Δ u ′ | t = t k = u ′ ( t k + ) − u ′ ( t k − ), where u ′ ( t k + ) and u ′ ( t k − ) represent the right and left limits of u ′ ( t ) at t = t k, respectively.

Moreover, there exist finite right limit and left limit at discontinuity point.

The impulsive moment (t_{k}) ((k=1, 2,ldots)) satisfies (0< t_{1}< t_{2}<cdots<t_{k}<cdots) and (lim_{krightarrow+infty} t_{k}=+infty), (I_{k}: mathbb{H} rightarrowmathbb{H}), (bigtriangleup x t) = x(t^) - x(t^)), where (x(t^)) and (x(t^)) denote the right and left limit of x at t, respectively.

where d > 0,L t, ω) is a stochastic Lévy process which trajectories are right-continuous and have left limits.

"However, years of falling shop prices and higher costs have left limited scope for retailers to continue absorbing this pressure, and everyone in the supply chain will need to play their part in maintaining low prices for consumers".

One possible reason is that the baselines in our study were higher since the morphologic and functional macular impairments in early AMD did not seriously affect visual functions, and this may have left limited space for improvement.