Sentence examples for bounds with respect to from inspiring English sources
A similar issue is faced in online learning when one tries to prove tracking expert regret bounds, i.e., bounds with respect to a shifting expert.
The failure probability and its sensitivity to input parameters are obtained together with confidence bounds with respect to uncertain input quantities.
Using the definition of Laplace transform, it is shown that time-domain integral equalities have to be satisfied by the step response of SISO (single-input, single-output) stable system, and shown that the maximum magnitude of its step response has some lower bounds with respect to the achievable settling time of the systems.
Note that the operations + and · on valuations give least upper bounds and greatest lower bounds with respect to this partial order.
We assume there that the symbol a satisfies elliptic type bounds with respect to a rather general comparison function f that is positive, increasing, and has suitable growth order.
Another closely related work that deals with mobile asynchronous sensors is [12], where the estimation bounds with respect to both sensor synchronization and position errors are developed and validated by Monte Carlo simulations.
Then the nonlinear dynamical system (3.1), (3.2) is ultimately bounded with respect to uniformly in with ultimate bound where.
If, in addition, and is a class- function, then the nonlinear dynamical system (3.1) and (3.2) is globally ultimately bounded with respect to uniformly in with ultimate bound.
(ii) The discrete-time nonlinear dynamical system (3.1) and (3.2) is ultimately bounded with respect to uniformly in with ultimate bound if there exists such that, for every, there exists such that implies,.
Hence, it follows from Theorem 3.3 that the closed-loop system given by (4.17), (A.2), and (A.3) is globally ultimately bounded with respect to uniformly in with ultimate bound given by, where.
The discrete-time nonlinear dynamical system (3.1) and (3.2) is globally ultimately bounded with respect to uniformly in with ultimate bound if, for every, there exists such that implies,. .
