Sentence examples for and generalized in from inspiring English sources
The proximal point algorithm, can be traced back to Martinet [33] in the context of convex minimization and Rockafellar [34] in the general setting of maximal monotone operators, has been extended and generalized in different directions by using novel and innovative techniques and ideas.
The important generalization of variational inequalities, called variational inclusions, have been extensively studied and generalized in different directions to study a wide class of problems arising in mechanics, optimization, nonlinear programming, economics, finance and applied sciences, and so forth; see, for example [1 7] and references theirin.
The convex sets and convex functions have been extended and generalized in different directions using innovative ideas to study different problems in a general and unified frame work; see [15 21].
Thereafter it has been proved and generalized in various ways by many authors.
Hence for this particular canal the head-middle-tail inequity often reported and generalized in the literature is not observed.
Hence, these are feasible, economical and effective methods and will be applied and generalized in industry in the future.
The relativistic law of energy-momentum conservation thus combines and generalizes in one relativistically invariant expression the separate conservation laws of prerelativistic physics: the conservation of mass, the conservation of momentum, and the conservation of energy.
The following consequences of Theorem 2, which are also illustrated by Example 4 below, improve and generalize in several directions the Banach contraction principle.
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Convergence rates have been derived in [6] for linear equations in Hilbert spaces and later generalized in [34] to non-linear equations in Banach spaces.
The Nelson consensus method, first described in [ 36] and then generalized in [ 37], relies on the graph theory techniques to find maximum cliques of mutually compatible clusters.
