Remark 3.1 Our results improve the ones of Sedghi et al. [18] as follows: (i) from kt to ϕ ( t ) ; (ii) the functions F and g are not required to be commutable. . from kt to ϕ ( t ) ; the functions F and g are not required to be commutable.
If A − 1 is a doubly stochastic matrix, then A e = e, A T e = e, where e = ( 1, 1, …, 1 ) T. In this section, we give two new lower bounds for τ ( A ∘ A − 1 ) which improve the ones in [8] and [9].
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Strauss offered Pietersen a role as an advisor on a board to improve the one-day side, which was turned down.
Fang's results improve the one of Jachymski (Nonlinear Anal. 73:2010-2203, 2010) by relaxing the restriction on the gauge function φ.
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"A robust Republican minority in the Senate has kept many bad things from happening this year, as well as improved the ones worth improving," Mr. McConnell said in a speech to the Federalist Society.
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This result improves the ones known in the literature.
