We shall prove, in the case of rank one projection operators, that the generating operators for such multipliers are either Riesz bases (exact frames) or not frames for their closed linear spans.
In other words, arithmetic is incomplete, i.e. some propositions of arithmetic can be understood as valid; yet they cannot be proven within the frame of arithmetic.
In this frame, convergence and quasi-optimality of the adaptive algorithm has first been proved in [33], while the necessary technical tools have independently been developed in [93].
As mentioned above, it is proved in [7] that the 2D Parseval frame of shearlets ψ ~ μ, μ ∈ M provides essentially optimal approximations for functions of two variables which are C2 regular away from discontinuities along C2 curves.
To explain this lack of effectiveness, it has been postulated that these preventive strategies have usually only proved to be effective in the frame of well controlled trials, usually in specialized clinics, but have failed to show any benefit when applied to the "real world" [ 12].
In this section, we prove coincidence point results in the frame work of partially ordered metric spaces in terms of a -distance.
The error rate is generally the highest in frames −1 and −2 of experimentally proven CDSs and the frame where its maximum is found varies according to the species and sequence size (columns 14, Fr−1 and 15, Fr−2).
His 51 proved decisive in the final frame, with O'Sullivan failing to seize the chance to force a respotted black when he got a kick on the penultimate red and missed a thin cut on the black that followed.
But is every formula that is valid on all transitive conversely well-founded frames also provable in GL? Unaware of the arithmetical significance of GL, K. Segerberg proved in 1971 that GL is indeed complete with respect to transitive conversely well-founded frames; D. de Jongh and S. Kripke independently proved this result as well.
