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That ({mathrm{Tr}} P-Q) in {mathbb {Z} P-Qas finst proven by Effros [134] and can also be proven using the Krein spectral shift [616, Problem 5.9.1].

If the formalism is applied on a compact metric space, then it is not required for large contractions the fulfilment of the boundedness condition of Theorem 2.12 ii - iii) from Edelstein fixed point theorem [8] which can be proven using the Meir-Keeler theorem [9] as observed in [10].

This converse also appears in Euclid's Elements (Book I, Proposition 48): It can be proven using the law of cosines or as follows: Let ABC be a triangle with side lengths a, b, and c, with a2 + b2 = c2.

On the one hand, one could conclude that risk signatures developed on a mixture of samples perform better, but on the other hand this hypothesis cannot be proven using the HD4 dataset since the signature had been developed on it.

The last equality can be proved using the polar decomposition.

In the same way, the following result can be proved using the ( O, M ) -compatibility involved in Theorem 33.

This assertion can easily be proved using the Painlevé and Liouville theorems in the Clifford analysis setting, see [1, 15].

For the fractional-order linear time invariant systems, the stability can easily be proved using the method proposed by Matignon [24].

Further results can be proved using the same plots as those of the earlier theorems in this paper, so we omit them.

Part (a) can be proved using the Poisson summation formula; see, e.g., [75, Chapter VII, Theorem 2.4 and Corollary 2.6]; a recent proof is given in [22].

Notice that the upper estimate in (3.54) and (3.55) can also be proved using the bounds for Jacobi-Sobolev polynomials given in Corollary 3.5.