For the nonlinearities (activation functions), the first assumptions of boundedness,

monotonicity, and differentiability have been all weakened to a Lipschitz condition.

Losonczi, Ratio of Stolarsky means:

monotonicity and comparison, Publ.

Now, from the first equation of system (1) and the

monotonicity of y(t), we have

This lemma is stated for the smaller and simpler probability 1/(n ln n), but applies also to larger p by

monotonicity.

In this paper, a direct approach based on the

monotonicity principle is used [24].

Roorda LD, Houwink A, Smits W, Molenaar IW, Geurts AC (2011): Measuring upper limb capacity in poststroke patients: development, fit of the monotone homogeneity model, unidimensionality, fit of the double

monotonicity model, differential item functioning, internal consistency, and feasibility of the stroke upper limb capacity scale, SULCS.

Monotonicity ([E.sub.C] > [E.sub.D]) holds when [p(C | C) - p(C | D)](T - P) > P - S + 9(C | C)[(T - P) - (R - S)].

The arguments for considering an explanation on the basis of

monotonicity on the part-whole relation as a solution to the puzzle posed by the data are found in Section 5, and Section 6 argues that

monotonicity on the partwhole relation cannot explain the Estonian data.

Together with Theorem 2.2 we deduce that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is not contained in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](K), hence by the

monotonicity of the index, the test-map (3) does not exist, which finishes the proof.

Finally, deviations with a' [greater than or equal to] a and d' [less than or equal to]d can only increase the payment made by a winning agent by

monotonicity (since the critical value must increase) and will have no effect on the outcome for a losing agent.

The

monotonicity of the function N[a, b; p, q; r, [alpha], [mu]] is proved in the following theorem.

This means the strong convexity of the functional, which implies the uniform

monotonicity of the nonlinear elliptic operator, defined by (3.1).