monotonicity

The above definition of monotonicity is relevant in these cases as well.

The notion of monotonicity does not mean anything if states are portrayed as terms.

This can be seen since it has neither the translation property or monotonicity.

All of these procedures can be examined for various properties like monotonicity.

The definition used here is closer to the notion of monotonicity in logic.

The general idea of examining monotonicity does not depend on calculus.

The most fundamental condition that occurs in this context is monotonicity.

Hence, all the special preservation properties stated above induce monotonicity.

However this modification may violate monotonicity and the partitioning property of the mean.

This second change (a removal caused by an addition) violates the condition of monotonicity.

In this general case the inequality is called Monotonicity of quantum relative entropy.

By the monotonicity property of the integral, it is immediate that:

This means that monotonicity is preserved for this case.

The monotonicity was enforced as , for 2 k .

In order to verify the monotonicity property, we set and , where for .

There is no single straightforward generalization of polygon monotonicity to higher dimensions.

If monotonicity must be strict then must have a value strictly greater than zero.

If "f" and "g" are of opposite monotonicity, then the above inequality works in the reverse way.

From these properties, one can also conclude monotonicity of the adjoints immediately.

This is classic monotonicity violation: the 321 who voted for the Liberals took part in hurting their own candidate.

Intuitively, monotonicity indicates that learning a new piece of knowledge cannot reduce the set of what is known.

That is for all , and a simple counterexample for monotonicity can be found.

To prevent overshoot and ensure monotonicity, at least one of the following conditions must be met:

Using the monotonicity property of measures, we can continue the above equalities as follows:

We have shown as a byproduct of the proof of monotonicity that .