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In this context, the term "monotonic transformation" refers to a positive monotonic transformation and is intended to distinguish it from a "negative monotonic transformation," which reverses the order of the numbers. [6]
In the case of a completely monotonic function, the function and its derivatives must be alternately non-negative and non-positive in its domain of definition which would imply that function and its derivatives are alternately monotonically increasing and monotonically decreasing functions.
In particular, infinite sums of non-negative numbers converge to the supremum of the partial sums if and only if the partial sums are bounded. For sums of non-negative increasing sequences 0 ≤ a i , 1 ≤ a i , 2 ≤ ⋯ {\displaystyle 0\leq a_{i,1}\leq a_{i,2}\leq \cdots } , it says that taking the sum and the supremum can be interchanged.
When it is necessary to distinguish the usual measures which take non-negative values from generalizations, the term positive measure is used. Positive measures are closed under conical combination but not general linear combination , while signed measures are the linear closure of positive measures.
A function : defined on an interval is said to be operator monotone if whenever and are Hermitian matrices (of any size/dimensions) whose eigenvalues all belong to the domain of and whose difference is a positive semi-definite matrix, then necessarily () where () and () are the values of the matrix function induced by (which are matrices of the same size as and ).
As explained in Riesz & Sz.-Nagy (1990), every non-decreasing non-negative function F can be decomposed uniquely as a sum of a jump function f and a continuous monotone function g: the jump function f is constructed by using the jump data of the original monotone function F and it is easy to check that g = F − f is continuous and monotone. [10]
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For example, the logical disjunction function or with boolean values used for true (1) and false (0) is positive unate. Conversely, Exclusive or is non-unate, because the transition from 0 to 1 on input x0 is both positive unate and negative unate, depending on the input value on x1.