He shows that the stretch factor is strictly decreasing at each of its local maxima, except for the case of the two ends of a diameter of a circle in which case the stretch factor is constant at.
12.
The array starts with strictly increasing subsequence ( every consecutive elemnt is stricly larger than the previous element ), and then appears a strictly decreasing subsequence ( every consecutive elemnt is stricly smaller than the previous element ).
13.
Amortized O ( 1 ) time appends ( or prepends ) are achieved by introducing a new inner node type called " Append ", and using it to encode a logarithmic-length list of Conc-trees, strictly decreasing in height.
14.
It is also sometimes referred to as the two-parameter Poisson Dirichlet process, after the two-parameter generalization of the Poisson Dirichlet distribution which describes the joint distribution of the sizes of the atoms in the random measure, sorted by strictly decreasing order.
15.
The distance d between points A and B on the sphere staisfies cos ( d ) = A . B, and since we are requiring 0 d " d d " ? where cosine is strictly decreasing, if the dot product is smaller then the distance is larger.
16.
But " P " ( " m " ) is strictly decreasing and the standard order < on ordinals is well-founded, therefore an infinite strictly decreasing sequence cannot exist, or equivalently, every strictly decreasing sequence of ordinals does terminate ( and cannot be infinite ).
17.
But " P " ( " m " ) is strictly decreasing and the standard order < on ordinals is well-founded, therefore an infinite strictly decreasing sequence cannot exist, or equivalently, every strictly decreasing sequence of ordinals does terminate ( and cannot be infinite ).
18.
But " P " ( " m " ) is strictly decreasing and the standard order < on ordinals is well-founded, therefore an infinite strictly decreasing sequence cannot exist, or equivalently, every strictly decreasing sequence of ordinals does terminate ( and cannot be infinite ).
19.
Note that in the first two cases, " f " is not required to be strictly increasing or strictly decreasing to the left or right of " x ", while in the last two cases, " f " is required to be strictly increasing or strictly decreasing.
20.
Note that in the first two cases, " f " is not required to be strictly increasing or strictly decreasing to the left or right of " x ", while in the last two cases, " f " is required to be strictly increasing or strictly decreasing.
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