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Stable sort algorithms sort equal elements in the same order that they appear in the input. For example, in the card sorting example to the right, the cards are being sorted by their rank, and their suit is being ignored. This allows the possibility of multiple different correctly sorted versions of the original list.
Sometimes the list labeling problem is presented where S is not a set of values but rather a set of objects subject to a total order. In this setting, when an item is inserted into S, it is specified to be the successor of some other item already in S. For example, this is the way that list labeling is used in the order-maintenance problem. The ...
A constant-recursive sequence is any sequence of integers, rational numbers, algebraic numbers, real numbers, or complex numbers,,,, … (written as () = as a shorthand) satisfying a formula of the form
List or describe a set of sentences in the language L σ, called the axioms of the theory. Give a set of σ-structures, and define a theory to be the set of sentences in L σ holding in all these models. For example, the "theory of finite fields" consists of all sentences in the language of fields that are true in all finite fields. An L σ ...
In C and C++, the + operator is not associated with a sequence point, and therefore in the expression f()+g() it is possible that either f() or g() will be executed first. The comma operator introduces a sequence point, and therefore in the code f(),g() the order of evaluation is defined: first f() is called, and then g() is called.
In order for the solution method to work, as in linear equations, it is necessary to express every term in the nonlinear equation as a power series so that all of the terms may be combined into one power series. As an example, consider the initial value problem ″ + ′ + ′ =; = , ′ = which describes a solution to capillary-driven flow in ...
Furthermore, the mean value of the series can be calculated via: /: ¯ = +. The formula is essentially the same as the formula for the mean of a discrete uniform distribution, interpreting the arithmetic progression as a set of equally probable outcomes.
Here input is the input array to be sorted, key returns the numeric key of each item in the input array, count is an auxiliary array used first to store the numbers of items with each key, and then (after the second loop) to store the positions where items with each key should be placed, k is the maximum value of the non-negative key values and ...