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For example, JavaScript's loose equality rules can cause equality to be intransitive (i.e., a == b and b == c, but a != c), or make certain values be equal to their own negation. [ 2 ] A strict equality operator is also often available in those languages, returning true only for values with identical or equivalent types (in PHP, 4 === "4" is ...
Under intensional equality, two functions f and g are considered equal if they have the same "internal structure". This kind of equality could be implemented in interpreted languages by comparing the source code of the function bodies (such as in Interpreted Lisp 1.5) or the object code in compiled languages .
The closest pair of points problem or closest pair problem is a problem of computational geometry: given points in metric space, find a pair of points with the smallest distance between them. The closest pair problem for points in the Euclidean plane [ 1 ] was among the first geometric problems that were treated at the origins of the systematic ...
In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value () of some function. An important class of pointwise concepts are the pointwise operations, that is, operations defined on functions by applying the operations to function values separately for each point in the domain of definition.
It was common into the 18th century to use an abbreviation of the word equals as the symbol for equality; examples included æ and œ , from the Latin aequālis. [9] Diophantus's use of ἴσ , short for ἴσος (ísos 'equals'), in Arithmetica (c. 250 AD) is considered one of the first uses of an equals sign. [10]
To see this, note that the two constraints x 1 (x 1 − 1) ≤ 0 and x 1 (x 1 − 1) ≥ 0 are equivalent to the constraint x 1 (x 1 − 1) = 0, which is in turn equivalent to the constraint x 1 ∈ {0, 1}. Hence, any 0–1 integer program (in which all variables have to be either 0 or 1) can be formulated as a quadratically constrained ...
When minimizing a function f in the neighborhood of some reference point x 0, Q is set to its Hessian matrix H(f(x 0)) and c is set to its gradient ∇f(x 0). A related programming problem, quadratically constrained quadratic programming , can be posed by adding quadratic constraints on the variables.
A popular example combines a 7-point Gauss rule with a 15-point Kronrod rule (Kahaner, Moler & Nash 1989, §5.5). Because the Gauss points are incorporated into the Kronrod points, a total of only 15 function evaluations are needed.