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For example, 15 is a composite number because 15 = 3 · 5, but 7 is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example 60 = 3 · 20 = 3 · (5 · 4) .
If the decaying quantity, N(t), is the number of discrete elements in a certain set, it is possible to compute the average length of time that an element remains in the set. This is called the mean lifetime (or simply the lifetime ), where the exponential time constant , τ {\displaystyle \tau } , relates to the decay rate constant, λ, in the ...
For example, the p i may be the factors of the square-free factorization of g. When K is the field of rational numbers , as it is typically the case in computer algebra , this allows to replace factorization by greatest common divisor computation for computing a partial fraction decomposition.
In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur functions as a linear combination of other Schur functions. These coefficients are natural numbers, which the Littlewood–Richardson rule describes as counting certain skew tableaux.
Such a number is algebraic and can be expressed as the sum of a rational number and the square root of a rational number. Constructible number: A number representing a length that can be constructed using a compass and straightedge. Constructible numbers form a subfield of the field of algebraic numbers, and include the quadratic surds.
We can now define the decreasing rearrangment (or, sometimes, nonincreasing rearrangement) of as the function : [,) [,] by the rule = {[,]: ()}. Note that this version of the decreasing rearrangement is not symmetric, as it is only defined on the nonnegative real numbers.
For example, () = = is the number of different podiums—assignments of gold, silver, and bronze medals—possible in an eight-person race. On the other hand, x ( n ) {\displaystyle x^{(n)}} is "the number of ways to arrange n {\displaystyle n} flags on x {\displaystyle x} flagpoles", [ 8 ] where all flags must be used and each flagpole can ...
A chain in this partial order is a monotonically increasing subsequence, and an antichain is a monotonically decreasing subsequence. By Mirsky's theorem, either there is a chain of length r , or the sequence can be partitioned into at most r − 1 antichains; but in that case the largest of the antichains must form a decreasing subsequence with ...