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This definition allows us to state Bézout's theorem and its generalizations precisely. This definition generalizes the multiplicity of a root of a polynomial in the following way. The roots of a polynomial f are points on the affine line, which are the components of the algebraic set defined by the polynomial.
However the multiplicity equals the number of spin orientations only if S ≤ L. When S > L there are only 2L+1 orientations of total angular momentum possible, ranging from S+L to S-L. [ 2 ] [ 3 ] The ground state of the nitrogen atom is a 4 S state, for which 2S + 1 = 4 in a quartet state, S = 3/2 due to three unpaired electrons.
Multiplicity (chemistry), multiplicity in quantum chemistry is a function of angular spin momentum; Multiplicity (informatics), a type of relationship in class diagrams for Unified Modeling Language used in software engineering; Multiplicity (mathematics), the number of times an element is repeated in a multiset
This extended multiplicity function is commonly called simply the multiplicity function, and suffices for defining multisets when the universe containing the elements has been fixed. This multiplicity function is a generalization of the indicator function of a subset , and shares some properties with it.
However, it is useful as an intermediate step to calculate multiplicity as a function of and . This approach shows that the number of available macrostates is N + 1 . For example, in a very small system with N = 2 dipoles, there are three macrostates, corresponding to N ↑ = 0 , 1 , 2. {\displaystyle N_{\uparrow }=0,1,2.}
For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; and they are used in calculus and ...
In number theory, a multiplicative function is an arithmetic function f(n) of a positive integer n with the property that f(1) = 1 and = () whenever a and b are coprime.. An arithmetic function f(n) is said to be completely multiplicative (or totally multiplicative) if f(1) = 1 and f(ab) = f(a)f(b) holds for all positive integers a and b, even when they are not coprime.
The multiplicity of a prime factor p in n, that is the largest exponent m for which p m divides n. a 0 (n) – the sum of primes dividing n counting multiplicity, sometimes called sopfr(n), the potency of n or the integer logarithm of n (sequence A001414 in the OEIS). For example: a 0 (4) = 2 + 2 = 4 a 0 (20) = a 0 (2 2 · 5) = 2 + 2 + 5 = 9 a ...