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We can also define the multiplicity of the zeroes and poles of a meromorphic function. If we have a meromorphic function =, take the Taylor expansions of g and h about a point z 0, and find the first non-zero term in each (denote the order of the terms m and n respectively) then if m = n, then the point has non-zero value.
The goal is to determine the multiplicity as a function of U; from there, the entropy and other thermodynamic properties of the system can be determined. However, it is useful as an intermediate step to calculate multiplicity as a function of N ↑ {\displaystyle N_{\uparrow }} and N ↓ . {\displaystyle N_{\downarrow }.}
The multiplicity is often equal to the number of possible orientations of the total spin [3] relative to the total orbital angular momentum L, and therefore to the number of near–degenerate levels that differ only in their spin–orbit interaction energy. For example, the ground state of a carbon atom is 3 P (Term symbol).
If n > 0, then is a pole of order (or multiplicity) n of f. If n < 0, then is a zero of order | | of f. Simple zero and simple pole are terms used for zeroes and poles of order | | = Degree is sometimes used synonymously to order.
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; Multiplicity (software), a software application which allows a user to control two or more computers from one mouse and keyboard
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.
The concept of multiplicity is fundamental for Bézout's theorem, as it allows having an equality instead of a much weaker inequality. Intuitively, the multiplicity of a common zero of several polynomials is the number of zeros into which the common zero can split when the coefficients are slightly changed.
In statistics, the multiple comparisons, multiplicity or multiple testing problem occurs when one considers a set of statistical inferences simultaneously [1] or estimates a subset of parameters selected based on the observed values. [2] The larger the number of inferences made, the more likely erroneous inferences become.