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The main application of statistical power is "power analysis", a calculation of power usually done before an experiment is conducted using data from pilot studies or a literature review. Power analyses can be used to calculate the minimum sample size required so that one can be reasonably likely to detect an effect of a given size (in other ...
Critical exponents are defined in terms of the variation of certain physical properties of the system near its phase transition point. These physical properties will include its reduced temperature τ {\displaystyle \tau } , its order parameter measuring how much of the system is in the "ordered" phase, the specific heat , and so on.
Probability generating functions are often employed for their succinct description of the sequence of probabilities Pr(X = i) in the probability mass function for a random variable X, and to make available the well-developed theory of power series with non-negative coefficients.
In mathematics, a power series (in one variable) is an infinite series of the form = = + + + … where represents the coefficient of the nth term and c is a constant called the center of the series. Power series are useful in mathematical analysis , where they arise as Taylor series of infinitely differentiable functions .
One of the major discoveries in the study of critical phenomena is that mean field theory of critical points is only correct when the space dimension of the system is higher than a certain dimension called the upper critical dimension which excludes the physical dimensions 1, 2 or 3 in most cases. The problem with mean field theory is that the ...
The convergence criteria of the power series then apply, requiring ‖ ‖ to be sufficiently small under the appropriate matrix norm. For more general problems, which cannot be rewritten in such a way that the two matrices commute, the ordering of matrix products produced by repeated application of the Leibniz rule must be tracked.
In statistical hypothesis testing, a uniformly most powerful (UMP) test is a hypothesis test which has the greatest power among all possible tests of a given size α. For example, according to the Neyman–Pearson lemma , the likelihood-ratio test is UMP for testing simple (point) hypotheses.
In d=2, the two-dimensional critical Ising model's critical exponents can be computed exactly using the minimal model,. In d=4, it is the free massless scalar theory (also referred to as mean field theory). These two theories are exactly solved, and the exact solutions give values reported in the table.