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Though there are many approximate solutions (such as Welch's t-test), the problem continues to attract attention [4] as one of the classic problems in statistics. Multiple comparisons: There are various ways to adjust p-values to compensate for the simultaneous or sequential testing of hypotheses. Of particular interest is how to simultaneously ...
So in this case the solution to the Hamburger moment problem is unique and μ, being the spectral measure of T, has finite support. More generally, the solution is unique if there are constants C and D such that, for all n, | m n | ≤ CD n n! (Reed & Simon 1975, p. 205). This follows from the more general Carleman's condition.
Central limit theorem; Characterization of probability distributions; Cochran's theorem; Complete class theorem; Continuous mapping theorem; Cox's theorem; Cramér's decomposition theorem; Craps principle
The first HK theorem demonstrates that the ground-state properties of a many-electron system are uniquely determined by an electron density that depends on only three spatial coordinates. It set down the groundwork for reducing the many-body problem of N electrons with 3 N spatial coordinates to three spatial coordinates, through the use of ...
Example: Given the mean and variance (as well as all further cumulants equal 0) the normal distribution is the distribution solving the moment problem. In mathematics , a moment problem arises as the result of trying to invert the mapping that takes a measure μ {\displaystyle \mu } to the sequence of moments
The formal foundation of TDDFT is the Runge–Gross (RG) theorem (1984) [1] – the time-dependent analogue of the Hohenberg–Kohn (HK) theorem (1964). [2] The RG theorem shows that, for a given initial wavefunction, there is a unique mapping between the time-dependent external potential of a system and its time-dependent density.
Yamabe problem. Yamabe claimed a solution in 1960, but Trudinger discovered a gap in 1968, and a complete proof was not given until 1984. Mordell conjecture over function fields. Manin published a proof in 1963, but Coleman (1990) found and corrected a gap in the proof. In 1973 Britton published a 282-page attempted solution of Burnside's problem.
Further, a foundation can be used to explain statistical paradoxes, provide descriptions of statistical laws, [1] and guide the application of statistics to real-world problems. Different statistical foundations may provide different, contrasting perspectives on the analysis and interpretation of data, and some of these contrasts have been ...