In this paper, a clear expression of the null distribution function of the likelihood ratio test statistics U that is considered by Hawkins (1977) is derived when σ2 is known. The numerical tables of the distribution funtion are given.
A new expression of the quantitative phase analysis by X-ray without anyrelative calibration standard has been derived in the case of every sample,inwhich the composition of one phase is known,among those able to meet therequirements of Zevin's method.
In this paper it is shown that the basic inequality G(a)≤A(a),Cauchy inequality,Tchebychef inequality,Holder inequality,Lyapunov inequality,triangle inequality and Minkowski inequality are all equivalent to the simple proposition that a~2≥0 for every a∈R(set of real number),provided the proposition that the closure of the set of rational number equals R is known.
The sufficient and necessary conditions of existence of the univalent harmonic orientationpreserving mappings f(z) are given,and f(z) satisfies the equation f =a(z)f z for given Beltrami coefficient a(z)= e iθ (z-a 11- a 1 z)(z-a 21- a 2 z) and maps unit disk U onto a given polygon P .
It is given that the sufficient and necessary conditions of existence of the univalent harmonic orientationpreserving mapping f(z) which satisfies the equation f =a(z)f z for given Beltrami coefficient a(z)= e iθ ((z-α)/(1-z)) m,m=1,2 and maps unit disk U onto a given polygon P .
It is known [M4] that K?-orbits S and G?-orbits S' on a complex flag manifold are in one-to-one correspondence by the condition that S ∩ S' is nonempty and compact.
When the characteristic of k is 0, it is known that the invariants of d vectors, d ≥ n, are obtained from those of n vectors by polarization.
It is known thatT (A, D) tiles?n by some subset of?n.
It is known  that dualizing a form of the Poisson summation formula yields a pair of linear transformations which map a function ? of one variable into a function and its cosine transform in a generalized sense.
Implementing this point of view, Poisson Summation Formulas are proved in several spaces including integrable functions of bounded variation (where the result is known) and elements of mixed norm spaces.
In the recent years, the theory related to friction-lines has been independently developed both in Europe~ and here~[2-8], according the information~ reached the author in 1956. These investigations are based on the condition of least frictional resistance at a point.