The components of chi-squared statistics for goodness-of-fit tests

One obtains a formula for the transformation of a vector with a given covariance matrix. The sum of the squares of the components of the transformed vector is equal to a chi-square type quadratic form, constructed with the aid of the initial vector. Taking into account this property, one finds the components of the Pearson statistic, Pearson-Fisher statistic, etc.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 337–350, 1986.     

Synthesis of methyl ethers of methyl 2-acetamido-2-deoxy-α-D-glucopyranoside

A convenient method is proposed for the synthesis of all the individual methyl ethers of methyl 2-acetamido-2-deoxy-α-D-glucopyranoside based on the partial methylation of 2-acetamido-2-deoxy-α-D-glucopyranoside with dimethyl sulfate in an alkaline medium followed by preparative liquid column chromatography on silica gel of the resulting mixture of methyl ethers.     

Transport as a consequence of state-dependent diffusion

Overdamped particles subject to a drift in a force field with sinusoidal space dependence and also a sinusoidally modulated space-dependent diffusion, with the same period as the drift, experience a net driving force. The resulting current depends on the amplitude of the modulation of the diffusion and is a periodic function of the phase difference between the sinusoidal drift and the sinusoidal modulation of the diffusion. For small modulation amplitudes a particle subject to state-dependent noise behaves the same way as a particle subject to thermal noise but with a drift which, in addition to the sinusoidal term, contains a net force term.Dedicated to Professor Harry Thomas on the occasion of his 60th birthday     

The matrix equation AX-YB=C and related problems

The main result of the paper is a theorem, using which a new proof of Roth’s theorem is obtained, a new solvability criterion for the matrix equation AX-YB = C is proved, a formula for a particular solution of the latter is derived, and the least of the orders of square nonsingular matrices containing a given rectangular matrix as a submatrix is determined. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 15–23.