Ukr. Mat. Zh. - 2013. - 65, № 12. - pp. 1700–1711
Let p: X → X/A be a quotient map, where A is a subspace of X. We study the conditions under which p ∗(π 1 qtop (X, x 0)) is dense in π 1 qtop (X/A,∗)), where the fundamental groups have the natural quotient topology inherited from the loop space and p * is a continuous homomorphism induced by the quotient map p. In addition, we present some applications in order to determine the properties of π 1 qtop (X/A,∗). In particular, we establish conditions under which π 1 qtop (X/A,∗) is an indiscrete topological group.
An admissible estimator for the r th power of a bounded scale-parameter in a subclass of the exponential family under entropy loss function
Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1138-1147
We consider an admissible estimator for the rth power of a scale parameter that is lower or upper bounded in a subclass of the scale-parameter exponential family under entropy loss function. An admissible estimator of a bounded parameter in the family of transformed chi-square distributions is also given.