Mathematical Researches

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Let be a differentiable action of a Lie group on a differentiable manifold and consider the orbit space with the quotient topology.  Dimension of is called the cohomogeneity of the action of  on . If is a differentiable manifold  of  cohomogeneity one under the action of  a compact and connected Lie group, then the orbit space is homeomorphic to one of the spaces , , or . In this paper we suppose that the hyperbolic space  is of cohomogeneity  two under the action of , a connected and closed subgroup of  Then we prove that its orbit space is homeomorphic to  or  Also we prove that either all orbits are diffeomorphic to  or there are nonnegative integers   such that some orbits are diffeomorphic to , and the other orbits are diffeomorphic to , where may be a sphere, a homogeneous hypersurface of sphere or a helix in some Euclidean space.
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We classify orbit spaces of cohomogeneity two isometric actions on Riemannian manifolds of constant negative curvature.

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Stratified sampling is one of the most widely used sampling designs. In some cases, it is up to the researcher to determine the boundaries of the strata, and in some cases, the population is already stratified. The optimal classification is obtained for a situation of strata boundries, where the variance of the population mean (or total) estimator reaches its lowest value. In traditional methods, the variance of the estimator is considered as a function of the strata boiundries for the response variable, in order to reach the minimum of the variance, equations are obtained which are often solved by numerical methods. The first deficiency of this method is not considering all auxiliary variables. For example, in estimating the average income, classifying the society based on factors such as gender and job history can not only increase the efficiency of the estimator, but also make the interpretability and generalizability of the results easier. The second one is complex equations that do not have a closed and understandable solutions
n this paper, we have tried to construct the optimal classification based on a new criterion that is a combination of variance and a penalty for increasing the number of strata, so that important auxiliary variables in the formation of the decision tree determine the boundries of the strata. The classification process starts from the saturated tree and with successive pruning until reaching the root node, the number of strata decreases, the optimal stratification is achieved based on the introduced combined criterion.
 

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The homogeneity of variances test is a prerequisite for many statistical methods. In this article, a recently introduced test based on the jackknife approach is compared with common tests such as Levene's and Bartlett's tests, as well as two tests by James and Alexander-Govern, in terms of their ability to maintain the first type error rate and test power for several distributions. The permutation versions of these tests were also examined. The results indicate that the performance of the tests significantly improves in the permutation version. To evaluate the performance of the tests in the real world, the tests were applied to two real data sets, and the results are presented.