We show that in the generic case the smooth noncommutative tori associated with two$n$×$n$real skew-symmetric matrices are Morita equivalent if and only if the matrices are in the same orbit of the natural SO($n$,$n$∣$Z$) action.

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We reduced the unsolved problem of Evans triangles to the problem of nding rational points on a class of elliptic curves, the twisted Edwards curves En. We proved that the set of all Evans triangles with a given ratio is isomorphic to a quotient group of En(Q). After this, using Sage, we can solve the Evans problem for an amount of cases within the capacity of computer. Even without the help of a computer, we can still obtain a large class of new Evans triangles, and nd an eective way to test if the rank of an Edwards curve is positive or not.

In this paper we start the classification of strongly bracket-generated sub-symmetric spaces. We prove a structure theorem and analyze the nilpotent case.

Variable selection is vital to statistical data analyses. Many of procedures in use are ad hoc stepwise selection procedures, which are computationally expensiveandignore stochastic errors in the variable selection process of previous steps. An automatic and simultaneous variable selection procedure can be obtained by using a penalized likelihood method. In traditional linear models, the best subset selection and stepwise deletion methods coincide with a penalized leastsquares method when design matrices are orthonormal. In this paper, we propose a few new approaches to selecting variables for linear models, robust regression models and generalized linear models based on a penalized likelihood approach. A family of thresholding functions are proposed. The LASSO proposed by Tibshirani (1996) is a member of the penalized leastsquares with the # #-penalty. A smoothly clipped absolute deviation (SCAD) penalty function is introduced to ameliorate the properties of # #-penalty. A unied algorithm is introduced, which is backed up by statistical theory. The new approaches are compared with the ordinary leastsquares methods, the garrote method by Breiman (1995) and the LASSO method by Tibshirani (1996). Our simulation results show that the newly proposed methods compare favorably with other approaches as an automatic variable selection technique. Because of simultaneous selection of variables and estimation of parameters, we are able to give a simple estimated standard error formula, which is tested to be accurate enough for practical applications. Two real data examples illustrate the versatility and eectiveness of the