We give the following representation theorem for a class containing quasianalytic ultradistributions and all the non-quasianalytic ultradistributions: Every ultradistribution in this class can be written as $$u = P(\Delta )g(x) + h(x)$$ where$g(x)$is a bounded continuous function,$h(x)$is a bounded real analytic function and$P(d/dt)$is an ultradifferential operator. Also, we show that the boundary value of every heat function with some exponential growth condition determines an ultradistribution in this class. These results generalize the theorem of Matsuzawa [M] for the above class of quasianalytic ultradistributions and partially solve a question of A. Kaneko [Ka]. Our interest lies in the quasianalytic case, although the theorems do not exclude non-quasianalytic classes.

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A class of variable selection procedures for parametric models via nonconcave penalized likelihood was proposed by Fan and Li to simultaneously estimate parameters and select important variables. They demonstrated that this class of procedures has an oracle property when the number of parameters is finite. However, in most model selection problems the number of parameters should be large and grow with the sample size. In this paper some asymptotic properties of the nonconcave penalized likelihood are established for situations in which the number of parameters tends to as the sample size increases. Under regularity conditions we have established an oracle property and the asymptotic normality of the penalized likelihood estimators. Furthermore, the consistency of the sandwich formula of the covariance matrix is demonstrated. Nonconcave penalized likelihood ratio statistics are discussed, and their

In this paper, we introduce the notation of bi-shift of biprojections in subfactor theory to unimodular Kac algebras. We characterize the minimizers of the Hirschman-Beckner uncertainty principle and the Donoho-Stark uncertainty principle for unimodular Kac algebras with biprojections and prove Hardy’s uncertainty principle in terms of the minimizers.

We introduce a new notion of an angle between intermediate subfactors and prove various interesting properties of the angle and relate it to the Jones index. We prove a uniform 60 to 90 degree bound for the angle between minimal intermediate subfactors of a finite index irreducible subfactor. From this rigidity we can bound the number of minimal (or maximal) intermediate subfactors by the kissing number in geometry. As a consequence, the number of intermediate subfactors of an irreducible subfactor has at most exponential growth with respect to the Jones index. This answers a question of Longo’s published in 2003.