probability 1. (Note that we do not assume any relation of dependence or indepenaence between X(r) and X(s) for m $ n.) 1 See M. Frechet, Recherches theoriques modernes, Vol. 1, Paris, 1937, p. 27. 2 See H. Cram&r, Mathematical methods of statistics, Princeton, 1946, p. 93. Compare F. P. Cantelli, Considerazioni sulla legge uniforme dei grandi numeri ecc., Giornale dell 'Istituto Italiano degli...

Abstract deformations of the CR structure a compact strictly pseudoconvex hypersurface M in $${\mathbb {C}}^2$$ are encoded by complex functions on M. In sharp contrast with higher dimensional case, natural integrability condition for 3-dimensional structures is vacuous, and generic $$M\subseteq {\mathbb not embeddable even {C}}^N$$ any N. A fundamental (and difficult) problem to characterize w...

We show that in two dimensions or higher, the Mordukhovich-loffe approximate subdifferential and Clarke subdifferential may differ almost everywhere for real-valued Lipschitz functions. Uncountably many Frechet differentiable vector-valued Lipschitz functions differing by more than constants can share the same Mordukhovich-Ioffe coderivatives. Moreover, the approximate Jacobian associated with ...

Consider two (n−1)-dimensional manifolds, S and S in R. We say that they are normalcompatible when the closest projection of each one onto the other is a homeomorphism. We give a tight condition under which S and S are normal-compatible. It involves the minimum feature size of S and of S and the Hausdorff distance between them. Furthermore, when S and S are normal-compatible, their Frechet dist...

Optogenetically evoked local field potential (LFP) recorded from the medial prefrontal cortex (mPFC) of mice during basal conditions and following a systemic cocaine administration were analyzed. Blue light stimuli were delivered to mPFC through a fiber optic every 2 s and each trial was repeated 100 times. As in the previous study, we used a surrogate data method to check that nonlinearity was...