A noncentral limit theorem for quadratic forms of Gaussian stationary sequences

We examine the limit behavior of quadratic forms of stationary Gaussian sequences with long-range dependence. The matrix that characterizes the quadratic form is Toeplitz and the Fourier transform of its entries is a regularly varying function at the origin. The spectral density of the stationary sequence is also regularly varying at the origin. We show that the normalized quadratic form converges inD[0, 1] to a new type of non-Gaussian self-similar process, which can be represented as a Wiener-Itô integral onR ².     

Central and Non-central Limit Theorems in a Free Probability Setting

Long-range dependence in time series may yield non-central limit theorems. We show that there are analogous time series in free probability with limits represented by multiple Wigner integrals, where Hermite processes are replaced by non-commutative Tchebycheff processes. This includes the non-commutative fractional Brownian motion and the non-commutative Rosenblatt process.     

On nonlinear higher spin curvature

We present the first nonlinear term of the higher spin curvature which is covariant with respect to deformed gauge transformations that are linear in the field. We consider the case of spin 3 after presenting spin 2 as an example, and then construct the general spin s quadratic term of the de Wit–Freedman curvature.     

The nonclassical noncovalent interactions control: A case study of the crystal structure of 3,5-dibromo-2-amino-4,6-dimethylpyridinium tetrahalocuprate [3,5-DBr-2-A-4,6-DMPH]2CuX4 (X = Cl, and Br)

The compound [3,5-dibromo-2-amino-4,6-dimethylpyridinium]₂CuCl₄, (1), was prepared from acidic hydrochloric acetonitrilic medium containing CuCl₂ and 3,5-dibromo-2-amino-4,6-dimethylpyridinium bromide while [3,5-dibromo-2-amino-4,6-dimethylpyridinium]₂CuBr₄, (2), from acidic HBr/Br₂ ethanolic medium containing 4,6-dimethylpyridine and CuBr₂. The ring got dibrominated at 3 and 5 positions in situ. Single-crystal X-ray diffraction proved the compounds to be isomorphous in P − 1. A systematic study of the crystallographic parameters in dibrominated and nonbrominated 2-amino-4,6-dimethylpyridine salts will explain the role of nonclassical noncovalent interaction between aryl bromine and ionic metal bromide in the control of structure.