This paper deals with the covariance matrix (CM) of two-mode Gaussian states, which, together mean vector, fully describes these states. In (ordinary) CM is a real symmetric order 4; therefore, it depends on 10 variables. However, there very efficient representation called standard form (SF) that reduces degrees freedom to four variables, while preserving all relevant information state. The SF ...

Let σ be a constant in the interval (0, 1), and let A be an infinite set of positive integers which contains at least c1x σ and at most c2x σ elements in the interval [1, x] for some constants c2 > c1 > 0 independent of x and each x ≥ x0. We prove that then the sumset A + A has more elements than A (counted up to x) by a factor c(σ) √ log x/ log log x for x large enough. An example showing that...

Let T be a d×d matrix with real coefficients. Then determines self-map of the d-dimensional torus Td=Rd/Zd. {En}n∈N sequence subsets Td and let W(T,{En}) set points x∈Td such that Tn(x)∈En for infinitely many n∈N. For large class (namely, those satisfying so called bounded property (B) which includes balls, rectangles, hyperboloids) we show Lebesgue measure shrinking target is zero (resp. one) ...

This paper discusses recent research on methods for estimating configuration parameters for the Matrix Comparator used for linking unstandardized or heterogeneously standardized references. The matrix comparator computes the aggregate similarity between the tokens (words) in a pair of references. The two most critical parameters for the matrix comparator for obtaining the best linking results a...