Let R be a commutative Noetherian ring. We give criteria for flatness of R-modules in terms of associated primes and torsion-freeness of certain tensor products. This allows us to develop a criterion for regularity if R has characteristic p, or more generally if it has a locally contracting endomorphism. Dualizing, we give criteria for injectivity of R-modules in terms of coassociated primes an...

In this article, we try to explain and unify standard divisibility tests found in various books. We then look at recurring decimals, and list a few of their properties. We show how to compute the number of digits in the recurring part of any fraction. Most of these results are accompanied by a proof (along with the assumptions needed), that works in a Euclidean domain. We then ask some question...

Robin’s criterion states that the Riemann hypothesis is true if and only inequality $$\sigma (n) < e^{\gamma } \times n \log n$$ holds for all natural numbers $$n > 5040$$ , where (n)$$ sum-of-divisors function of $$\gamma \approx 0.57721$$ Euler–Mascheroni constant. We show Robin are not divisible by some prime between 2 1771559. prove when $$\frac{\pi ^{2}}{6} n' \le $$n>5040$$ $$n'$$ square ...

Three equivalence relations are considered on the set of n x n matrices with elements in Fo' an · abelian group with absorbing zero adjoined. They are the relations of diagonal similarity, diagonal equivalence, and restricted diagonal equivalence. These relations are usually considered for matrices with elements in a field, but only multiplication is involved. Thus our formulation in terms of a...

Quantum dynamical maps provide suitable mathematical representation of quantum evolutions. When representing states by density operators, the evident requirements for any map are positivity and trace-preservation. However, these properties not consistent with mechanics composite systems. It is very notion complete which provides a proper evolution gives rise to powerful generalization unitary c...

Thorin classes T (R+),κ > 0, of infinitely divisible distributions on R+ are defined and characterized. Poisson, Karlin and Bessel transforms of Thorin classes are investigated. The extended Thorin classes T (κ)(Rd), κ > 0, are also considered. Canonical representation and selfdecomposability properties of the Thorin subordinated Gaussian Lévy processes are discussed. As an example, the subordi...

We consider the independent sum of a given random variable with a Gaussian variable and an infinitely divisible one. We find a novel tight upper bound on the entropy of the sum which still holds when the variable possibly has an infinite second moment. The proven bound has several implications on both information theoretic problems and infinitely divisible noise channels’ transmission rates.