In 1924, Helmut Hasse established a local-to-global principle for representations of rational quadratic forms. Unfortunately, an analogous local-to-global principle does not hold for representations over the integers. A quadratic polynomial is called regular if such a principle exists; that is, if it represents all the integers which are represented locally by the polynomial itself over Zp for ...

The paper considers algebraic directed-complete partial orders with a semi-regular Scott topology, called regular domains. As is well known, the category of Scott domains and continuous maps is Cartesian closed. This is no longer true, if the domains are required to be regular. Two Cartesian closed subcategories of the regular Scott domains are exhibited: regular dI-domains with stable maps and...

In this paper, we introduce semi-symmetric metric Koszul forms and non-metric on singular semi-Riemannian manifolds. Semi-symmetric their curvature of semi-regular warped products are expressed in terms those the factor We also associated to almost product structure Furthermore, generalize results \cite{St2} multiply products.