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In this paper, we examine the behavior of ideal-adic separatedness and completeness under certain ring extensions using trace map. Then we prove that adic completeness of a base ring is hereditary to its ring extension under reasonable conditions. We aim to give many results on ascent and descent of certain ring theoretic properties under completion. As an application, we give conceptual details to the proof of the almost purity theorem for Witt-perfect rings by Davis and Kedlaya. Witt-perfect rings have the advantage that one does not need to assume that the rings are complete and separated. [ABSTRACT FROM AUTHOR]

Davis and Jedwab (1997) established a great construction theory unifying many previously known constructions of difference sets, relative difference sets and divisible difference sets. They introduced the concept of building blocks, which played an important role in the theory. On the other hand, Polhill (2010) gave a construction of Paley type partial difference sets (conference graphs) based on a special system of building blocks, called a covering extended building set, and proved that there exists a Paley type partial difference set in an abelian group of order 9 i v 4 for any odd positive integer v > 1 and any i = 0 , 1. His result covers all orders of abelian non p -groups in which Paley type partial difference sets exist. In this paper, we give new constructions of strongly regular Cayley graphs on abelian groups by extending the theory of building blocks. The constructions are large generalizations of Polhill's construction. In particular, we show that for a positive integer m and elementary abelian groups G i , i = 1 , 2 , ... , s , of order q i 4 such that 2 m | q i + 1 , there exists a decomposition of the complete graph on the abelian group G = G 1 × G 2 × ⋯ × G s by strongly regular Cayley graphs with negative Latin square type parameters (u 2 , c (u + 1) , − u + c 2 + 3 c , c 2 + c) , where u = q 1 2 q 2 2 ⋯ q s 2 and c = (u − 1) / m. Such strongly regular decompositions were previously known only when m = 2 or G is a p -group. Moreover, we find one more new infinite family of decompositions of the complete graphs by Latin square type strongly regular Cayley graphs. Thus, we obtain many strongly regular graphs with new parameters. [ABSTRACT FROM AUTHOR]

• A new explanation to shock instability is proposed. • Some theoretical research results for viscous shock structure with Davis' wave velocity algorithm are provided. • A simple but effective cure strategy which is based on the insight of shock instability is suggested. This paper concentrates on numerical shock instabilities of some approximate Riemann solvers which stem from the HLLE method, including HLLC-type and HLLEM-type flux functions. Carbuncle phenomena are illustrated for these numerical schemes, especially for those believed to be carbuncle-free, such as the HLLE scheme. A linear matrix stability analysis shows that there exists a threshold independent of shock strength to trigger instability for a given numerical scheme. In this instability mechanism, a specific cell interface in the numerical structure of shock layer is regarded as the source of instability if the density ratio between both sides of this interface exceeds the threshold. The modification by adjusting nonlinear signal velocities in these numerical schemes can change solution behaviors in the shock layer, and reduce the degree of density transition on the specific interface. A very simple cure strategy for the HLLE-type Riemann solvers is presented based on the in-depth understanding of the carbuncle onset of viscous shock. [ABSTRACT FROM AUTHOR]