The association of algebraic objects to forms has had many important applications in number theory. Gauss, over two centuries ago, studied quadratic rings and ideals associated to binary quadratic forms, and found that ideal classes of quadratic rings are exactly parametrized by equivalence classes of integral binary quadratic forms. Delone and Faddeev, in 1940, showed that cubic rings are para...

By first describing the fixed maximal ideals of RL and those of its bounded part, R∗L, we show that every fixed maximal ideal of the bigger ring contracts to a fixed maximal ideal of the smaller ring, and every fixed maximal ideal of the smaller ring extends to a fixed maximal ideal of the bigger ring. However, the only instance where every maximal ideal of R∗L extends to a maximal ideal of RL ...

This paper considers PI controller tuning for Integral Plus Dead Time plant (IPDT) by new Matlab/Simulink tool based on the performance portrait method. It enables to guarantee transient responses with specified deviations from ideal shapes at the plant output and input and to fulfill additional optimality specification, defined in terms of the minimal IAE (Integral of Absolute Error) values we...

• Remarks and observations: (1) The tail of the van der Waals attractive potential (∝ r) extends to very long sep­ arations. Yet, its integral in eq.(V.39) is dominated by contributions from the short scales r0. In this limited context, the van der Waals potential is short-ranged, and results in corrections to the ideal gas behavior that are analytical in density n, leading to the virial series...

Abstract. This extends a theorem of Davenport and Erdös [7] on sequences of rational integers to sequences of integral ideals in arbitrary number fields K. More precisely, we introduce a logarithmic density for sets of integral ideals in K and provide a formula for the logarithmic density of the set of so-called A -free ideals, i.e. integral ideals that are not multiples of any ideal from a fix...

We define the Segre numbers of an ideal as a generalization of the multiplicity of an ideal of finite colength. We prove generalizations of various theorems involving the multiplicity of an ideal such as a principle of specialization of integral dependence, the Rees-Böger theorem, and the formula for the multiplicity of the product of two ideals. These results are applied to the study of variou...

We refine the definition of the fractional Galois ideal introduced in [Paul Buckingham. The canonical fractional Galois ideal at s = 0. J. Number Theory, 128(6):1749–1768, 2008] which was based on Snaith’s fractional ideal, allowing us to give a general relationship of this object with the Stark elements appearing in Rubin’s integral sharpening of Stark’s Conjecture. We motivate this by using a...