This paper introduces a conceptual, yet quantifiable, architecture framework by extending the notion of system modularity in its broadest sense. Acknowledging that modularity is not a binary feature and comes in various types and levels, the proposed framework introduces higher levels of modularity that naturally incorporate decentralized architecture on the one hand and autonomy in agents and subsystems on the other. This makes the framework suitable for modularity decisions in Systems of Systems and for analyzing the impact of modularity on broader surrounding ecosystems. The stages of modularity in the proposed framework are naturally aligned with the level of variations and uncertainty in the system and its environment, a relationship that is central to the benefits of modularity. The conceptual framework is complemented with a decision layer that makes it suitable to be used as a computational architecture decision tool to determine the appropriate stage and level of modularity of a system, for a given profile of variations and uncertainties in its environment. We further argue that the fundamental systemic driving forces and trade-offs of moving from monolithic to distributed architecture are essentially similar to those for moving from integral to modular architectures. The spectrum, in conjunction with the decision layer, could guide system architects when selecting appropriate parameters and building a system-specific computational tool from a combination of existing tools and techniques.To demonstrate the applicability of the framework, a case for fractionated satellite systems based on a simplified demo of the DARPA F 6 program is presented where the value of transition from a monolithic architecture to a fractionated architecture, as two consecutive levels of modularity in the proposed spectrum, is calculated and ranges of parameters where fractionation increases systems value are determined.
Let (R, m R ) be a local domain, with quotient field K. Suppose that ν is a valuation of K with valuation ring (V, m V ), and that ν dominates R; that is, R ⊂ V and m V ∩R = m R . The possible value groups Γ of ν have been extensively studied and classified, including in the papers MacLane [8], MacLane and Schilling [9], Zariski and Samuel [12], and Kuhlmann [7]. Γ can be any ordered abelian group of finite rational rank (Theorem 1.1 [7]). The semigroupis however not well understood, although it is known to encode important information about the topology and resolution of singularities of Spec(R) and the ideal theory of R.In Zariski and Samuel's classic book on Commutative Algebra [12], two general facts about the semigroup S R (ν) are proven (in Appendix 3 to Volume II).1. S R (ν) is a well ordered subset of the positive part of the value group Γ of ν of ordinal type at most ω h , where ω is the ordinal type of the well ordered set N, and h is the rank of ν. 2. The rational rank of ν plus the transcendence degree of V /m V over R/m R is less than or equal to the dimension of R.The second condition is the Abhyankar inequality [1]. The only semigroups which are realized by a valuation on a one dimensional regular local ring are isomorphic to the natural numbers. The semigroups which are realized by a valuation on a regular local ring of dimension 2 with algebraically closed residue field are much more complicated, but are completely classified by Spivakovsky in [10]. A different proof is given by Favre and Jonsson in [5], and the theorem is formulated in the context of semigroups by Cutkosky and Teissier [3].In [3], Teissier and the first author give some examples showing that some surprising semigroups of rank > 1 can occur as semigroups of valuations on noetherian domains, and raise the general questions of finding new constraints on value semigroups and classifying semigroups which occur as value semigroups.In this paper, we consider semigroups of rank 1 valuations. We show in Theorem 2.1 that the Hilbert polynomial of R gives a bound on the growth of the valuation semigroup S R (ν). This allows us to give (in Corollary 2.4) a very simple example of a well ordered subsemigroup of Q + of ordinal type ω, which is not a value semigroup of a local domain. This shows that the above conditions 1 and 2 do not characterize value semigroups on local domains.The simple bound of Theorem 2.1 of this paper is extended in the article [4] of Teissier and the first author to give a very general bound on the growth of a value semigroup of arbitrary rank, from which a rough description of the (extremely bizzare) shape of a higher rank valuation semigroup is derived.
We study the dependence of graded Betti numbers of monomial ideals on the characteristic of the base field. The examples we describe include bipartite ideals, Stanley-Reisner ideals of vertex-decomposable complexes and ideals with componentwise linear resolutions. We give a description of bipartite graphs and, using discrete Morse theory, provide a way of looking at the homology of arbitrary simplicial complexes through bipartite ideals. We also prove that the Betti table of a monomial ideal over the field of rational numbers can be obtained from the Betti table over any field by a sequence of consecutive cancellations.
Given two finitely generated R-modules A and B, what can we say about the number of generators of the module Hom R (A, B)? In this paper we seek uniform bounds for the number of generators of Hom R (A, B) in terms of the numerical invariants of A and B. We will show that in many cases polynomial bounds in terms of specific invariants of A and B are possible.
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