Kleene Star Research Articles

Semistar Linkedness and Flatness, Prüfer Semistar Multiplication Domains#

In 1994, Matsuda and Okabe introduced the notion of semistar operation, extending the “classical” concept of star operation. In this paper, we introduce and study the notions of semistar linkedness and semistar flatness which are natural generalizations, to the semistar setting, of their corresponding “classical” concepts. As an application, among other results, we obtain a semistar version of Davis' and Richman's overring-theoretical theorems of characterization of Prüfer domains for Prüfer semistar multiplication domains.

Energy–momentum conservation in pre-metric electrodynamics with magnetic charges

A necessary and sufficient condition for energy-momentum conservation is proved within a topological, pre-metric approach to classical electrodynamics including magnetic as well as electric charges. The extended Lorentz force, consisting of mutual actions by F=(E, B) on the electric current and G=(H, D) on the magnetic current, can be derived from an energy-momentum "potential" if and only if the constitutive relation G=G(F) satisfies a certain vanishing condition. The electric-magnetic reciprocity introduced by Hehl and Obukhov is seen to define a complex structure on the tensor product of 2-form pairs (F,G) which is independent of but consistent with the Hodge star operator defined by any Lorentzian metric. Contrary to a recent claim in the literature, it does not define a complex structure on the space of 2-forms itself.

Stellar activity on the lower main sequence in Praesepe

AbstractAn αΩ dynamo is considered responsible for magnetic activity in late K/early M main sequence stars, which is expected to be enhanced in later types as the surface convection zone deepens. At about spectral type M3, where the star presumably becomes fully convective, the magnetic field is theorized to change in character, switching to a more uniform, turbulence‐generated surface field. As a consequence, the nature of activity is expected to change at later spectral types. In field stars, age, mass, rotation and perhaps metallicity play a role in determining the activity level, but the effects are difficult to disentangle. Therefore, open clusters with a more homogeneous sample can provide valuable information on the dynamo operation and magnetic activity of lower main sequence stars. We present preliminary results of our spectroscopic study for activity indicators among the lower main sequence stars of the intermediate age (700 My) open cluster Praesepe. Chromospheric activity as manifested by the presence/absence of Hα in late K/M stars is presented, and other activity indicators are discussed. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Rule formats for compositional non-interference properties

We introduce the transition rule formats rooted SBSNNI and CP_BNDC. We prove that the non-interference property rooted SBSNNI introduced in the present paper, and the already known non-interference property CP_BNDC, are preserved by constructs of all process algebras with SOS transition rules respecting the restrictions of the formats rooted SBSNNI and CP_BNDC, respectively. To show that our formats have practical applications, we prove that a slight variant of Focardi and Gorrieri's Security Process Algebra, the Kleene star recursion construct, the replication construct of polyadic π-calculus, and a process algebra extending BPA ϵτ to deal with two level systems, respect both formats. By means of some counterexamples, we prove also that all restrictions of the formats are necessary.

Rule formats for compositional non-interference properties*1

We introduce the transition rule formats rooted SBSNNI and CP_BNDC. We prove that the non-interference property rooted SBSNNI introduced in the present paper, and the already known non-interference property CP_BNDC, are preserved by constructs of all process algebras with SOS transition rules respecting the restrictions of the formats rooted SBSNNI and CP_BNDC, respectively. To show that our formats have practical applications, we prove that a slight variant of Focardi and Gorrieri's Security Process Algebra, the Kleene star recursion construct, the replication construct of polyadic π-calculus, and a process algebra extending BPAϵτ to deal with two level systems, respect both formats. By means of some counterexamples, we prove also that all restrictions of the formats are necessary.

Inductive [formula omitted]-semirings

One of the most well-known induction principles in computer science is the fixed point induction rule, or least pre-fixed point rule. Inductive ∗ -semirings are partially ordered semirings equipped with a star operation satisfying the fixed point equation and the fixed point induction rule for linear terms. Inductive ∗ -semirings are extensions of continuous semirings and the Kleene algebras of Conway and Kozen. We develop, in a systematic way, the rudiments of the theory of inductive ∗ -semirings in relation to automata, languages and power series. In particular, we prove that if S is an inductive ∗ -semiring, then so is the semiring of matrices S n× n , for any integer n⩾0, and that if S is an inductive ∗ -semiring, then so is any semiring of power series S〈〈A ∗〉〉 . As shown by Kozen, the dual of an inductive ∗ -semiring may not be inductive. In contrast, we show that the dual of an iteration semiring is an iteration semiring. Kuich proved a general Kleene theorem for continuous semirings, and Bloom and Ésik proved a Kleene theorem for all Conway semirings. Since any inductive ∗ -semiring is a Conway semiring and an iteration semiring, as we show, there results a Kleene theorem applicable to all inductive ∗ -semirings. We also describe the structure of the initial inductive ∗ -semiring and conjecture that any free inductive ∗ -semiring may be given as a semiring of rational power series with coefficients in the initial inductive ∗ -semiring. We relate this conjecture to recent axiomatization results on the equational theory of the regular sets.

Logics for Classes of Boolean Monoids

This paper presents the algebraic and Kripke modelsoundness and completeness ofa logic over Boolean monoids. An additional axiom added to thelogic will cause the resulting monoid models to be representable as monoidsof relations. A star operator, interpreted as reflexive, transitiveclosure, is conservatively added to the logic. The star operator isa relative modal operator, i.e., one that is defined in terms ofanother modal operator. A further example, relative possibility,of this type of operator is given. A separate axiom,antilogism, added to the logic causes the Kripke models to support acollection of abstract topological uniformities which become concretewhen the Kripke models are dual to monoids of relations. The machineryfor the star operator is shownto be a recasting of Scott-Montague neighborhood models. An interpretationof the Kripke frames and properties thereof is presented in terms ofcertain CMOS transister networks and some circuit transformation equivalences.The worlds of the Kripke frame are wires and the Kripke relation is a specializedCMOS pass transistor network.

Grassmann electrodynamics and general relativity

The aim of this paper is to present a short introduction to supergeometry on pure odd supermanifolds. (Pseudo)differential forms, Cartan calculus (DeRham differential, Lie derivative, interior product), metric, “inner” product, Killing’s vector fields, Hodge star operator, integral forms, co-differential and connection on odd Riemannian supermanifolds are introduced. The electrodynamics and Einstein relativity with anti-commuting variables only are formulated modifying the geometry beyond classical (even, bosonic) theories appropriately. Extension of these ideas to general supermanifolds is straightforward.

Grassmann electrodynamics and general relativity

The aim of this paper is to present a short introduction to supergeometry on pure odd supermanifolds. (Pseudo)differential forms, Cartan calculus (DeRham differential, Lie derivative, interior product), metric, “inner” product, Killing’s vector fields, Hodge star operator, integral forms, co-differential and connection on odd Riemannian supermanifolds are introduced. The electrodynamics and Einstein relativity with anti-commuting variables only are formulated modifying the geometry beyond classical (even, bosonic) theories appropriately. Extension of these ideas to general supermanifolds is straightforward.

Star operations and pullbacks

In this paper we study the star operations on a pullback of integral domains. In particular, we characterize the star operations of a domain arising from a pullback of “a general type” by introducing new techniques for “projecting” and “lifting” star operations under surjective homomorphisms of integral domains. We study the transfer in a pullback (or with respect to a surjective homomorphism) of some relevant classes or distinguished properties of star operations such as v−, t−, w−, b−, d−, finite type, e.a.b., stable, and spectral operations. We apply part of the theory developed here to give a complete positive answer to a problem posed by D.F. Anderson in 1992 concerning the star operations on the “ D+ M” constructions.

Arithmetic Complexity, Kleene Closure, and Formal Power Series

In this paper we show how two fundamental operations used in formal language theory provide useful tools for the investigation of arithmetic complexity classes. More precisely, we use Kleene closure of languages and inversion of formal power series to investigate subclasses of the complexity class GapL. GapL is the complexity class that characterizes the complexity of computing the determinant; it corresponds to counting the number of accepting and rejecting paths of nondeterministic logspace-bounded Turing machines.) We define a counting version of Kleene closure and show that it is intimately related to inversion within the complexity classes GapL and GapNC^1. In particular, we prove that Kleene closure and inversion are both hard operations in the following sense There is a set in AC^0 for which Kleene closure is NL-complete and inversion is GapL-complete. There is a finite set for which Kleene closure is NC^1-complete and inversion is GapNC^1-complete. Furthermore, we classify the complexity of the Kleene closure of finite languages. We formulate the problem in terms of finite monoids and relate its complexity to the internal structure of the monoid.

More lead stars

The standard model for the operation of the s-process in asymptotic giant branch (AGB) stars predicts that low-metallicity ([Fe/H] <= -1) AGB stars should exhibit large overabundances of Pb and Bi as compared to other s-elements. Eight CH stars (with [Fe/H] ranging from -1.5 to -2.5) are studied in the present paper using high-resolution spectra (R = lambda/Delta lambda = 135 000), and five of them appear to be enriched in lead (with [Pb/Ce] ~ 0.7). Abundances of heavy elements in these `lead stars' (defined as stars enriched in s-elements and with [Pb/hs] >= 1, hs being any of Ba, La or Ce) are consistent with the predictions for the s-process operating in low-metallicity AGB stars as a consequence of the `partial mixing' of protons below the convective hydrogen envelope. Two among the studied stars (HD 189711 and V Ari) add to a growing number of low-metallicity stars which do not conform to these predictions, however. Variations on the canonical proton-mixing scenario for the operation of the s-process in low-metallicity stars, that could account for these discrepant stars, are briefly discussed.

A note on an expressiveness hierarchy for multi-exit iteration

Multi-exit iteration is a generalization of the standard binary Kleene star operation. The addition of this construct to Basic Process Algebra (BPA) yields a more expressive language than that obtained by augmenting BPA with the standard binary Kleene star. This note offers an expressiveness hierarchy, modulo bisimulation equivalence, for the family of multi-exit iteration operators proposed by Bergstra, Bethke and Ponse.

The stable set problem and the lift-and-project ranks of graphs

We study the lift-and-project procedures for solving combinatorial optimization problems, as described by Lovasz and Schrijver, in the context of the stable set problem on graphs. We investigate how the procedures' performances change as we apply fundamental graph operations. We show that the odd subdivision of an edge and the subdivision of a star operations (as well as their common generalization, the stretching of a vertex operation) cannot decrease the N 0 -, N-, or N + -rank of the graph. We also provide graph classes (which contain the complete graphs) where these operations do not increase the N 0 - or the N-rank. Hence we obtain the ranks for these graphs, and we also present some graph-minor like characterizations for them. Despite these properties we give examples showing that in general most of these operations can increase these ranks. Finally, we provide improved bounds for N + -ranks of graphs in terms of the number of nodes in the graph and prove that the subdivision of an edge or cloning a vertex can increase the N + -rank of a graph.

PRÜFER ⋆-MULTIPLICATION DOMAINS AND SEMISTAR OPERATIONS

Starting from the notion of semistar operation, introduced in 1994 by Okabe and Matsuda [49], which generalizes the classical concept of star operation (cf. Gilmer's book [27]) and, hence, the related classical theory of ideal systems based on the works by W. Krull, E. Noether, H. Prüfer, P. Lorenzen and P. Jaffard (cf. Halter–Koch's book [32]), in this paper we outline a general approach to the theory of Prüfer ⋆-multiplication domains (or P⋆MDs), where ⋆ is a semistar operation. This approach leads to relax the classical restriction on the base domain, which is not necessarily integrally closed in the semistar case, and to determine a semistar invariant character for this important class of multiplicative domains (cf. also J. M. García, P. Jara and E. Santos [25]). We give a characterization theorem of these domains in terms of Kronecker function rings and Nagata rings associated naturally to the given semistar operation, generalizing previous results by J. Arnold and J. Brewer ]10] and B. G. Kang [39]. We prove a characterization of a P⋆MD, when ⋆ is a semistar operation, in terms of polynomials (by using the classical characterization of Prüfer domains, in terms of polynomials given by R. Gilmer and J. Hoffman [28], as a model), extending a result proved in the star case by E. Houston, S. J. Malik and J. Mott [36]. We also deal with the preservation of the P⋆MD property by ascent and descent in case of field extensions. In this context, we generalize to the P⋆MD case some classical results concerning Prüfer domains and PvMDs. In particular, we reobtain as a particular case a result due to H. Prüfer [51] and W. Krull [41] (cf. also F. Lucius [43] and F. Halter-Koch [34]). Finally, we develop several examples and applications when ⋆ is a (semi)star given explicitly (e.g. we consider the case of the standardv-, t-, b-, w-operations or the case of semistar operations associated to appropriate families of overrings).

Nagata Rings, Kronecker Function Rings, and Related Semistar Operations

In 1994, Matsuda and Okabe introduced the notion of semistar operation. This concept extends the classical concept of star operation (cf. for instance, Gilmer's book (Gilmer, R. (1972). Multiplicative Ideal Theory. New York: Marcel Dekker) and, hence, the related classical theory of ideal systems based on the works by W. Krull, E. Noether, H. Prüfer and P. Lorenzen from 1930's. Fontana and Loper investigated properties of the Kronecker function rings which arise from arbitrary semistar operations on an integral domain D (Fontana M., Loper K. A. (2001a). Kronecker function rings: a general approach. In Anderson, D. D., Papick, I. J., eds. Ideal Theoretic Methods in Commutative Algebra. Lecture Notes Pure Appl. Math. 220, Marcel Dekker, pp. 189–205 and Fontana, M., Loper, K. A. (2001b). A Krull-type theorem for the semistar integral closure of an integral domain. ASJE Theme Issue “Commutative Algebra” 26:89–95). In this paper we extend that study and also generalize Kang's notion of a star Nagata ring to the semistar setting (Kang, B. G. (1987). ⋆-Operations on Integral Domains. Ph.D. dissertation, Univ. Iowa and Kang, B. G. (1989). Prüfer v-multiplication domains and the ring R[X] N v , J. Algebra 123: 151–171). Our principal focuses are the similarities between the ideal structure of the Nagata and Kronecker semistar rings and between the natural semistar operations that these two types of function rings give rise to on D.

Kronecker Function Rings and Generalized Integral Closures

We provide an axiomatic concept of Kronecker function rings and apply it to associate a Kronecker function ring to any integral domain D and any ideal system (star operation) on D. We investigate its behavior in algebraic field extensions and its connection with the defining valuation domains.

Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework

The spectral problem associated with the linearization about solitary waves of the generalized fifth-order KdV equation is formulated in terms of the Evans function, a complex analytic function whose zeros correspond to eigenvalues. A numerical framework, based on a fast robust shooting algorithm on exterior algebra spaces is introduced. The complete algorithm has several new features, including a rigorous numerical algorithm for choosing starting values, a new method for numerical analytic continuation of starting vectors, the role of the Grassmannian G 2( C 5) in choosing the numerical integrator, and the role of the Hodge star operator for relating ⋀ 2( C 5) and ⋀ 3( C 5) and deducing a range of numerically computable forms for the Evans function. The algorithm is illustrated by computing the stability and instability of solitary waves of the fifth-order KdV equation with polynomial nonlinearity.

A Note on an Expressiveness Hierarchy for Multi-exit Iteration

Multi-exit iteration is a generalization of the standard binary Kleene star operation that allows for the specification of agents that, up to bisimulation equivalence, are solutions of systems of recursion equations of the form&lt;br /&gt;X_1 = P_1 X_2 + Q_1 &lt;br /&gt;&lt;br /&gt; X_n = P_n X_1 + Q_n &lt;br /&gt;&lt;br /&gt; where n is a positive integer, and the P_i and the Q_i are process terms. The addition of multi-exit iteration to Basic Process Algebra (BPA) yields a more expressive language than that obtained by augmenting BPA with the standard binary Kleene star. This note offers an expressiveness hierarchy, modulo bisimulation equivalence, for the family of multi-exit iteration operators proposed by Bergstra, Bethke and Ponse.

Locally Closed Semirings

We call a semiring S locally closed if for all a ∈ S there is some integer k such that 1 + a + ⋯ + a k =1 + a + ⋯ + a k + 1 . In any locally closed semiring we may define a star operation a ↦ a *, where a * is the above finite sum. We prove that when S is locally closed and commutative, then S is an iteration semiring.