Kleene Star Research Articles

NUCLEOSYNTHESIS PREDICTIONS FOR INTERMEDIATE-MASS ASYMPTOTIC GIANT BRANCH STARS: COMPARISON TO OBSERVATIONS OF TYPE I PLANETARY NEBULAE

Type I planetary nebulae (PNe) have high He/H and N/O ratios and are thought to be descendants of stars with initial masses of ∼3–8 M� . These characteristics indicate that the progenitor stars experienced proton-capture nucleosynthesis at the base of the convective envelope, in addition to the slow neutron capture process operating in the He-shell (the s-process). We compare the predicted abundances of elements up to Sr from models of intermediate-mass asymptotic giant branch (AGB) stars to measured abundances in Type I PNe. In particular, we compare predictions and observations for the light trans-iron elements Se and Kr, in order to constrain convective mixing and the s-process in these stars. A partial mixing zone is included in selected models to explore the effect of a 13 C pocket on the s-process yields. The solar-metallicity models produce enrichments of [(Se, Kr)/Fe] 0.6, consistent with Galactic Type I PNe where the observed enhancements are typically 0.3 dex, while lower metallicity models predict larger enrichments of C, N, Se, and Kr. O destruction occurs in the most massive models but it is not efficient enough to account for the 0.3 dex O depletions observed in some Type I PNe. It is not possible to reach firm conclusions regarding the neutron source operating in massive AGB stars from Se and Kr abundances in Type I PNe; abundances for mores-process elements may help to distinguish between the two neutron sources. We predict that only the most massive (M 5 M� ) models would evolve into Type I PNe, indicating that extra-mixing processes are active in lower-mass stars (3–4 M� ), if these stars are to evolve into Type I PNe.

Note on Star Operations Over Polynomial Rings

This article studies the notions of star and semistar operations over a polynomial ring. It aims at characterizing when every upper to zero in R[X] is a *-maximal ideal and when a *-maximal ideal Q of R[X] is extended from R, that is, Q = (Q ∩ R)[X] with Q ∩ R ≠0, for a given star operation of finite character * on R[X]. We also answer negatively some questions raised by Anderson–Clarke by constructing a Prüfer domain R for which the v-operation is not stable.

Efficient inclusion for a class of XML types with interleaving and counting

Inclusion between XML types is important but expensive, and is much more expensive when unordered types are considered. We prove here that inclusion for XML types with interleaving and counting can be decided in polynomial time in the presence of two important restrictions: no element appears twice in the same content model, and Kleene star is only applied to disjunctions of single elements. Our approach is based on the transformation of each such content model into a set of constraints that completely characterizes the generated language. We then reduce inclusion checking to constraint implication. We exhibit a quadratic algorithm to perform inclusion checking on a RAM machine.

Theoretical Limitations of Discrete Exterior Calculus in the Context of Computational Electromagnetics

Discrete exterior calculus (DEC) attempts to mimic basic operations on differential forms in a discrete setting. This paper considers two concrete instances of how the completion of the DEC program is unachievable, and outlines the practical implications for computational electromagnetics. The two problems are the Commutative Cochain Problem (CCP) and the Discrete Star Localization Problem (DSLP). This paper elaborates on how the CCP is related to problems in computational magnetohydrodynamics, and how DSPL is related to material modeling and constructing discrete Hodge star operators whose matrix representations are maximally sparse in some concrete sense. A final section elaborates on the problem of Poincare duality on the level of cochains, and how the metric-free Whitney form finite element discretization of helicity functionals provides a model of how some of these theoretical obstacles can be side-stepped in three dimensions and, more generally, in dimensions.

Infinitary Action Logic: Complexity, Models and Grammars

Action logic of Pratt [21] can be presented as Full Lambek Calculus FL [14, 17] enriched with Kleene star *; it is equivalent to the equational theory of residuated Kleene algebras (lattices). Some results on axiom systems, complexity and models of this logic were obtained in [4, 3, 18]. Here we prove a stronger form of *-elimination for the logic of *-continuous action lattices and the $${\Pi_{1}^{0}}$$ ---completeness of the equational theories of action lattices of subsets of a finite monoid and action lattices of binary relations on a finite universe. We also discuss possible applications in linguistics.

Efficient Calculation of the Transition Matrix in a Max-Plus Linear State-Space Representation

This research considers an efficient method for calculating the transition matrix in an MPL (Max-Plus Linear) state-space representation. This matrix can be generated by applying the Kleene star operator to an adjacency matrix. The proposed method, based on the idea of a topological sort in graph theory and block splitting, is able to calculate the transition matrix efficiently.

Complete Axiomatization for Divergent-Sensitive Bisimulations in Basic Process Algebra with Prefix Iteration

We study the divergent-sensitive spectrum of weak bisimulation equivalences in the setting of process algebra. To represent the infinite behavior, we consider the prefix iteration extension of a fragment of Milner's CCS. The prefix iteration operator is a variant on the binary version of the Kleene star operator obtained by restricting the first argument to be an atomic action and allows us to capture the notion of recursion in a pure algebraic way. We investigate four typical divergent-sensitive weak bisimulation equivalences, namely divergent, stable, completed and divergent stable weak bisimulation equivalences from an axiomatic perspective. A lattice of distinguishing axioms is developed and thus pure equational axiomatizations for these congruences are obtained. A large part of the current paper is devoted to a considerable complicated proof for completeness. This work, to some extent, sheds light on distinct semantics of divergence.

Topological and Geometrical Considerations for Maxwell's Equations on Unstructured Meshes

A discrete differential form approach to solving Maxwell's equations on unstructured meshes is presented. Discrete representations of differential forms, the underlying manifolds, and associated operators are developed. The discrete boundary, coboundary, and hodge star operators are shown to maintain divergence-free regions. With the construction of a cell complex, its dual complex, and the associated discrete operators, we have determined the numerical update equations for the electromagnetic fields on unstructured meshes with second-order accuracy. The discrete differential form approach generalizes to the Yee algorithm on an orthogonal complex and to the discrete surface integral algorithm on a parallelepiped complex.

On geometric discretization of elasticity

This paper presents a geometric discretization of elasticity when the ambient space is Euclidean. This theory is built on ideas from algebraic topology, exterior calculus, and the recent developments of discrete exterior calculus. We first review some geometric ideas in continuum mechanics and show how constitutive equations of linearized elasticity, similar to those of electromagnetism, can be written in terms of a material Hodge star operator. In the discrete theory presented in this paper, instead of referring to continuum quantities, we postulate the existence of some discrete scalar-valued and vector-valued primal and dual differential forms on a discretized solid, which is assumed to be a triangulated domain. We find the discrete governing equations by requiring energy balance invariance under time-dependent rigid translations and rotations of the ambient space. There are several subtle differences between the discrete and continuous theories. For example, power of tractions in the discrete theory is written on a layer of cells with a nonzero volume. We obtain the compatibility equations of this discrete theory using tools from algebraic topology. We study a discrete Cosserat medium and obtain its governing equations. Finally, we study the geometric structure of linearized elasticity and write its governing equations in a matrix form. We show that, in addition to constitutive equations, balance of angular momentum is also metric dependent; all the other governing equations are topological.

A general approach for building combinational P automata

In traditional automata theory, the closure with respect to certain operations is one of the most investigated properties, especially the methods of constructing automata for certain operations among the given sub-automata. However, up to now less effort has been addressed to this question for P automata. As an improvement on earlier results, we introduce P automata with communication and active membrane rules working in the initial mode (CAIP). We present methods for constructing automata that recognize the Union, the Concatenation, the Kleene Closure, or the ω Closure of the given languages which are represented by some P automata. We also show that for any language denoted by a regular expression, we can readily construct a CAIP automaton corresponding to it.

Some remarks on Prüfer ⋆-multiplication domains and class groups

Let D be an integral domain with quotient field K and let X be an indeterminate over D. Also, let T : = { T λ | λ ∈ Λ } be a defining family of quotient rings of D and suppose that * is a finite type star operation on D induced by T . We show that D is a P*MD (respectively, P vMD) if and only if ( c D ( f g ) ) * = ( c D ( f ) c D ( g ) ) * (respectively, ( c D ( f g ) ) w = ( c D ( f ) c D ( g ) ) w ) for all 0 ≠ f , g ∈ K [ X ] . A more general version of this result is given in the semistar operation setting. We give a method for recognizing P vMD's which are not P*MD's for a certain finite type star operation *. We study domains D for which the *-class group Cl * ( D ) equals the t-class group Cl t ( D ) for any finite type star operation *, and we indicate examples of P vMD's D such that Cl * ( D ) ⊊ Cl t ( D ) . We also compute Cl v ( D ) for certain valuation domains D.

Joint relations on elements of the symmetric group

We introduce two methods which reduce the checking of the joint relations on element pairs x , y (i.e., checking if μ ( x , y ) ≠ 0 , see 1.2) inside a two-sided cell of the symmetric group S n to some special cases up to star operations. One involves the involutions in S n , and the other involves the set Σ λ for λ ∈ Λ n (see 3.1 and 5.1). We give a detailed investigation for the properties of the set Σ λ in terms of standard tableaux. Also, we study the sets U ( S ∞ ) and U ′ ( S ∞ ) and give an affirmative answer to a question of N.H. Xi (see Proposition 2.9).

GEOMETRIC ALGEBRAS AND EXTENSORS

This is the first paper in a series (of four) designed to show how to use geometric algebras of multivectors and extensors to a novel presentation of some topics of differential geometry which are important for a deeper understanding of geometrical theories of the gravitational field. In this first paper we introduce the key algebraic tools for the development of our program, namely the euclidean geometrical algebra of multivectors [Formula: see text] and the theory of its deformations leading to metric geometric algebras [Formula: see text] and some special types of extensors. Those tools permit obtaining, the remarkable golden formula relating calculations in [Formula: see text] with easier ones in [Formula: see text] (e.g. a noticeable relation between the Hodge star operators associated to G and GE). Several useful examples are worked in details for the purpose of transmitting the "tricks of the trade".

Unique representation domains, II

Given a star operation ∗ of finite type, we call a domain R a ∗ -unique representation domain ( ∗ -URD) if each ∗ -invertible ∗ -ideal of R can be uniquely expressed as a ∗ -product of pairwise ∗ -comaximal ideals with prime radical. When ∗ is the t -operation we call the ∗ -URD simply a URD. Any unique factorization domain is a URD. Generalizing and unifying results due to Zafrullah [M. Zafrullah, On unique representation domains, J. Nat. Sci. Math. 18 (1978) 19–29] and Brewer–Heinzer [J.W. Brewer, W.J. Heinzer, On decomposing ideals into products of comaximal ideals, Comm. Algebra 30 (2002) 5999–6010], we give conditions for a ∗ -ideal to be a unique ∗ -product of pairwise ∗ -comaximal ideals with prime radical and characterize ∗ -URD’s. We show that the class of URD’s includes rings of Krull type, the generalized Krull domains introduced by El Baghdadi and weakly Matlis domains whose t -spectrum is treed. We also study when the property of being a URD extends to some classes of overrings, such as polynomial extensions, rings of fractions and rings obtained by the D + X D S [ X ] construction.

Uppers to zero and semistar operations in polynomial rings

Given a stable semistar operation of finite type ⋆ on an integral domain D, we show that it is possible to define in a canonical way a stable semistar operation of finite type [ ⋆ ] on the polynomial ring D [ X ] , such that D is a ⋆-quasi-Prüfer domain if and only if each upper to zero in D [ X ] is a quasi- [ ⋆ ] -maximal ideal. This result completes the investigation initiated by Houston–Malik–Mott [E. Houston, S. Malik, J. Mott, Characterizations of ∗-multiplication domains, Canad. Math. Bull. 27 (1984) 48–52, Section 2. [17]] in the star operation setting. Moreover, we show that D is a Prüfer ⋆-multiplication (respectively, a ⋆-Noetherian; a ⋆-Dedekind) domain if and only if D [ X ] is a Prüfer [ ⋆ ] -multiplication (respectively, a [ ⋆ ] -Noetherian; a [ ⋆ ] -Dedekind) domain. As an application of the techniques introduced here, we obtain a new interpretation of the Gabriel–Popescu localizing systems of finite type on an integral domain D (Problem 45 of [S.T. Chapman, S. Glaz, One hundred problems in commutative ring theory, in: S.T. Chapman, S. Glaz (Eds.), Non-Noetherian Commutative Ring Theory, Kluwer Academic Publishers, 2000, pp. 459–476. [4]]), in terms of multiplicatively closed sets of the polynomial ring D [ X ] .

Finite turns and the regular closure of linear context-free languages

Turn bounded pushdown automata with different conditions for beginning a new turn are investigated. Their relationships with closures of the linear context-free languages under regular operations are studied. For example, automata with an unbounded number of turns that have to empty their pushdown store up to the initial symbol in order to start a new turn are characterized by the regular closure of the linear languages. Automata that additionally have to re-enter the initial state are (almost) characterized by the Kleene star closure of the linear languages. For both a bounded and an unbounded number of turns, requiring to empty the pushdown store is a strictly stronger condition than requiring to re-enter the initial state. Several new language families are obtained which form a double-stranded hierarchy. Closure properties of these families under AFL operations are derived. The regular closure of the linear languages share the strong closure properties of the context-free languages, i.e., the family is a full AFL. Interestingly, three natural new language families are not closed under intersection with regular languages and inverse homomorphism. Finally, an algorithm is presented parsing languages from the new families in quadratic time.

On ∗– λ-semirings

A ∗– λ-semiring is an ordered semiring S equipped with a star operation such that for any a, b ∈ S, a ∗ b is the least fixed point of the linear mapping x ↦ ax + b over S. The notion of ∗– λ-semirings is a generalization of several important concepts such as continuous semirings, (weak) inductive ∗-semirings and the Kleene algebras of Conway and Kozen. We investigate several basic properties of ∗– λ-semirings and obtain results on ∗– λ-semirings in relation to preorders, duality and formal power series. Some of these results can be seen as generalizations of relevant results on inductive ∗-semirings.

The Origin of Carbon Enhancement and the Initial Mass Function of Extremely Metal‐poor Stars in the Galactic Halo

It is known that the carbon-enhanced, extremely metal-poor (CEMP) stars constitute a substantial proportion in the extremely metal-poor (EMP) stars of the Galactic Halo, by far larger than CH stars in Population II stars. We investigate their origin with taking into account an additional evolutionary path to the surface carbon-enrichment, triggered by hydrogen engulfment by the helium flash convection, in EMP stars of $[Fe/H] \lesssim -2.5$. This process is distinct from the third dredge-up operating in more metal-rich stars and also in EMP stars. In binary systems of EMP stars, the secondary stars become CEMP stars through mass transfer from the primary stars of low and intermediate masses, which have developed the surface carbon-enhancement. Our binary scenario can predict the variations in the abundances not only for carbon but also for nitrogen and s-process elements and reasonably explain the observed properties such as the stellar distributions with respect to the carbon abundances, the binary periods, and the evolutionary stages. Furthermore, from the observed frequencies of CEMP stars with and without s-process element enhancement, we demonstrate that the initial mass function of EMP stars need to give the mean mass $~10\msun$ under the reasonable assumptions on the distributions of orbital separations and mass ratio of binary components. This also indicates that the currently observed EMP stars were exclusively born as the secondary members of binaries, making up $\sim 10%$ remnants of EMP binary systems of mass $~10^8\msun$ in total; in addition to CEMP stars with white dwarf companions, a significant fraction of them have experienced supernova explosions of their companions. We discuss the implications of the present results in relation to the formation of Galactic halo.

Cochain algebra on manifolds and convergence under refinement

In this paper we develop several algebraic structures on the simplicial cochains of a triangulated manifold and prove they converge to their differential-geometric analogues as the triangulation becomes small. The first such result is for a cochain cup product converging to the wedge product on differential forms. Moreover, we show any extension of this product to a C ∞ -algebra also converges to the wedge product of forms. For cochains equipped with an inner product, we define a combinatorial star operator and show that for a certain cochain inner product this operator converges to the smooth Hodge star operator.

Noncommutative Instantons from Twisted Conformal Symmetries

We construct a five-parameter family of gauge-nonequivalent SU (2) instantons on a noncommutative four sphere \({S_{\theta}^4}\) and of topological charge equal to 1. These instantons are critical points of a gauge functional and satisfy self-duality equations with respect to a Hodge star operator on forms on \({S_{\theta}^4}\) . They are obtained by acting with a twisted conformal symmetry on a basic instanton canonically associated with a noncommutative instanton bundle on the sphere. A completeness argument for this family is obtained by means of index theorems. The dimension of the “tangent space” to the moduli space is computed as the index of a twisted Dirac operator and turns out to be equal to five, a number that survives deformation.