Kleene Star Research Articles

Note on characterizations of semistar operations and star operations on an integral domain

We study semistar operations and star operations on an integral domain and we give some new characterizations of both a semistar operation and a star operation.

Staged Abdominal Repair Surgery in Abdominal Compartment Syndrome- An Observational Study

Objective:Staged Abdominal Repair Surgery in Abdominal Compartment Surgery.Research design and methods: The study was conducted on 45 patients who underwent laparotomy due to acute abdominal conditions and abdomen was closed in staged manner using staged abdominal repair surgery also known as STAR operation.Statistical analysis was performed and results analysed.Results: The present series comprising of 45 cases had low rate of grave complications like entero-cutaneous fistulae formation (< 16%), intraabdominal infection (approx 17%), development of bedsore (approx 13%), chest infection (approx 22%). Average hospital stay was of 3 weeks and postoperative mortality of approximately 11% .Conclusions: We thus conclude that the STAR technique has definite edge over previous techniques of temporary abdominal closure and shows a new path of abdominal closure to the surgical fraternity . The effects of increased abdominal pressure on various organ systems has been noted over the past century. Intra-abdominal hypertension (IAH) and abdominal compartment syndrome (ACS) have been identified as a cause of organ dysfunction and mortality in several subsets of critically ill patients. The staged abdominal repair operation popularly known as STAR operation is a technique of temporary closue of abdomen during primary surgery followed by a second surgery 24 to 48 hours later ending in final aponeurosis to aponeurosis closure of abdomen.. In abdominal compartment syndrome;this operation, is gaining popularity because it is life saving in critically ill patients and has lower complication rate as well. Intra-abdominal pressure (IAP) is the steady pressure within the abdominal cavity.For critically ill patients, an IAP of 5 to 7 mm of Hg is considered normal. Intra-abdominal pressure (IAP) is graded as follows :

The semistar operations on certain Prüfer domain, II

Let D be a 1-dimensional Prüfer domain with exactly two maximal ideals. We completely determine the star operations and the semistar operations on D.

A modification of the Hodge star operator on manifolds with boundary

If M is a smooth compact oriented Riemannian manifold of dimension n=4k+2, with or without boundary, and F is a vector bundle on M with an inner product and a flat connection, we construct a modification of the Hodge star operator on the parabolic cohomology H^{2k+1}_{par}(M;F). The operator gives a canonical complex structure on H^{2k+1}_{par}(M;F) compatible with the symplectic form \omega given by the wedge product of forms in the middle dimension. In case when k=0 that gives a canonical almost complex structure on the non-singular part of the moduli space of flat connections on a Riemann surface with or without boundary and monodromies along boundary components belonging to fixed conjugacy classes. The almost complex structure is compatible with the standard symplectic form \omega on the moduli space.

On Local ☆-Completely Integrally Closed Domains

Let ☆ be a star operation on an integral domain R. The domain R is a ☆-CICD if (AA −1)☆ = R for all nonzero (fractional) ideals A of R. In this article, we prove that, if the maximal ideal of a local ☆-CICD is a ☆-ideal, then R is ☆-principal ideal domain. We also establish that any ☆-CICD R is locally a PID when ☆ is induced by the localizations at prime ideals of R.

Omega-Rational Expressions with Bounded Synchronization Delay

In 1965 Schutzenberger published his famous result that star-free languages ( $\operatorname{SF}$ ) and aperiodic languages ( $\operatorname{AP}$ ) coincide over finite words, often written as $\operatorname{SF}= \operatorname {AP}$ . Perrin generalized $\operatorname{SF} = \operatorname{AP}$ to infinite words in the mid 1980s. In 1973 Schutzenberger presented another (and less known) characterization of aperiodic languages in terms of rational expressions where the use of the star operation is restricted to prefix codes with bounded synchronization delay and no complementation is used. We denote this class of languages by $\operatorname{SD}$ . In this paper, we present a generalization of $\operatorname{SD}= \operatorname{AP}$ to infinite words. This became possible via a substantial simplification of the proof for the corresponding result for finite words. Moreover, we show that $\operatorname{SD}= \operatorname{AP}$ can be viewed as more fundamental than $\operatorname{SF}= \operatorname{AP}$ in the sense that the classical 1965 result of Schutzenberger and its 1980s extension to infinite words by Perrin are immediate consequences of $\operatorname{SD}= \operatorname{AP}$ .

Quotient Complexity of Closed Languages

A language L is prefix-closed if, whenever a word w is in L, then every prefix of w is also in L. We define suffix-, factor-, and subword-closed languages in an analogous way, where by factor we mean contiguous subsequence, and by subword we mean scattered subsequence. We study the state complexity (which we prefer to call quotient complexity) of operations on prefix-, suffix-, factor-, and subword-closed languages. We find tight upper bounds on the complexity of the subword-closure of arbitrary languages, and on the complexity of boolean operations, concatenation, star, and reversal in each of the four classes of closed languages. We show that repeated applications of positive closure and complement to a closed language result in at most four distinct languages, while Kleene closure and complement give at most eight.

Deciding the unguarded modal -calculus

The modal -calculus extends basic modal logic with second-order quantification in terms of arbitrarily nested fixpoint operators. Its satisfiability problem is EXPTIME-complete. Decision procedures for the modal -calculus are not easy to obtain though since the arbitrary nesting of fixpoint constructs requires some combinatorial arguments for showing the well-foundedness of least fixpoint unfoldings. The tableau-based decision procedures so far also make assumptions on the unfoldings of fixpoint formulas, e.g., explicitly require formulas to be in guarded normal form. In this paper we present a tableau calculus for deciding satisfiability of arbitrary, i.e., not necessarily guarded -calculus formulas. It is based on a new unfolding rule for greatest fixpoint formulas which allows unguarded formulas to be handled without an explicit transformation into guarded form, thus avoiding a (seemingly) exponential blow-up in formula size. We prove soundness and completeness of the calculus, and compare it empirically to using guarded transformation instead. The new unfolding rule can also be used to handle nested star operators in PDL formulas correctly.

Composition and orbits of language operations: finiteness and upper bounds

We consider a set of eight natural operations on formal languages (Kleene closure, positive closure, complement, prefix, suffix, factor, subword, and reversal), and compositions of them. If x and y are compositions, we say x is equivalent to y if they have the same effect on all languages L. We prove that the number of equivalence classes of these eight operations is finite. This implies that the orbit of any language L under the elements of the monoid is finite and bounded, independent of L. This generalizes previous results about complement, Kleene closure, and positive closure. We also estimate the number of distinct languages generated by various subsets of these operations.

Comments on Arithmetic Complexity, Kleene Closure, and Formal Power Series

In reference to our earlier work [1], Pierre McKenzie and Sambuddha Roy pointed out that the proofs of statements (b) and (c) in Theorem 7.3 are buggy. The main flaw is that the identity e of the group F may not be the identity of the monoid, and so the claim that w ∈ (AF,r )∗ ⇐⇒w ∈ Test does not work. Herewith, we show: • With a slight change to Definition 7.1, the statement of Theorem 7.3 holds unchanged. In our opinion, this is the most interesting way to correct the error in the original paper. We present a complete proof below. For completeness, we also mention another way to correct the error: • Leaving Definition 7.1 unchanged, a weaker version of Theorem 7.3 holds (with only minor adjustments to the proof given in the paper).

Residuated Park theories

When $L$ is a complete lattice, the collection $\Mon_L$ of all monotone functions $L^p \to L^n$, $n,p \geq 0$, forms a Lawvere theory. We enrich this Lawvere theory with the binary supremum operation $\vee$, an operation of (left) residuation $\res$ and the parameterized least fixed point operation $^\dagger$. We exhibit a system of \emph{equational} axioms which is sound and proves all valid equations of the theories $\Mon_L$ involving only the theory operations, $\vee$ and $^\dagger$, i.e., all valid equations not involving residuation. We also present an alternative axiomatization, where $^\dagger$ is replaced by a star operation, and provide an application to regular tree languages.

THE RINGS $D((\mathcal{X}))_i$ AND $D\{\{\mathcal{X}\}\}_i$

Let D be an integral domain, [Formula: see text] be an infinite set of indeterminates over D, and [Formula: see text] be the i th type of power series ring over D for i = 1, 2, 3. For [Formula: see text], let c(f) denote the ideal of D generated by the coefficients of f. For a star operation * on D, put [Formula: see text], where *f is the star operation of finite type on D induced by *. Let [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. We show that [Formula: see text], and that D is a Noetherian domain if and only if [Formula: see text] is a Noetherian domain. We also show that D is a Krull domain if and only if [Formula: see text] is a Dedekind domain, if and only if [Formula: see text] is a Prüfer domain, and that D is a Dedekind domain if and only if [Formula: see text] is a Dedekind domain, if and only if [Formula: see text] is a Prüfer domain.

ON ⋆-COLON-MULTIPLICATION DOMAINS

Let ⋆ be a star operation on an integral domain R. An ideal A is a ⋆-colon-multiplication ideal if A⋆ = (B(A : B))⋆ for all fractional ideal B of R. We prove that every maximal ideal of R is a ⋆-colon-multiplication ideal if and only if R is a ⋆-CICD or R is a local ⋆-MTP domain. It is also shown that every ideal of R is ⋆-colon-multiplication if and only if R is a ⋆-CICD.

On the cardinality of star operations on a pseudo-valuation domain

On the cardinality of star operations on a pseudo-valuation domain

The Möbius function of generalized factor order

We use discrete Morse theory to determine the Möbius function of generalized factor order. Ordinary factor order on the Kleene closure A∗ of a set A is the partial order defined by letting u≤w if w contains u as a subsequence of consecutive letters. Generalized factor order takes into account a partial order PA on the alphabet A, that is, u≤w whenever w contains a subsequence w(i+1)⋯w(i+|u|) such that for each j, u(j)≤w(i+j) in A. Using Babson and Hersh’s application of Robin Forman’s discrete Morse theory to poset order complexes, we are able to give a recursive formula for the Möbius function in the case where each element of A covers a unique letter in PA.

Formulation of gauge theories on transitive Lie algebroids

In this paper we introduce and study some mathematical structures on top of transitive Lie algebroids in order to formulate gauge theories in terms of generalized connections and their curvature: metrics, Hodge star operator and integration along the algebraic part of the transitive Lie algebroid (its kernel). Explicit action functionals are given in terms of global objects and in terms of their local description as well. We investigate applications of these constructions to Atiyah–Lie algebroids and to derivations on a vector bundle. The obtained gauge theories are discussed with respect to ordinary and to similar noncommutative gauge theories.

Delaunay Hodge star

We define signed dual volumes at all dimensions for circumcentric dual meshes. We show that for pairwise Delaunay triangulations with mild boundary assumptions these signed dual volumes are positive. This allows the use of such Delaunay meshes for Discrete Exterior Calculus (DEC) because the discrete Hodge star operator can now be correctly defined for such meshes. This operator is crucial for DEC and is a diagonal matrix with the ratio of primal and dual volumes along the diagonal. A correct definition requires that all entries be positive. DEC is a framework for numerically solving differential equations on meshes and for geometry processing tasks and has had considerable impact in computer graphics and scientific computing. Our result allows the use of DEC with a much larger class of meshes than was previously considered possible.

Decidability of Downward XPath

We investigate the satisfiability problem for downward-XPath, the fragment of XPath that includes the child and descendant axes, and tests for (in)equality of attributes’ values. We prove that this problem is decidable, EXPTIME-complete. These bounds also hold when path expressions allow closure under the Kleene star operator. To obtain these results, we introduce a Downward Data automata model (DD automata) over trees with data, which has a decidable emptiness problem. Satisfiability of downward-XPath can be reduced to the emptiness problem of DD automata and hence its decidability follows. Although downward-XPath does not include any horizontal axis, DD automata are more expressive and can perform some horizontal tests. Thus, we show that the satisfiability remains in EXPTIME even in the presence of the regular constraints expressible by DD automata. However, the same problem in the presence of any regular constraint is known to have a nonprimitive recursive complexity. Finally, we give the exact complexity of the satisfiability problem for several fragments of downward-XPath.

Determination of an electromagnetic medium from the Fresnel surface

We study Maxwell’s equations on a 4-manifold where the electromagnetic medium is described by a suitable antisymmetric -tensor κ with real components. In this setting, the Tamm–Rubilar tensor density determines a polynomial surface of fourth order in each cotangent space. This surface is called the Fresnel surface and acts as a generalization of the null cone determined by a Lorentz metric; the Fresnel surface parameterizes electromagnetic wavespeed as a function of direction. We show that if (a) κ has no skewon and no axion component, (b) κ is invertible and (c) the Fresnel surface is pointwise a Lorentz null cone, then the tensor κ is proportional to a Hodge star operator of a Lorentz metric and κ represents an isotropic medium. In other words, in a suitable class of media one can recognize isotropic media from wavespeed alone. What is more, we study the nonunique dependence between the tensor κ, its Tamm–Rubilar tensor density and its Fresnel surface. For example, we show that if κ is invertible, then κ and κ−1 have the same Fresnel surfaces.

Stability and Clifford Regularity with Respect to Star Operations

In the last few years, the concepts of stability and Clifford regularity have been fruitfully extended by using star operations. In this article we deepen the study of star stable and star regular domains and relate these two classes of domains to each other.