On pseudo-fractional integral inequalities related to Hermite–Hadamard type

General versions of Hermite–Hadamard type inequality for pseudo-fractional integrals of the order \(\alpha >0\) on a semiring \(\left( \left[ a,b\right] ,\oplus ,\odot \right) \) are studied. These inequalities include both pseudo-integral and fractional integral. The well-known previous results are shown to be special cases of our results. Finally, two open problems for further investigations are given.     

Approximate Comparison of Functions Computed by Distance Automata

Distance automata are automata weighted over the semiring \((\mathbb {N}\cup \{\infty \},\min , +)\) (the tropical semiring). Such automata compute functions from words to \(\mathbb {N}\cup \{\infty \}\). It is known from Krob that the problems of deciding ‘ f≤g’ or ‘ f=g’ for f and g computed by distance automata is an undecidable problem. The main contribution of this paper is to show that an approximation of this problem is decidable. We present an algorithm which, given ε>0 and two functions f,g computed by distance automata, answers “yes” if f≤(1?ε)g, “no” if f≦?g, and may answer “yes” or “no” in all other cases. The core argument behind this quasi-decision procedure is an algorithm which is able to provide an approximated finite presentation of the closure under products of sets of matrices over the tropical semiring. Lastly, our theorem of affine domination gives better bounds on the precision of known decision procedures for cost automata, when restricted to distance automata.     

The semiring of matrices over a finite chain

Let L_m denote the chain {0,1,2,…,m-1} with the usual ordering and M_n(L_m) the matrix semiring of all n×n matrices with elements in L_m. We firstly introduce some order-preserving semiring homomorphisms from M_n(L_m) to M(L_k). By using these homomorphisms, we show that a matrix over the finite chain L_m can be decomposed into the sum of some matrices over the finite chain L_k, where k<m. As a result, cut matrices decomposition theorem of a fuzzy matrix (Theorem 4 in [Z.T. Fan, Q.S. Cheng, A survey on the powers of fuzzy matrices and FBAMs, International Journal of Computational Cognition 2 (2004) 1-25 (invited paper)]) is generalized and extended. Further, we study the index and periodicity of a matrix over a finite chain and get some new results. On the other hand, we introduce a semiring embedding mapping from the semiring M_n(L_m) to the direct product of the h copies of the semiring M_n(L_k) and discuss Green’s relations on the multiplicative semigroup of the semiring M_n(L_m). We think that some results obtained in this paper is useful for the study of fuzzy matrices.     

Free semiring-representations and nondeterminism

Semirings provide a simple abstract model of the syntax for a nondeterministic programming language. Each element of a semiring is a nondeterministic program segment, and the semiring operations (+ and ?) correspond to nondeterministic or and program composition. This is analogous to using an algebraic theory for the abstract syntax of a deterministic language. In the case of algebraic theories, an algebra provides the semantics, and free algebras (which always exist) are particularly important. For a semiring, semantics is provided by a representation as a system of relations. This paper examines the question of when free representations exist. Unlike free algebras, free representations do not always exist. It is shown that a semiring has free representations generated by arbitrary sets of variables iff it has a free representation generated by a single variable. Examples of semirings are given that do not have free representations. However, for an important class of semirings, free representations are always available. This class consists of semirings which arise when nondeterminism is freely added to a deterministic programming language.     

A sketch of a dynamic epistemic semiring

This paper proposes a semiring formulation for reasoning about an agent’s changing beliefs: a dynamic epistemic semiring (DES). A DES is a modal semiring extended with epistemic-action operators. The paper concentrates on the revision operator by proposing an axiomatisation, developing a basic calculus and deriving the classical AGM revision axioms in the algebra. Iterated action is also considered.     

A complete axiom system for rational sets with multiplicity

By extending the axiom system for equalities between rational (or recognizable) sets by Salomaa, we introduce an axiom system for equalities between rational K-subsets (sets recognized by finite state automata with multiplicity in semiring, K). The system is consistent and if K is a field it is complete; that is, all and only equalities between rational K-subsets are derivable in the system.     

Free inductive K-semialgebras

We consider rational power series over an alphabet Σ with coefficients in a ordered commutative semiring K and characterize them as the free ordered semialgebras in various classes defined by the least pre-fixed point rule and its dual. The results are generalizations of Kozen’s axiomatization of regular languages.     

Searching a polygonal region by a group of stationary k-searchers

We study the problem of searching for mobile intruders in a polygonal region by stationary searchers having various levels of vision given by the number of flashlights that a searcher carries. We show that (2g−1) 1-searchers (i.e., 2g−1 searchers with one flashlight each) are always sufficient, and sometimes necessary, to search a simple polygonal region having a guard number g, which is the size of a minimum guard set. We also show that g(h+1)-searchers (i.e., g searchers with h+1 flashlights each), and consequently g(h+1) 1-searchers as well, can always search a polygonal region with h?1 holes having a guard number g.     

Two methods for the calculation of the degree of an approximate greatest common divisor of two inexact polynomials

The calculation of the degree d of an approximate greatest common divisor of two inexact polynomials f(y) and g(y) reduces to the determination of the rank loss of a resultant matrix, the entries of which are functions of the coefficients of f(y) and g(y). This paper considers this issue by describing two methods to calculate d, such that knowledge of the noise level imposed on the coefficients of f(y) and g(y) is not assumed. One method uses the residual of a linear algebraic equation whose coefficient matrix and right hand side vector are derived from the Sylvester resultant matrix S(f,g), and the other method uses the first principal angle between a line and a hyperplane, the equations of which are calculated from S(f,g). Computational results on inexact polynomials whose exact forms have multiple roots of high degree are shown and very good results are obtained. These results are compared with the rank loss of S(f,g) for the calculation of d, and it is shown that this method yields incorrect results for these examples.     

An application of H differentiability to generalized complementarity problems over symmetric cones

In this paper, we focus on a generalized complementarity problems over symmetric cone GSCCP(f,g) when the underlying functions f and g are H-differentiable. By introducing the concepts of relatively uniform Cartesian P-property, relatively Cartesian P(P₀)-property, the Cartesian semimonotone (E₀)-property (strictly Cartesian semimonotone (E)-property), and the relatively regular point with respect to the merit function Ψ(x), we extend various similar results proved in GCP(f,g) to generalized complementarity problems over symmetric cone GSCCP(f,g) and establish various conditions on f and g to get a solution to GSCCP(f,g).     

Test sets and checking words for homomorphism equivalence

Given a language L over an alphabet Σ and two homomorphisms g and h, defined on $\text{Σ}{}^1$, we want to decide whether or not g and h are equivalent on L, i.e., whether or not g(w) = h(w) holds for all words w in L. We prove the following results' for the case where Σ consists of two letters. Every language L possesses a finite subset L, such that, for any pair (g, h), g and h are equivalent on L if and only if they are equivalent on L₁. For every language L (with the exception of some trivial cases), there is a word w (not necessarily in L) such that, for any pair (g, h), g and h are equivalent on L if and only if g(w) = h(w). Our constructions are, in general, noneffective. Also some related notions are discussed in the paper.     

Equality sets of prefix morphisms and regular star languages

We consider equality sets of prefix morphisms, that is, sets E(g₁,g₂)={w|g₁(w)=g₂(w)}, where g₁ and g₂ are prefix morphisms. Recall that a morphism g is prefix if, for all different letters a and b, g(a) is not a prefix of g(b). We prove a rather surprising equality on families of languages, namely, that the family of regular star languages coincides with the family of languages of form πA(E(g₁,g₂)) for some prefix morphisms g₁ and g₂, and a projection πA which deletes the letters not in A.     

On packing shortest cycles in graphs

We study the problems to find a maximum packing of shortest edge-disjoint cycles in a graph of given girth g (g-ESCP) and its vertex-disjoint analogue g-VSCP. In the case g=3, Caprara and Rizzi (2001) have shown that g-ESCP can be solved in polynomial time for graphs with maximum degree 4, but is APX-hard for graphs with maximum degree 5, while g-VSCP can be solved in polynomial time for graphs with maximum degree 3, but is APX-hard for graphs with maximum degree 4. For g∈{4,5}, we show that both problems allow polynomial time algorithms for instances with maximum degree 3, but are APX-hard for instances with maximum degree 4. For each g?6, both problems are APX-hard already for graphs with maximum degree 3.     

An algebraic characterization of some principal regulated rational cones

The aim of this paper is to deal with formal power series over a commutative semiring A. Generalizing Wechler's pushdown automata and pushdown transition matrices yields a characterization of the A-semi-algebraic power series in terms of acceptance by pushdown automata. Principal regulated rational cones generated by cone generators of a certain form are characterized by algebraic systems given in certain matrix form. This yields a characterization of some principal full semi-AFL's in terms of context-free grammars. As an application of the theory, the principal regulated rational cone of one-counter “languages” is considered.     

Evaluation of thermodynamic properties from alloy phase diagram with miscibility gap using non-random two-liquid equation

The non-random two-liquid equation has been applied to evaluate the thermodynamic properties of the liquid solution at elevated temperatures in a binary alloy system with a liquid phase miscibility gap. Only upon making use of the phase equilibrium data at the critical and monotectic points of the miscibility gap from a T-X phase diagram and thermochemical data, the parameters needed for the evaluation, i. e., (g₁₂ - g₂₂), g₂₁ - (g₁₁) and a of the non-random two-liquid solution approach, can be determined. The evaluation of thermodynamic properties was carried out numerically for three binary alloy systems, i. e., Al-Pb, Zn-Pb and Ga-Hg systems. The application of the non-random two-liquid equation to these three binary alloy systems shows that the evaluated results are close to the available experimental measurements.     

Weighted aggregation operators based on minimization

Minimization based aggregation operators A_g,D are discussed. Special attention is paid to weighting function g based cases related to some fixed dissimilarity function D. When D₂(x,y)=(x-y)², we recognize mixture operators and we recall some sufficient conditions for g ensuring the monotonicity of A_g,D. For D₁(x,y)=|x-y| and non-decreasing (non-increasing) g, A_g,D is shown to be the upper (lower) median whenever A_g,D is an aggregation operator.     

Why-provenance information for RDF,rules, and negation

The provenance (i.e., origins) of derived information on the Web is crucial in many applications to allow information quality assessment, trust judgments, accountability, as well as understanding the temporal and spatial status of the information. On the other hand, the inclusion of negative information in knowledge representation both in the form of negation-as-failure and explicit negation is also important to allow various forms of reasoning, provided that weakly negated information is associated with the sources (contexts) in which it holds. In this work, we consider collections of g-RDF ontologies, distributed over the web, along with a set of conflict statements expressing that information within a pair of g-RDF ontologies cannot be combined together for deriving new information. A g-RDF ontology is the combination of (i) a g-RDF graph G (i.e., a set of positive and strongly negated RDF triples, called g-RDF triples) and (ii) a g-RDF program P containing derivation rules with possibly both explicit and scoped weak negation. Information can be inferred through the g-RDF graphs or the derivation rules of the g-RDF ontologies, or through the RDFS derivation rules. We associate each derived grounded g-RDF triple [¬] p(s, o) with the set of names S of the g-RDF ontologies that contributed to its derivation. To achieve this, we define the provenance stable models of a g-RDF ontology collection. We show that our provenance g-RDF semantics faithfully extends RDFS semantics. Finally, we provide an algorithm based on Answer Set Programming that computes all provenance stable models of a g-RDF ontology collection and provides the answer to various kinds of queries. Various complexity results are provided.     

Linear and nonlinear resistive instability studies

The linear stability properties of the resistive ‘g’ mode are examined. The effects of including a full stress tensor have been examined for this mode. The nonlinear ‘g’ mode has also been examined and a potential saturation mechanism identified. The results of 2D calculations for the m = 0 and m = 1 ‘g’ modes in the reverse field pinch (RFP) are presented. Ergodic field line behaviour is found as a result of the interaction of mixed helicity ‘g’ modes.     

Inductive semimodules and the vector modules over them

The notion of an inductive semimodule over an ordered *-semiring is introduced and some related properties are investigated. Inductive semimodules are extensions of several important algebraic structures such as Kleene modules, Kleene algebras and inductive *-semirings. We prove that an inductive semimodule over an ordered *-semiring K is a Kleene module if and only if K is a Kleene algebra. Moreover, we establish that the vector module of an inductive semimodule over an ordered Conway semiring is again an inductive semimodule over the matrix semiring. Consequently, in an inductive semimodule over an ordered Conway semiring, least solutions to linear inequation systems can be denoted by linear expressions, avoiding the least fixed point operator. In addition, we also introduce a related notion called weak inductive semimodules, and propose several open problems on them.

Periodic orbits on discrete dynamical systems

In this paper, we discuss the discrete dynamical system x_n+1 = βx_n − g(x_n), n = 0, 1, …, arising as a discrete-time network of single neuron, where β is the internal decay rate, g, is a signal function. First, we consider the case where g is of McCulloch-Pitts nonlinearity. Periodic orbits are discussed according to different range of β. Moreover, we can construct periodic orbits. Then, we consider the case where g is a sigmoid function. Sufficient conditions are obtained for (1) has periodic orbits of arbitrary periods and an example is also given to illustrate the theorem.