The band method for positive and strictly contractive extension problems: An alternative version and new applications

A novel decision-making method using R-Norm concept and VIKOR approach under picture fuzzy environment

We propose a new method for conducting Single transferable vote (STV) elections and provide a unified method for describing classic STV procedures (the Gregory Method, the inclusive Gregory method and the Weighted inclusive Gregory method) as an iterative procedure. We also propose a modification for quota definition that improves the theoretical properties of the procedures. The method is justified by utilising a new set of axioms. We show that this method extends the Weighted inclusive Gregory method with the modified definition of quota and random equiprobable selection of a winning coalition in each iteration. The results are extended to the methods, allowing fractional numbers of votes.

A kernel-trace description of right congruences on an inverse semigroup is developed. It is shown that the trace mapping is a complete ∩ \cap homomorphism but not a ∨ \vee -homomorphism. However, the trace classes are intervals in the complete lattice of right congruences. In contrast, each kernel class has a maximum element, namely the principal right congruence on the kernel, but in general there is no minimum element in a kernel class. The kernel mapping preserves neither intersections nor joins. The set of axioms presented in [7] for right kernel systems is reviewed. A new set of axioms is obtained as a consequence of the fact that a right congruence is the intersection of the principal right congruences on the idempotent classes. Finally, it is shown that even though a congruence on a regular semigroup is the intersection of the principal congruences on the idempotent classes, the situation is not the same for right congruences on a regular semigroup. Right congruences on a regular, even orthodox, semigroup are not, in general, determined by their idempotent classes.

A theoretical development on the entropy of interval-valued fuzzy sets based on the intuitionistic distance and its relationship with similarity measure

This paper is devoted to an entropy-independent measure of knowledge in the context of intuitionistic fuzzy sets (IFSs). We point out and justify that there are at least two facets of knowledge associated with an IFS, namely the information content and the information clarity. Having this in mind, we put forward a novel axiomatic definition of knowledge measure for IFSs. More specifically, a set of new axioms is presented with which knowledge measure should comply in the context of IFSs. A parametric model following these axioms is then developed to realize this measure. Both facets mentioned above are simultaneously taken into account in the axioms and the model so as to better capture the unique features of an IFS. In particular, we suggest the concept of the amount of potential knowledge related to the hesitancy or non-specificity of an IFS. This allows us to introduce the idea of an attitudinal-based knowledge measure for IFSs. We believe that the knowledge measure provided in this manner could truly reflect the nature of IFSs, and what people really want with different attitudes towards the unknown. Finally, a practical application of the developed technique to decision making under uncertainty is illustrated. As such, the developed measure could be considered as a safe and effective alternative to help tackle some special problems that are difficult to handle by using entropy alone, especially when dealing with the complex situation in which different attitudes of users have to be considered.

User-friendly formal specification and verification of concurrent and distributed systems for various domains, such as automatic control systems, telecommunications, and business processes, are active research topics due to their practical significance. In this paper, we present methods of developing verification-oriented domain-specific process ontologies used to describe concurrent systems of subject domains. One of the advantages of such ontologies is their formal semantics, which provides formal verification of the described systems. Our method is based on an abstract verification-oriented process ontology. We use two methods of domain specialization of the abstract process ontology. The declarative method relies on specializing classes of the original ontology, introducing new declarative classes, and using a new set of axioms to set restrictions on classes and relations of the abstract ontology. The constructive method uses semantic markup and pattern matching techniques to link domain concepts to classes of the abstract process ontology. We provide detailed ontological specifications for these techniques. Our methods preserve the formal semantics of the original process ontology; therefore, formal verification methods can be applied to the resulting domain-specific process ontologies. We demonstrate that the constructive method is a refined version of the declarative method. We illustrate our methods on the example of constructing an ontology for standard elements of automatic control systems: we develop declarative descriptions of the classes and restrictions of the domain-specific ontology in the Protégé system in the Web Ontology Language (OWL) using inference rules written in the Semantic Web Rule Language (SWRL) and construct a system of semantic markup patterns that implements standard elements of automatic control systems.

The Grassmannian approach is used to develop a new addition of the band method. This addition allows one to obtain a linear fractional representation of all solutions of a completion problem from special extensions that are not necessarily band extensions (for the positive case) or triangular extensions (for the contractive case). Also linear fractional representations are obtained for all solutions of a completion problem of non-band type.KeywordsBand StructureArbitrary ElementMultiplication TablePositive Real PartOperator MatriceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We show that extension problems for generalized Toeplitz kernels may be completely reduced to extension problems for positive definite functions, where the solution is well known.These considerations in particular imply that generalized Toeplitz kernels may be represented as Fourier transforms of positive operator-valued measures.The notion of generalized Toeplitz kernels (g.T.k.) was introduced in [5], where also a generalized Bochner theorem was proved in the discrete case.Since then many papers concerning this topic appeared, for example, [2,3].The extension problem for the discrete case was discussed in [1] and for the continuous case, for example, in [4].The proofs for the extendibility and for the generalized Bochner theorem use mainly lifting theorems for families of operators.In this paper we want to show that these questions can be reduced directly to appropriate questions for positive definite operator-valued functions.Fix Hilbert spaces HX,H2, and let 0 < a < co, 1(a) := (-2a, 2a), Ix(a) := (-2a, 0), /2(a) := (0, 2a).An operator-valued generalized Toeplitz kernel K on 1(a) x 1(a) is an operator-valued function with K(s, t) e L(Ha, HB) if s e Ia(a), t e Ip(a) and if there are four weakly continuous functions KBo: Ia(a)-IB(a)-> L(Ha,Hp), a, 8 = 1,2, such that K(s, t) = KBa(s-1) for 5 e Ia(a), t e IB(a), and Kx2(-t) = K2x(t)* for ?e (0,4a); i.e., Kxx: (-2a, 2a) -L(HX), Kx2: (-4a, 0) -L(H2, //,), K2X: (0, 4a) -L(//,, H2), K22: (-2a, 2a) -L(H2).Afor all pairs of functions tpa: I(a(a) -> Ha, a = 1, 2, with finite support.Recall that an operator-valued function F: (-2a, 2a) -> L(H) with some Hilbert space H is called positive definite if(1) Y (F(s-t)<p(s),4>(t))H>0s,t€{-a,a)

In the first part l1r of this paper we stated the multidimensional nonlinear Schur parametrization problem for higher-order (and non-Gaussian) stochastic sequences as a generalization of the linear prediction/innovations filter problem for second-order stochastic sequences. The subject of this paper is a multidimensional nonlinear generalization of the linear Schur parametrization procedure for second-order stochastic sequences. We state and solve the problem geometrically, in the space of generalized (multi-indexed) matrices. We derive multidimensional nonlinear Levinson and Schur recursions, resulting in a complete parametrization schema for higher-order nonstationary stochastic sequences and show its connection to solution the non-linear orthogonal approximate filter problem of the Volterra–Wiener class. In the third part l2r we propose a low-complexity nonlinear Schur parametrization procedure, following from a multidimensional nonlinear generalization of the staircase matrix extension problem.

The paper studies the extension problem of continuous maps between axiomatic locally finite spaces (for short, ALF spaces). Indeed, an ALF space is a topological space satisfying a set of axioms suggested in Kovalevsky [Axiomatic digital topology, J. Math. Imag. Vis. 26 (2006), pp. 41–58; Geometry of Locally Finite Spaces, Monograph, Berlin, 2008]. Further, an ALF space is defined by using a special kind of neighbourhood different from the topological neighbourhood in classical topology so that the continuity of maps between ALF spaces can be defined by preserving the neighbourhood relation (see Definition 10). Therefore, it is necessary to develop the notions of continuity, homeomorphism and local homeomorphism for such spaces by using the neighbourhood relation, which can be applicable in computer science. In the study of a deformation of an ALF space, we can develop a special kind of retract on ALF spaces. By using the retract, we can efficiently deal with the extension of continuous maps between ALF spaces.

The band method for positive definite extension problems of non-band type is specified for the bordered case. A fractional type description of all positive extensions is new. We illustrate the result for Fredholm integral operators. The notion of Schur complement plays a crucial role.

Notation and terminology.- Generalized Kothe-Toeplitz duals.- Characterization of matrix classes.- Tauberian theorems.- Consistency theorems.- Operator Norlund means.