Axiomatisation of an Interval Calculus for Theorem Proving

We provide an axiomatisation of the Timed Interval Calculus, a set-theoretic notation for expressing properties of time intervals. We implement the axiomatisation in the Ergo theorem prover in order to allow the machine-checked proof of laws for reasoning about predicates expressed using interval operators. These laws can be then used in the machine-assisted verification of real-time applications.

SAS/OWA:ordered weighted averaging in SAS optimization

This paper explores the use of the optimization procedures in SAS/OR software with application to the ordered weight averaging (OWA) operators of decision-making units (DMUs). OWA was originally introduced by Yager (IEEE Trans Syst Man Cybern 18(1):183-190, 1988) has gained much interest among researchers, hence many applications such as in the areas of decision making, expert systems, data mining, approximate reasoning, fuzzy system and control have been proposed. On the other hand, the SAS is powerful software and it is capable of running various optimization tools such as linear and non-linear programming with all type of constraints. To facilitate the use of OWA operator by SAS users, a code was implemented. The SAS macro developed in this paper selects the criteria and alternatives from a SAS dataset and calculates a set of OWA weights. An example is given to illustrate the features of SAS/OWA software. © Springer-Verlag 2009.

Parametric aggregation in ordered weighted averaging

Incorporating further information into the ordered weighted averaging (OWA) operator weights is investigated in this paper. We first prove that for a constant orness the minimax disparity model [13] has unique optimal solution while the modified minimax disparity model [16] has alternative optimal OWA weights. Multiple optimal solutions in modified minimax disparity model provide us opportunity to define a parametric aggregation OWA which gives flexibility to decision makers in the process of aggregation and selecting the best alternative. Finally, the usefulness of the proposed parametric aggregation method is illustrated with an application in metasearch engine. © 2011 Elsevier Inc. All rights reserved.

Improvements on the Juxtaposing Theorem

A new class of binary constant weight codes is presented. We establish new lower bound and exact values on A(n1 +n2; 2(a1 +a2); n2) ≥ min {M1;M2}+1, if A(n1; 2a1; a1 +b1) = M1 and A(n2; 2b2; a2 +b2) = M2, in particular, A(30; 16; 15) = 16 and A(33; 18; 15) = 11.

On the Various Bisection Methods Derived from Vincent’s Theorem

In 2000 A. Alesina and M. Galuzzi presented Vincent’s theorem “from a modern point of view” along with two new bisection methods derived from it, B and C. Their profound understanding of Vincent’s theorem is responsible for simplicity — the characteristic property of these two methods. In this paper we compare the performance of these two new bisection methods — i.e. the time they take, as well as the number of intervals they examine in order to isolate the real roots of polynomials — against that of the well-known Vincent-Collins-Akritas method, which is the first bisection method derived from Vincent’s theorem back in 1976. Experimental results indicate that REL, the fastest implementation of the Vincent-Collins-Akritas method, is still the fastest of the three bisection methods, but the number of intervals it examines is almost the same as that of B. Therefore, further research on speeding up B while preserving its simplicity looks promising.

On a New Approach to Williamson's Generalization of Pólya's Enumeration Theorem

Pólya’s fundamental enumeration theorem and some results from Williamson’s generalized setup of it are proved in terms of Schur- Macdonald’s theory (S-MT) of “invariant matrices”. Given a permutation group W ≤ Sd and a one-dimensional character χ of W , the polynomial functor Fχ corresponding via S-MT to the induced monomial representation Uχ = ind|Sdv/W (χ) of Sd , is studied. It turns out that the characteristic ch(Fχ ) is the weighted inventory of some set J(χ) of W -orbits in the integer-valued hypercube [0, ∞)d . The elements of J(χ) can be distinguished among all W -orbits by a maximum property. The identity ch(Fχ ) = ch(Uχ ) of both characteristics is a consequence of S-MT, and is equivalent to a result of Williamson. Pólya’s theorem can be obtained from the above identity by the specialization χ = 1W , where 1W is the unit character of W.

Indirect ties in knowledge networks:a social network analysis with ordered weighted averaging operators

This PhD thesis analyses networks of knowledge flows, focusing on the role of indirect ties in the knowledge transfer, knowledge accumulation and knowledge creation process. It extends and improves existing methods for mapping networks of knowledge flows in two different applications and contributes to two stream of research. To support the underlying idea of this thesis, which is finding an alternative method to rank indirect network ties to shed a new light on the dynamics of knowledge transfer, we apply Ordered Weighted Averaging (OWA) to two different network contexts. Knowledge flows in patent citation networks and a company supply chain network are analysed using Social Network Analysis (SNA) and the OWA operator. The OWA is used here for the first time (i) to rank indirect citations in patent networks, providing new insight into their role in transferring knowledge among network nodes; and to analyse a long chain of patent generations along 13 years; (ii) to rank indirect relations in a company supply chain network, to shed light on the role of indirectly connected individuals involved in the knowledge transfer and creation processes and to contribute to the literature on knowledge management in a supply chain. In doing so, indirect ties are measured and their role as means of knowledge transfer is shown. Thus, this thesis represents a first attempt to bridge the OWA and SNA fields and to show that the two methods can be used together to enrich the understanding of the role of indirectly connected nodes in a network. More specifically, the OWA scores enrich our understanding of knowledge evolution over time within complex networks. Future research can show the usefulness of OWA operator in different complex networks, such as the on-line social networks that consists of thousand of nodes.

A Remark On S.M. Bates' Theorem

In his paper [1], Bates investigates the existence of nonlinear, but highly smooth, surjective operators between various classes of Banach spaces. Modifying his basic method, he obtains the following striking results.

Analog of Favard's Theorem for Polynomials Connected with Difference Equation of 4-th Order

Orthonormal polynomials on the real line {pn (λ)} n=0 ... ∞ satisfy the recurrent relation of the form: λn−1 pn−1 (λ) + αn pn (λ) + λn pn+1 (λ) = λpn (λ), n = 0, 1, 2, . . . , where λn > 0, αn ∈ R, n = 0, 1, . . . ; λ−1 = p−1 = 0, λ ∈ C. In this paper we study systems of polynomials {pn (λ)} n=0 ... ∞ which satisfy the equation: αn−2 pn−2 (λ) + βn−1 pn−1 (λ) + γn pn (λ) + βn pn+1 (λ) + αn pn+2 (λ) = λ2 pn (λ), n = 0, 1, 2, . . . , where αn > 0, βn ∈ C, γn ∈ R, n = 0, 1, 2, . . ., α−1 = α−2 = β−1 = 0, p−1 = p−2 = 0, p0 (λ) = 1, p1 (λ) = cλ + b, c > 0, b ∈ C, λ ∈ C. It is shown that they are orthonormal on the real and the imaginary axes in the complex plane ...

Cerami (C) Condition and Mountain Pass Theorem for Multivalued Mappings

Partially supported by Sapientia Foundation.

Projectively Solid Sets and an N-dimensional Piccard's Theorem

We discuss functions f : X × Y → Z such that sets of the form f (A × B) have non-empty interiors provided that A and B are non-empty sets of second category and have the Baire property.

Three Spaces Problem for Lyapunov Theorem on Vector Measure

It is proved that a Banach space X has the Lyapunov property if its subspace Y and the quotient space X/Y have it.