CONCEPTS OF FUZZY INFORMATION MEASURES ON CONTINUOUS DOMAINS

Measures of information based on fuzzy sets (possibility distributions) had been defined only for finite domains of discourse. This paper presents a method of defining such information functions on a continuous universe of discourse—a domain which is a measurable space of measure 1. The method is based on the concept of “rearangement” of a function, used in lieu of sorting discrete possibility values. For technical reasons, it is preferred to express information value as information distance to the most “uninformed” (constant possibility 1) distribution. The final form of the information for possibility distribution f is

The paper then discusses related information distances and approximations using discrete information functions.     

Book reviews

Patent Harmonization Harold C. Wegner London, Sweet & Maxwell, 1993 xx + 376 pp., £70.00 (hardback)

Of Authors and Origins: essays on copyright law Brad Sherman & Alain Strowel (Eds) Oxford, Clarendon Press, 1994 xiv + 313 pp., £30.00 (hardback)

Future Air Navigation Systems Werner Guldimann & Stefan Kaiser Dordrecht, Martinus Nijhoff, 1993 Utrecht Studies in Air and Space Law, Vol. 13 viii + 281 pp., £76.50 (hardback)

The Law of Copyright Terence Prime London, Fourmat Publishing, 1992, xxiv + 313 pp., £30.00 (paperback)     

A sufficient condition for pancyclic graphs

In 2005, Rahman and Kaykobad proved that if G is a connected graph of order n such that d(x)+d(y)+d(x,y)n+1 for each pair x, y of distinct nonadjacent vertices in G, where d(x,y) is the length of a shortest path between x and y in G, then G has a Hamiltonian path [Inform. Process. Lett. 94 (2005) 37–41]. In 2006 Li proved that if G is a 2-connected graph of order n3 such that d(x)+d(y)+d(x,y)n+2 for each pair x,y of nonadjacent vertices in G, then G is pancyclic or G=K_n/2,n/2 where n4 is an even integer [Inform. Process. Lett. 98 (2006) 159–161]. In this work we prove that if G is a 2-connected graph of order n such that d(x)+d(y)+d(x,y)n+1 for all pairs x, y of distinct nonadjacent vertices in G, then G is pancyclic or G belongs to one of four specified families of graphs.     

Asymptotic behaviour of the solutions of second order functional differential equations

This paper presents integral criteria to determine the asymptotic behaviour of the solutions of second order nonlinear differential equations of the type y^″(x)+q(x)f(y(x))=0, with q(x)>0 and f(y) odd and positive for y>0, as x tends to +∞. It also compares them with the results obtained by Chanturia (1975) in [11] for the same problem.     

An upper bound for the distance between a zero and a critical point of a solution of a second order linear differential equation

This paper presents an upper bound for the distance between a zero and a critical point of a solution of the second order linear differential equation (p(x)y^′)^′+q(x)y(x)=0, with p(x),q(x)>0. It also compares it with previous results.     

ON THE REPRESENTATION OF ACTIVITY ARRAYS†

A general methodology for empirical investigaiion is described which consists of: (i) the selection of a system on the object of investigation with respect to the purpose and constraints of investigation, (ii) the gathering of data for the system and organizing them in the form of activity arrays, (iii) the processing of the data through which certain time-invariant properties representing the data are determined, (iv) the interpretation of these time-invariant data representations from the standpoint of the purpose of investigation.

It is shown that each activity array yields many different representations. A procedure through which the various representations can be obtained is described that consists of: (1) the selection of a mask specifying a pattern for sampling the data, (2) the sampling procedure through which certain types of time-invariant relations are determined for the given mask, (3) the simplification of the time-invariant relations, if necessary or desirable.

A basis for the objective comparison of possible representations and ascertainment of the representation with the highest possible degree of determinism, subject to given constraints, is developed.

The procedure is applicable to both well defined and fuzzy variables of any scale. No classification into input and output variables is required, although it is not prohibited.     

Improved upper bounds on the L(2,1) -labeling of the skew and converse skew product graphs

An L(2,1)-labeling of a graph G is a function f from the vertex set V(G) to the set of all nonnegative integers such that |f(x)−f(y)|≥2 if d(x,y)=1 and |f(x)−f(y)|≥1 if d(x,y)=2, where d(x,y) denotes the distance between x and y in G. The L(2,1)-labeling number λ(G) of G is the smallest number k such that G has an L(2,1)-labeling with max{f(v):vV(G)}=k. Griggs and Yeh conjecture that λ(G)≤Δ² for any simple graph with maximum degree Δ≥2. This paper considers the graph formed by the skew product and the converse skew product of two graphs with a new approach on the analysis of adjacency matrices of the graphs as in [W.C. Shiu, Z. Shao, K.K. Poon, D. Zhang, A new approach to the L(2,1)-labeling of some products of graphs, IEEE Trans. Circuits Syst. II: Express Briefs (to appear)] and improves the previous upper bounds significantly.     

Developing measures for information ergonomics in knowledge work

Information ergonomics is an evolving application domain of ergonomics focusing on the management of workload in the real-world contexts of information-intensive tasks. This study introduces a method for the evaluation of information ergonomics in knowledge work. To this end, five key dimensions of information ergonomics were identified: contextual factors of knowledge work, multitasking, interruptions at work, practices for managing information load, and perceived job control and productivity. In total, 24 measures focusing on the above dimensions were constructed. The measures include, for example, the number of fragmented work tasks per work day. The measures were preliminarily tested in two Finnish organisations, making use of empirical data gathered by interviews, electronic questionnaires and log data applications tracking work processes on personal computers. The measures are applicable to the evaluation of information ergonomics, even though individual measures vary with regard to the amount of work and time needed for data analysis.

Practitioner Summary:

The study introduces a method for the evaluation of information ergonomics in knowledge work. To this end, 24 measures were constructed and tested empirically. The measures focus on contextual factors of knowledge work, multitasking, interruptions at work, practices for managing information load, and perceived job control and productivity.     

On input-output linearization of discrete-time nonlinear systems

We consider a nonlinear discrete-time system of the form Σ: x(t+1)=f(x(t), u(t)), y(t) =h(x(t)), where x ε R^N, u ε R^m, y ε R^q and f and h are analytic. Necessary and sufficient conditions for local input-output linearizability are given. We show that these conditions are also sufficient for a formal solution to the global input-output linearization problem. Finally, we show that zeros at infinity of ε can be obtained by the structure algorithm for locally input-output linearizable systems.     

Inapproximability of the Tutte polynomial of a planar graph

The Tutte polynomial of a graph G is a two-variable polynomial T(G; x, y) that encodes many interesting properties of the graph. We study the complexity of the following problem, for rationals x and y: given as input a planar graph G, determine T(G; x, y). Vertigan completely mapped the complexity of exactly computing the Tutte polynomial of a planar graph. He showed that the problem can be solved in polynomial time if (x, y) is on the hyperbola H _q given by (x ? 1)(y ? 1) = q for q = 1 or q = 2 or if (x, y) is one of the two special points (x, y) = (?1, ?1) or (x, y) = (1, 1). Otherwise, the problem is #P-hard. In this paper, we consider the problem of approximating T(G; x, y), in the usual sense of “fully polynomial randomized approximation scheme” or FPRAS. Roughly speaking, an FPRAS is required to produce, in polynomial time and with high probability, an answer that has small relative error. Assuming that NP is different from RP, we show that there is no FPRAS for the Tutte polynomial in a large portion of the (x, y) plane. In particular, there is no FPRAS if x > 1, y < ?1 or if y > 1, x < ?1 or if x < 0, y < 0 and q > 5. Also, there is no FPRAS if x < 1, y < 1 and q = 3. For q > 5, our result is intriguing because it shows that there is no FPRAS at (x, y) =?(1 ? q/(1 + ε), ?ε) for any positive ε but it leaves open the limit point ε =?0, which corresponds to approximately counting q-colorings of a planar graph.     

Stabilizability of two-dimensional linear systems via switched output feedback

The problem of stabilizing a second-order SISO LTI system of the form , y=Cx with feedback of the form u(x)=v(x)Cx is considered, where v(x) is real-valued and has domain which is all of . It is shown that, when stabilization is possible, v(x) can be chosen to take on no more than two values throughout the entire state-space (i.e., v(x){k₁,k₂} for all x and for some k₁,k₂), and an algorithm for finding a specific choice of v(x) is presented. It is also shown that the classical root locus of the corresponding transfer function C(sI-A)^-1B has a strong connection to this stabilization problem, and its utility is demonstrated through examples.     

The distance measures between q-rung orthopair hesitant fuzzy sets and their application in multiple criteria decision making

In this paper, we first introduce the concept of q-rung orthopair hesitant fuzzy set ($q$-ROHFS) and discuss the operational laws between any two $q$-ROHFSs. Then the distance measures between $q$-ROHFSs are proposed based on the concept of “multiple fuzzy sets”, and we develop the TOPSIS method to the proposed distance measures. The proposed distance measures not only retain the preference information expressed by $q$-ROHFSs, but also deal with the $q$-rung orthopair hesitant fuzzy decision information more objectively, In fact, the method can avoid the loss and distortion of the information in actual decision-making process. Furthermore, we give an illustrative example about the selection of energy projects to illustrate the reasonableness and effectiveness of the proposed method, which is also compared with other existing methods. Finally, we make the sensitivity analysis of the parameters in proposed distance measures about the selection of energy projects.     

On the Weighted Mean of a Pair of Strings

String matching and string edit distance are fundamental concepts in structural pattern recognition. In this paper, the weighted mean of a pair of strings is introduced. Given two strings, x and y, where d(x, y) is the edit distance of x and y, the weighted mean of x and y is a string z that has edit distances d(x, z) and d(z, y)to x and y, respectively, such that d(x, z) _ d(z, y) = d(x, y). We’ll show formal properties of the weighted mean, describe a procedure for its computation, and give practical examples. Received: 26 October 2000, Received in revised form: 27 April 2001, Accepted: 20 July 2001     

On a family of copulas constructed from the diagonal section

We characterize the class of copulas that can be constructed from the diagonal section by means of the functional equation C(x,y)+|x−y|=C(x∨y,x∨y), for all (x,y) in the unit square such that C(x,y)>0. Some statistical properties of this class are given.     

Sentences over Integral Domains and Their Computational Complexities

LetRbe a Hilbertian domain and letKbe its fraction field. Letψ(x₁, …, x_n, y) be a quantifier free arithmetical formula overR. We may also takeψ(x₁, …, x_n, y) to be an arithmetical formula overK[x₁, …, x_n] and write it asψ(y). In this paper we show that ifRhas enough non-units and x₁…x_n y ψ(x₁, …, x_n, y), called an ⁿ  sentence, is true inR, then y ψ(y) is true inK[x₁, …, x_n]. Also, ifR=K[T], whereKis an infinite integral domain andx₁…x_n y ψ(x₁, …, x_n, y)is true inR, then y ψ(y) is true inR[x₁, …, x_n]. These results are applied to find the upper and lower bounds of the time complexities of various decision problems on diophantine equations with parameters and arithmetical sentences. Some of the results are: 1. The decision problem of sentences and diophantine equations with parameters over the ring of integers of a global field are co-NP-complete. 2. The decision problem of sentences over the ring of integers of a global field is NP-complete. 3. LetKbe an infinite domain, the time complexities of the decision problems of equations with parameters and sentences over the polynomial ringK[t] are polynomial time reducible to factoring polynomials overK. 4. The decision problem of sentences over all algebraic integer rings is in P. 5. The decision problem of sentences over all integral domains with characteristic 0 is in P. 6. The time complexity of the decision problem of sentences over all integral domains is polynomial time reducible to factoring integers overZand factoring polynomials over finite fields.     

Book reviews

Industrial Design Law G Fellner London, Sweet & Maxwell, 1995 xxxiv + 302 pp., £58 (hardback). 0360–0834/96/030253–05

Terrell on the Law of Patents D. Young, A. Watson, S. Thorley & R. Miller London, Sweet & Maxwell, 1994 14th edn, £148 (hardback)     

DISCRETE EVENT-DYNAMIC SYSTEMS FOR MODELING DYNAMIC VALUED NETS

The last few years have seen the development of Discrete Event-Dynamic Net Systems^1,2 as instruments for modeling complex systems. They are able to achieve the following objectives:

—formality of the modeling methodology

—ability to model static and dynamic aspects

—ability to pass between levels of differently rich structures by morphisms

—uniform representation of the communication process as

—an information process

—a decision process and

—a control process

—homogeneity of the representation and modeling methods

—ability to derive qualitative and quantitative statements.

The foundation is provided by a Discrete Event-Dynamic Net System which includes the axiomatic declaration of general Petri nets. In order to calculate the structural and dynamic aspects, so-called Petri net machines are developed. It is shown that this approach can even be used to treat the following aspects:

—use of time during the process

—increase of costs during the generation and transportation of information

—augmentation, evaluation and transformation of information objects.

Recursive formulas are derived and some examples calculated.     

Two point hermite approximations for the solution of linear initial value and boundary value problems

Application of an idea originally due to Ch. Hermite allows the derivation of an approximate formula for expressing the integral ∫xixi?1y(x)dx as a linear combination of y(xi?1), y(xi), and their derivatives y(v)(xi?1) up to order v = α and y(v)(xi) up to order v = β. In addition to this integro-differential form a purely differential form of the 2-point Hermite approximation will be derived. Both types will be denoted by Hαβ-approximation. It will be shown that the well-known Obreschkoff-formulas contain no new elements compared to the much older Hαβ-method.The Hαβ-approximation will be applied to the solution of systems of ordinary differential equations of the type y'(x) = M(x)y(x) + q(x), and both initial value and boundary value problems will be treated. Function values at intermediate points x? (xi?1, xi) are obtained by the use of an interpolation formula given in this paper.An advantage of the Hαβ-method is the fact that high orders of approximation (α, β) allow an increase in step size hi. This will be demonstrated by the results of several test calculations.     

Relaxing the Triangle Inequality in Pattern Matching

Any notion of closeness in pattern matching should have the property that if A is close to B, and B is close to C, then A is close to C. Traditionally, this property is attained because of the triangle inequality (d(A, C) d(A, B) + d(B, C), where d represents a notion of distance). However, the full power of the triangle inequality is not needed for this property to hold. Instead, a relaxed triangle inequality suffices, of the form d(A, C) c(d(A, B) + d(B, C)), where c is a constant that is not too large. In this paper, we show that one of the measures used for distances between shapes in (an experimental version of) IBM's QBIC¹ ("Query by Image Content") system (Niblack et al., 1993) satisfies a relaxed triangle inequality, although it does not satisfy the triangle inequality.