Optimal approximations with Rough Sets and similarities in measure spaces

When arbitrary sets are approximated by more structured sets, it may not be possible to obtain an exact approximation that is equivalent to a given set. Presented here, is a new proposal for a ‘metric’ approach to Rough Sets. We assume some finite measure space is defined on a given universe, and then use it to define various similarity indexes. A set of axioms and the concept of consistency for similarity indexes are also proposed. The core of the paper is a definition of the ‘optimal’ or ‘best’ approximation with respect to any particular similarity index, and an algorithm to find this optimal approximation by using the Marczewski–Steinhaus Index. This algorithm is also shown to hold for a class of similarity indexes that are consistent with the Marczewski–Steinhaus Index.     

Normalising geometrical figures: dynamic manipulation and construction of meanings for ratio and proportion

Enlarging-shrinking geometrical figures by 13 year-olds is studied during the implementation of proportional geometric tasks in the classroom. Students worked in groups of two using ‘Turtleworlds’, a piece of geometrical construction software which combines symbolic notation, through a programming language, with dynamic manipulation of geometrical objects by dragging on sliders representing variable values. In this paper we study the students’ normalising activity, as they use this kind of dynamic manipulation to modify ‘buggy’ geometrical figures while developing meanings for ratio and proportion. We describe students’ normative actions in terms of four distinct Dynamic Manipulation Schemes (Reconnaissance, Correlation, Testing, Verification). We discuss the potential of dragging for mathematical insight in this particular computational environment, as well as the purposeful nature of the task which sets up possibilities for students to appreciate the utility of proportional relationships.     

E_{\infty}-structure for Q_*(R)

MacLane homology of a ring R is the Hochschild homology of the so called cubical construction , which is a chain-algebra. If we take a commutative ring R, the Dixmier product on is no longer commutative. The main result of this paper is that it is commutative up to higher homotopies, i.e. that it is an -algebra. The -operad which acts on is constructed by using the analogue of the -complex in the context of finite sets. For the precise notation of an operad action on these complexes the definition of an -monoidal functor is introduced. Received: 17 March 1998 / in revised form: 16 March 1999     

Relations among homogeneity,collapsibility and nonconfounding in distribution effects

In this paper, the concept of distribution effect is proposed without the causal diagram. Following the notation of Stone [11], we assume that the exposure treatment X is an unknown deterministic function of the confounder set Pa(X) and a random error ε. We discuss sufficient and necessary conditions for homogeneity, collapsibility and nonconfounding for distribution effects and discuss relations among them.     

Sense and sensibility: using a model to examine the relationship between public pre-school places and fertility

This paper presents a stochastic dynamic mathematical model, in which a Family Policy Index (XFPI) is included to measure and compare two different models of provision of resources to support families with children from 0 to 3 years old. The main variables in this model are the XFPI, fertility, mortality, emigration and immigration rates. This mathematical model was validated in two different countries, Spain and Norway, during the 2007–2015 period. A sensitivity analysis was applied to simulate the future trend (2016–2030), examining the influence of providing public pre-school services (0 to 3 years) on (XISF). The results obtained show that these services may indeed have an influence on fertility rates, as long as they are developed extensively.     

Challenging assumptions of notational transparency: the case of vectors in engineering mathematics

ABSTRACT

The notation for vector analysis has a contentious nineteenth century history, with many different notations describing the same or similar concepts competing for use. While the twentieth century has seen a great deal of unification in vector analysis notation, variation still remains. In this paper, the two primary notations used for expressing the components of a vector are discussed in historical and current context. Popular mathematical texts use the two notations as if they are transparent and interchangeable. In this research project, engineering students’ proficiency at vector analysis was assessed and the data were analyzed using the Rasch measurement method. Results indicate that the students found items expressed in unit vector notation more difficult than those expressed in parenthesis notation. The expert experience of notation as transparent and unproblematically symbolic of underlying processes independent of notation is shown to contrast with the student experience where the less familiar notation is experienced as harder to work with.     

Turing jumps in the Ershov hierarchy

We look at infinite levels of the Ershov hierarchy in the natural system of notation, which are proper for jumps of sets. It is proved that proper infinite levels for jumps are confined to D_a ^{- 1} \Delta_a^{ - 1} -levels, where a stands for an ordinal ωⁿ > 1.     

Multi-fuzzy sets: An extension of fuzzy sets

In this paper we propose a method to construct more general fuzzy sets using ordinary fuzzy sets as building blocks. We introduce the concept of multi-fuzzy sets in terms of ordered sequences of membership functions. The family of operations T, S, M of multi-fuzzy sets are introduced by coordinate wise t-norms, s-norms and aggregation operations. We define the notion of coordinate wise conjugation of multifuzzy sets, a method for obtaining Atanassov’s intuitionistic fuzzy operations from multi-fuzzy sets. We show that various binary operations in Atanassov’s intuitionistic fuzzy sets are equivalent to some operations in multi-fuzzy sets like M operations, 2-conjugates of the T and S operations. It is concluded that multi-fuzzy set theory is an extension of Zadeh’s fuzzy set theory, Atanassov’s intuitionsitic fuzzy set theory and L-fuzzy set theory.     

Numerical Solution of the Consistently Linearized Eigenproblem by Means of a Finite Difference Expression for Approximation of a Directional Derivative in MSC.MARC

More recently, a concept of energy-based categorization of buckling was proposed. It represents a symbiosis of mechanics of solids and spherical geometry. The fundamental mathematical background of this concept is the so-called consistently linearized eigenproblem. The numerical solution of this eigenproblem by means of the finite element method is a key issue of the mentioned concept. This solution is obtained with the help of the commercial finite element software MSC.MARC. An iterative process, involving the programming lanuages PYTHON and FORTRAN 77 and the software MATLAB, is devised, making use of the available element library, of different types of solvers, and of further modeling tools. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new definition of independence of uncertain sets

Uncertain sets are an effective tool to describe unsharp concepts like “young”, “tall” and “most”. As a key concept in uncertain set theory, the independence was first defined in the paper (Liu in Fuzzy Optim Decis Mak 11(4):387–410, 2012b). However, the definition is somewhat weak to deal with uncertain sets completely. In order to overcome this disadvantage, this paper presents a stronger definition of independence of uncertain sets and discusses its mathematical properties.     

The algebra of fuzzy logic

In the following, human thinking based on premises with no complete truth value is reviewed for controlling the algebra of fuzzy sets operations. Assuming a system may be developed in this sphere, it should be considered as the algebra of fuzzy sets, as the same algebra is satisfied by classical logic and sets. As will be proved, this algebra is not a lattice and consequently the Zadeh definitions do not constitute an adequate representation. The binary operations of my algebra are “interactive” types. An axiom system is given that, in my opinion, is the foundation of the conception, adequately and without redundancy. The agreement of the theorems deduced from the axiom system with the intuitive expectations is shown. A special arithmetical structure satisfying this algebra is given, and the relation between this structure and the theory of probability is analyzed.Adapting a process of classical logics, fuzzy quantifiers are defined on the basis of the operations of propositional algebra. A “qualifier” is also defined. The qualifier is functional; applying it to Ax we get the statement “usually Ax” s a middle cource between the statements “at least once Ax” and “always Ax”. The concept of entailment of fuzzy logics is introduced. This concept is an innovative generalization of the classical deduction theory, opposite to the concept of entailment of classical multi-valued logics. An important error of the abbreviated system of notation of the fuzzy theory [e.g. m(x, AvB)] appears: the functional type operations (e.g. quantifiers) cannot be interpreted in propositional calculus. Therefore a new system of symbols is proposed in this paper.     

Hierarchical sets in mathematical programming modeling languages

The mathematical notation commonly applied for the formulation of mathematical programming models is extended to include hierarchical structures. The proposed notation is related to hierarchical set concepts in the languages UIMP, AMPL, and LPL. With the proposed notation it is possible to aggregate and disaggregate over hierarchical structures. In addition, views are introduced to permit the use of hierarchical substructures and to create new hierarchies out of existing ones. The proposed notation for hierarchical sets and views is illustrated by applying it to the representation and estimation of social accounting matrices (SAMs).     

Book Reviews

The new, and prevalent, raised‐dash notation (as in ^–5) appearing in school mathematics texts is examined, especially for its effects on students' computational skills. A substantial testing programme ties students' inability to compute –5² correctly with the raised dash. Hence, the traditional centred dash is discussed as a background for studying the new notation. A treatment of the raised dash presents the mathematical and pedagogical viewpoints of the authors of textbooks in which it is used. Inherent difficulties with this usage are examined. In the conclusion, reasons for a return to the standard notation of the centred dash are presented.     

Cyclic Tableaux and Symmetric Functions

We introduce the notion of cyclic tableaux and develop involutions for Waring's formulas expressing the power sum symmetric function p_n in terms of the elementary symmetric function e_n and the homogeneous symmetric function h_n. The coefficients appearing in Waring's formulas are shown to be a cyclic analog of the multinomial coefficients, a fact that seems to have been neglected before. Our involutions also spell out the duality between these two forms of Waring's formulas, which turns out to be exactly the “duality between sets and multisets.” We also present an involution for permutations in cycle notation, leading to probably the simplest combinatorial interpretation of the Möbius function of the partition lattice and a purely combinatorial treatment of the fundamental theorem on symmetric functions. This paper is motivated by Chebyshev polynomials in connection with Waring's formula in two variables.     

Ordinal diagrams for recursively Mahlo universes

In this paper we introduce a recursive notation system of ordinals. An element of the notation system is called an ordinal diagram following G. Takeuti [25]. The system is designed for proof theoretic study of theories of recursively Mahlo universes. We show that for each in KPM proves that the initial segment of determined by is a well ordering. Proof theoretic study for such theories will be reported in [9]. Received: 13 January, 1998     

Operation properties and -equalities of complex fuzzy sets

A complex fuzzy set is a fuzzy set whose membership function takes values in the unit circle in the complex plane. This paper investigates various operation properties and proposes a distance measure for complex fuzzy sets. The distance of two complex fuzzy sets measures the difference between the grades of two complex fuzzy sets as well as that between the phases of the two complex fuzzy sets. This distance measure is then used to define δ-equalities of complex fuzzy sets which coincide with those of fuzzy sets already defined in the literature if complex fuzzy sets reduce to real-valued fuzzy sets. Two complex fuzzy sets are said to be δ-equal if the distance between them is less than 1-δ. This paper shows how various operations between complex fuzzy sets affect given δ-equalities of complex fuzzy sets. An example application of signal detection demonstrates the utility of the concept of δ-equalities of complex fuzzy sets in practice.     

Equilibrium and least element problems for multivalued functions

The principal aim of this paper is to extend some recent results which concern problems involving bifunctions to similar generalized problems for multivalued bifunctions. To this end, by using the appropriate notions of strict pseudomonotonicity we establish the relationships between generalized vector equilibrium problems and generalized minimal element problems of feasible sets. Moreover relationships between generalized least element problems of feasible sets and generalized vector equilibrium problems are studied by employing the concept of Z-multibifunctions.     

On Geodesic <Emphasis Type="Italic">E</Emphasis>-Convex Sets,Geodesic <Emphasis Type="Italic">E</Emphasis>-Convex Functions and <Emphasis Type="Italic">E</Emphasis>-Epigraphs

In this paper, we introduce a new class of sets and a new class of functions called geodesic E-convex sets and geodesic E-convex functions on a Riemannian manifold. The concept of E-quasiconvex functions on R ⁿ is extended to geodesic E-quasiconvex functions on Riemannian manifold and some of its properties are investigated. Afterwards, we generalize the notion of epigraph called E-epigraph and discuss a characterization of geodesic E-convex functions in terms of its E-epigraph. Some properties of geodesic E-convex sets are also studied.     

Sorting dinosaurs: an opportunity to teach mathematics with biology

With the growing desire for increased communication between mathematics teachers and science teachers there is a need for materials specifically written to help bring about communication at that level which also have potential in class. This paper is concerned with one example, a study of dinosaurs through the algebra of sets, simultaneously exploring and developing a topic in biology and a topic in mathematics by raising a series of questions about dinosaurs which are most effectively answered by means of mathematics.

Initially the questions asked can be answered by means of Venn diagrams, and set notation is no more than an alternative, brief way of expressing the same answer. But systematically the Venn diagrams become more involved and there is a growing dependence upon the algebra of sets so that ultimately answers are sought by that means and Venn diagrams are used (if at all) to reinforce the findings. Very many aspects of set notation and algebra are incorporated into the study and a number of set theories illustrated.

Some aspects of its possible use in class are considered.