Studies on some graph operators and related topics

Department of Mathematics, Cochin University of Science and Technology

Studies on Fuzzy Matroitls and Related Topics

The doctoral thesis focuses on the Studies on fuzzy Matroids and related topics.Since the publication of the classical paper on fuzzy sets by L. A. Zadeh in 1965.the theory of fuzzy mathematics has gained more and more recognition from many researchers in a wide range of scientific fields. Among various branches of pure and applied mathematics, convexity was one of the areas where the notion of fuzzy set was applied. Many researchers have been involved in extending the notion of abstract convexity to the broader framework of fuzzy setting. As a result, a number of concepts have been formulated and explored. However. many concepts are yet to be fuzzified. The main objective of this thesis was to extend some basic concepts and results in convexity theory to the fuzzy setting. The concept like matroids, independent structures. classical convex invariants like Helly number, Caratheodoty number, Radon number and Exchange number form an important area of study in crisp convexity theory. In this thesis, we try to generalize some of these concepts to the fuzzy setting. Finally, we have defined different types of fuzzy matroids derived from vector spaces and discussed some of their properties.

‘Studies on some topics in product graphs’

For routing problems in interconnection networks it is important to find the shortest containers between any two vertices, since the w-wide diameter gives the maximum communication delay when there are up to w−1 faulty nodes in a network modeled by a graph. The concept of ‘wide diameter’ was introduced by Hsu [41] to unify the concepts of diameter and The concept of ‘domination’ has attracted interest due to its wide applications in many real world situations [38]. A connected dominating set serves as a virtual backbone of a network and it is a set of vertices that helps in routing. In this thesis, we make an earnest attempt to study some of these notions in graph products. This include, the diameter variability, the diameter vulnerability, the component factors and the domination criticality.connectivity

A study on semigroup action on fuzzy subsets, inverse fuzzy automata and related topics

This thesis comprises five chapters including the introductory chapter. This includes a brief introduction and basic definitions of fuzzy set theory and its applications, semigroup action on sets, finite semigroup theory, its application in automata theory along with references which are used in this thesis. In the second chapter we defined an S-fuzzy subset of X with the extension of the notion of semigroup action of S on X to semigroup action of S on to a fuzzy subset of X using Zadeh's maximal extension principal and proved some results based on this. We also defined an S-fuzzy morphism between two S-fuzzy subsets of X and they together form a category S FSETX. Some general properties and special objects in this category are studied and finally proved that S SET and S FSET are categorically equivalent. Further we tried to generalize this concept to the action of a fuzzy semigroup on fuzzy subsets. As an application, using the above idea, we convert a _nite state automaton to a finite fuzzy state automaton. A classical automata determine whether a word is accepted by the automaton where as a _nite fuzzy state automaton determine the degree of acceptance of the word by the automaton. 1.5. Summary of the Thesis 17 In the third chapter we de_ne regular and inverse fuzzy automata, its construction, and prove that the corresponding transition monoids are regular and inverse monoids respectively. The languages accepted by an inverse fuzzy automata is an inverse fuzzy language and we give a characterization of an inverse fuzzy language. We study some of its algebraic properties and prove that the collection IFL on an alphabet does not form a variety since it is not closed under inverse homomorphic images. We also prove some results based on the fact that a semigroup is inverse if and only if idempotents commute and every L-class or R-class contains a unique idempotent. Fourth chapter includes a study of the structure of the automorphism group of a deterministic faithful inverse fuzzy automaton and prove that it is equal to a subgroup of the inverse monoid of all one-one partial fuzzy transformations on the state set. In the fifth chapter we define min-weighted and max-weighted power automata study some of its algebraic properties and prove that a fuzzy automaton and the fuzzy power automata associated with it have the same transition monoids. The thesis ends with a conclusion of the work done and the scope of further study.

Topics of divisibility

The set of integers forms a commutative ring whose elements admit a unique decomposition into primes. In this folder three lecture notes are bound, concerning topics, developed by dropping or replacing special properties of this most natural and most special ring. For short: investigated are groupoids w.r.t the interplay of multiplication - on the one hand - and divisibility, ideal decomposition and residuation, respectively, on the other hand.

Alternativas al curriculum ordinario : motivación a través de las nuevas tecnologías.

Ofrecer una alternativa al Curriculum Ordinario para el alumnado desmotivado, utilizando las Nuevas Tecnologías como recurso fundamental, con el objeto de lograr los objetivos generales de etapa y la consiguiente titulación.

Comparing european and asian studies : towards the renewal of curriculum programmes. 'Comparación de estudios europeos y asiáticos : hacia la renovación de los programas de estudio'.

Publicación financiada por el Programa Asia-Link de la Comisión Europea. - Notas a pie de página.

Ciencia, tecnología y sociedad en la enseñanza secundaria : hacia la penetración en el curriculum de los aspectos sociales de la ciencia y tecnología.

Probar mediante el análisis de diferentes materiales didácticos, libros de texto y Cuadernillos Argo, cómo cada uno de ellos responde a una concepción distinta del curriculum. Cinco libros de texto de diferentes editoriales y 7 cuadernillos del Grupo Argo de educación secundaria con contenidos en ciencia, tecnología y sociedad. En la primera parte se realiza una presentación en el ámbito internacional del enfoque Ciencia, Tecnología y Sociedad (CTS) que intenta contextualizar el origen intelectual, académico y educativo de la enseñanza CTS que se desarrolla en España. También se explica cómo se introduce la CTS en el curriculum español y cuáles son algunas de las lineas de trabajo representativas de los programas, y trabajos de investigación CTS en nuestro país. En la segunda parte, se pretende profundizar en las propuestas de enseñanza CTS a través del análisis de materiales didácticos, para terminar con las conclusiones generales. Tablas. La investigación en el curriculum de los aspectos sociales de la Ciencia y Tecnología debe intentar conseguir principalmente dos objetivos: relacionar las dos tradiciones CTS a través de casos de estudio que tengan relevancia social y hacerlo desde una perspectiva social crítica que fundamente los aspectos teóricos. Se debe optar por la investigación de un curriculum integrado en la enseñanza de las relaciones Ciencia, Tecnología y Sociedad en la Educación Secundaria, ya que el carácter interdisciplinar de los estudios CTS hace necesario un planteamiento didáctico coherente. Hay que abordar la investigación en el curriculum desde teorías socioculturales que fundamenten el aprendizaje del alumnado y que al mismo tiempo expliquen la construcción social de la ciencia y la tecnología.