Interval Linear Algebra

In this book we use only special types of intervals and introduce the notion of different types of interval linear algebras and interval vector spaces using the intervals of the form [0, a] where the intervals are from Zn or Z+ \cup {0} or Q+ \cup {0} or R+ \cup {0}. A systematic development is made starting from set interval vector spaces to group interval vector spaces. Vector spaces are taken as interval polynomials or interval matrices or just intervals over suitable sets or semigroups or groups. Main feature of this book is the authors have given over 350 examples. This book has six chapters. Chapter one is introductory in nature. Chapter two introduces the notion of set interval linear algebras of type one and two. Set fuzzy interval linear algebras and their algebras and their properties are discussed in chapter three. Chapter four introduces several types of interval linear bialgebras and bivector spaces and studies them. The possible applications are given in chapter five. Chapter six suggests nearly 110 problems of all levels.Comment: 247 page

Cuntz–Krieger Algebras and Wavelets on Fractals

We consider representations of Cuntz–Krieger algebras on the Hilbert space of square integrable functions on the limit set, identified with a Cantor set in the unit interval. We use these representations and the associated Perron–Frobenius and Ruelle operators to construct families of wavelets on these Cantor sets

Comparison of different algebras for inducing the temporal structure of texts

International audienceThis paper investigates the impact of using different temporal algebras for learning temporal relations between events. Specifically, we compare three interval-based algebras: Allen \shortcite{Allen83} algebra, Bruce \shortcite{Bruce72} algebra, and the algebra derived from the TempEval-07 campaign. These algebras encode different granularities of relations and have different inferential properties. They in turn behave differently when used to enforce global consistency constraints on the building of a temporal representation. Through various experiments on the TimeBank/AQUAINT corpus, we show that although the TempEval relation set leads to the best classification accuracy performance, it is too vague to be used for enforcing consistency. By contrast, the other two relation sets are similarly harder to learn, but more useful when global consistency is important. Overall, the Bruce algebra is shown to give the best compromise between learnability and expressive power

Interval valued (\in,\ivq)-fuzzy filters of pseudo BLBL-algebras

We introduce the concept of quasi-coincidence of a fuzzy interval value with an interval valued fuzzy set. By using this new idea, we introduce the notions of interval valued (\in,\ivq)-fuzzy filters of pseudo $BL$-algebras and investigate some of their related properties. Some characterization theorems of these generalized interval valued fuzzy filters are derived. The relationship among these generalized interval valued fuzzy filters of pseudo $BL$-algebras is considered. Finally, we consider the concept of implication-based interval valued fuzzy implicative filters of pseudo $BL$-algebras, in particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed