Euler diagrams form the basis of many visual languages. Such languages are formed by augmenting Euler diagrams with graphs or shading. However, tool support for creating augmented Euler diagrams is generally limited to generic diagram editing software using mouse and keyboard interaction. A more natural and convenient mode of entry is via a sketching interface which facilitates greater cognitiv...

We introduce notions of finiteness obstruction, Euler characteristic, L2-Euler characteristic, and Möbius inversion for wide classes of categories. The finiteness obstruction of a category Γ of type (FP) is a class in the projective class group K0(RΓ); the Euler characteristic and L2-Euler characteristic are respectively its RΓ-rank and L2-rank. We also extend the second author’s K-theoretic Mö...

In this paper we introduce the (I, J) similar method for incompressible two-dimensional Euler equations, and obtain a series of explicit (I, J) similar solutions to the incompressible two-dimensional Euler equations. These solutions include all of the twin wave solutions, some new singularity solutions, and some global smooth solutions with a finite energy. We also reveal that the twin wave sol...

The Euler-Lagrange equation plays an important role in the minimization problems of the calculus of variations. This paper employs the differential transformation method (DTM) for finding the solution of the Euler-Lagrange equation which arise from problems of calculus of variations. DTM provides an analytical solution in the form of an infinite power series with easily computable components. S...

we derive error estimates in the appropriate norms, for the streamlinediffusion (sd) finite element methods for steady state, energy dependent,fermi equation in three space dimensions. these estimates yield optimal convergencerates due to the maximal available regularity of the exact solution.high order sd method together with implicit integration are used. the formulationis strongly consistent...

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Euler diagrams have numerous application areas, with a large variety of languages based on them. In relation to software engineering, such areas encompass modelling and specification including from a formal perspective. In all of these application areas, it is desirable to provide tools to layout Euler diagrams, ideally in a nice way. Various notions of ‘niceness’ can be correlated with certain...

Euler diagrams are used in a wide variety of areas for representing information about relationships between collections of objects. Recently, several techniques for automated Euler diagram drawing have been proposed, contributing to the Euler diagram generation problem: given an abstract description, draw an Euler diagram with that description and which possesses certain properties, sometimes c...

Euler diagrams visually represent containment, intersection and exclusion using closed curves. They first appeared several hundred years ago, however, there has been a resurgence in Euler diagram research in the twenty-first century. This was initially driven by their use in visual languages, where they can be used to represent logical expressions diagrammatically. This work lead to the require...

Fuzzy set theory has entered into a large variety of disciplines of sciences,technology and humanities having established itself as an extremely versatileinterdisciplinary research area. Accordingly different notions of fuzzystructure have been developed such as fuzzy normed linear space, fuzzytopological vector space, fuzzy sequence space etc. While reviewing theliterature in fuzzy sequence sp...