Abstract-In fuzzy control, two approaches are mainly used: Mamdani's approach, in which we represent the knowledge base as a disjunction of statements Ai(x) & Bi(u) corresponding to individual rules, and logical approach, in which the knowledge based is represented as a conjunction of the rules themselves Ai(x) → Bi(u). Both approaches are known not to be perfect, so a natural question arises: what other approaches are possible? In this paper, we describe all possible approaches; alternative approaches use an "exclusive or" operation and correspond, e.g., to the fuzzy transform idea.
It is well known that an arbitrary continuous function on a bounded set-e.g., on an interval [a, b]-can be, with any given accuracy, approximated by a polynomial. Usually, polynomials are described as linear combinations of monomials. It turns out that in many computational problems, it is more efficient to represent a polynomial as Bernstein polynomialse.g., for functions of one variable, a linear combination of terms (x − a) k • (b − x) n−k. In this paper, we provide a simple fuzzybased explanation of why Bernstein polynomials are often more efficient, and we show how this informal explanation can be transformed into a precise mathematical explanation.
Protein structure is invariably connected to protein function. There are two important secondary structure elements: alpha-helices and beta-sheets (which sometimes come in a shape of beta-barrels). The actual shapes of these structures can be complicated, but in the first approximation, they are usually approximated by, correspondingly, cylindrical spirals and planes (and cylinders, for beta-barrels). In this paper, following the ideas pioneered by a renowned mathematician M. Gromov, we use natural symmetries to show that, under reasonable assumptions, these geometric shapes are indeed the best approximating families for secondary structures.
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