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In many practical applications, it is useful to represent a signal or an image by its average values on several fuzzy sets. The corresponding F-transform technique has many useful applications in signal and image processing. In principle, we can use different membership functions. Somewhat surprisingly, in many applications, the best results occur when we(More)
In many applications of interval computations, it turned out to be beneficial to represent polynomials on a given interval [x, x] as linear combinations of Bernstein polynomials (x − x) k · (x − x) n−k. In this paper, we provide a theoretical explanation for this empirical success: namely, we show that under reasonable optimality criteria, Bernstein(More)
One of the main problems with neural networks is that they are often very slow in learning the desired dependence. To speed up neural networks, Bruno Apolloni proposed to othogonalize neurons during training, i.e., to select neurons whose output functions are orthogonal to each other. In this paper, we use symmetries to provide a theoretical explanation for(More)
—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 or by a piece-wise polynomial function (spline). Usually, polynomials are described as linear combinations of monomials. It turns out that in many computational problems, it is more efficient(More)
In many applications, we have numerous molecules that are obtained from a " template " molecule like benzene C6H6 or cubane C8H8 by replacing some of its hydrogen atoms with other atoms or atom groups (called ligands). Depending on how many original atoms are replaced and which ones are replaced, we obtain a large number of different chemical substances. It(More)
In fuzzy control, two approaches are mainly used: Mamdani's approach, in which we represent the knowledge base as a disjunction of statements A<inf>i</inf>(x) &amp; B<inf>i</inf>(u) corresponding to individual rules, and logical approach, in which the knowledge base is represented as a conjunction of the rules themselves A<inf>i</inf>(x) &#x2192;(More)
In describing expert knowledge, it is often important to properly take into account hedges like " very " , " somewhat " , etc. In particular, fuzzy logic provides a consistent way of describing hedges. For some of the hedges, a repetition changes the meaning: e.g., " very very small " is smaller than " very small ". However, other hedges – like " somewhat "(More)
Many real-life dependencies can be reasonably accurately described by linear functions. If we want a more accurate description, we need to take non-linear terms into account. To take nonlinear terms into account, we can either explicitly add quadratic terms to the regression equation, or, alternatively, we can train a neural network with a non-linear(More)