Approximation by rational spline functions

Abstract We discuss the linear precision property of NURBS functions. The degree of approximation of continuous functions is studied. Keywords: NURBS functions; linear precision; approximation degree; modulus of smoothness Mathematics Subject Classification (1991): 41A15, 41A25, 41A28, 41A36, 41A63, 65D07, 65D17     

Structured Learning and Decomposition of Fuzzy Models for Robotic Control Applications

In this paper a specially designed structured-optimization procedure is used for learning the parameters of the Takagi–Sugeno (TS) type fuzzy models. It is well-known that the number of learning parameters increases exponentially with the number of model inputs. Therefore an appropriate learning scheme with preliminary structuring of the learning parameters into two groups: antecedent parameters and consequent parameters can be helpful for speeding-up the learning process. Two different optimization algorithms for tuning the antecedent and consequent parameters respectively are used in a sequence of repetitive loops (epochs). The stop criterion is defined as a number of repetitions of the loops or as a desired minimal error. Random walk algorithm with variable step size is used in this paper for tuning the antecedent parameters of the membership functions. For tuning the consequent parameters of the singletons, a specially proposed local learning algorithm is used. The problem of dimensionality reduction in fuzzy modeling is also considered in the paper from another viewpoint, namely as a hierarchical fuzzy model structure. It is accomplished by a decomposition of the complete fuzzy model into a feedforward hierarchical structure of sub-models called partial fuzzy models each one with two inputs and one output. Then the local models are learned separately in a preliminary specified and repetitive order. Such decomposition scheme has a potential for a significant reduction of the number of model parameters to be tuned thus reducing the total learning time. It has been experimentally shown that both concepts for dimensionality reduction in learning fuzzy models have benefits in learning speed and accuracy. A comparison with simultaneous optimization of all parameters of a single fuzzy model is also given. It shows that the proposed structured learning as well as the decomposition of the fuzzy model into a hierarchical fuzzy model structure lead to reducing the learning time and creating more accurate fuzzy models. Finally an application for learning a fuzzy controller of a two-link robot motion is shown and analysed.     

On multiplicativity of the Bernstein operator

For continuous functions f and g, we prove that the Bernstein operator B_n is multiplicative for all n≥1 and all x∈2[0,1] if and only if at least one of the functions f and g is a constant function. Some other variants of multiplicativity are also considered.     

Phase equilibria in the Ag4SSe-ZnTe system

The Ag₄SSe-ZnTe phase diagram has been determined on the basis of X-ray diffraction, differential thermal and metallographic analyses, as well as microhardness and density data. It has been divided into two subdiagrams of eutectic type by an intermediate A-phase with most probable composition of 2Ag₄SSe.ZnTe. The unit-cell parameters of the low-temperature -2Ag₄SSe·ZnTe modification have been determined (a=3.330 , b=3.010 , c=2.895 , =95.04°, =107.83°, =92.62°). It has been supposed that in the ZnTe-rich part a second intermediate B-phase with a composition of Ag₄SSe·2ZnTe is formed, which is stable in the range of 320÷500 °C.