By George A. Anastassiou (auth.), George A. Anastassiou, Oktay Duman (eds.)

Advances in utilized arithmetic and Approximation idea: Contributions from AMAT 2012 is a set of the easiest articles awarded at “Applied arithmetic and Approximation idea 2012,” a global convention held in Ankara, Turkey, might 17-20, 2012. This quantity brings jointly key paintings from authors within the box overlaying themes equivalent to ODEs, PDEs, distinction equations, utilized research, computational research, sign concept, confident operators, statistical approximation, fuzzy approximation, fractional research, semigroups, inequalities, precise capabilities and summability. the gathering should be an invaluable source for researchers in utilized arithmetic, engineering and statistics.

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Mr . 44) we get Gmr Gmr−1 . . 3984)e−m1 . 47) One can give easily many other interesting applications. Acknowledgement The author wishes to thank his colleague Professor Robert Kozma for introducing him to the general theory of cellular neural networks that motivated him to consider and study iterated neural networks. References 1. A. Anastassiou, Rate of convergence of some neural network operators to the unitunivariate case, J. Math. Anal. Appli. 212 (1997), 237–262. 2. A. Anastassiou, Rate of convergence of some multivariate neural network operators to the unit, J.

3984)e−m1 . 47) One can give easily many other interesting applications. Acknowledgement The author wishes to thank his colleague Professor Robert Kozma for introducing him to the general theory of cellular neural networks that motivated him to consider and study iterated neural networks. References 1. A. Anastassiou, Rate of convergence of some neural network operators to the unitunivariate case, J. Math. Anal. Appli. 212 (1997), 237–262. 2. A. Anastassiou, Rate of convergence of some multivariate neural network operators to the unit, J.

We mention from [7] the following result. 21. Let f ∈ Cm ∏Ni=1 [ai , bi ] , 0 < β < 1, n, m, N ∈ N. nmβ 1,m fα ∞ ∏Ni=1 αi ! 3984) b − a m! m ∞ fα max m ∞,m N e−n (1−β ) =: Mn . 22. Let f ∈ Cm ∏Ni=1 [ai , bi ] , 0<β <1, r, n, m, N∈N, x∈ ∏Ni=1 [ai , bi ] . Then |Grn ( f , x) − f (x)| ≤ Grn ( f ) − f ∞ ≤ r Gn ( f ) − f ∞ ≤ rMn . 43) One can have a similar result for the operator Fn but we omit it. Next we specialize on Lipschitz classes of functions. 23. Let f ∈ C ∏Ni=1 [ai , bi ] or f ∈ CB RN ; n, N, m1 , .