By Guy de Trė, Przemyslaw Grzegorzewski, Janusz Kacprzyk, Jan W. Owsiński, Wojciech Penczek, Slawomir Zadrożny

ISBN-10: 3319301640

ISBN-13: 9783319301648

ISBN-10: 3319301659

ISBN-13: 9783319301655

This quantity offers fresh examine, demanding difficulties and options in clever platforms– overlaying the next disciplines: synthetic and computational intelligence, fuzzy good judgment and different non-classic logics, clever database platforms, info retrieval, info fusion, clever seek (engines), information mining, cluster research, unsupervised studying, laptop studying, clever facts research, (group) determination aid structures, clever brokers and multi-agent platforms, knowledge-based structures, imprecision and uncertainty dealing with, digital trade, dispensed platforms, and so forth. The e-book defines a standard floor for often doubtless disparate difficulties and addresses them by utilizing the paradigm of greatly perceived clever structures. It provides a huge landscape of a large number of theoretical and useful difficulties which were effectively handled utilizing the paradigm of clever computing.

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**Additional info for Challenging Problems and Solutions in Intelligent Systems**

**Example text**

P − 1. Note that p−1 p β2 = 1 + i=1 p i is p i β, i 1 because β10 = 1 and p−1 p βj = i=1 p i β j−1 , i j = 3, 4, . . , since β 0j = 0, for j = 2, 3, . .. Notice that Fermat’s little theorem follows immediately from the above. Actually, let n, p be coprime numbers with prime p. Then p n = p i=0 n−1 p p p βi+1 ≡ β1 + (n − 1)β2 = 1 + n − 1 ≡ n i mod p. Let us now return to the issue of finding the sum of n initial terms of a k-arithmetic sequence. Assume that a polynomial an of the degree k having 50 A.

I −1 k 0 Therefore, we get (Δ1 Φ)1 = i=1 ψi+1 = ψ2 = a21 . Analogously, let us i−1 examine the form of a term of the mth difference sequence k Δm Φn = Δm−1 Φn+1 − Δm−1 Φn = i=m−1 k = i=m n n−1 − i −m+1 i −m+1 ψi+1 n−1 ψi+1 . i −m k 0 Hence we get (Δm Φ)1 = i=m ψi+1 = ψm+1 = am1 . Let us see what is the i−m nth term of the kth difference sequence. 40 A. Kołacz k Δk Φn = i=k n−1 n−1 ψk+1 = ψk+1 = r. ψi+1 = 0 i −k It is the difference of an . The sequences an and Φn have the same characterizing sequences, therefore, by Theorem 3, we conclude that an = Φn .

Proof Ad (a) We use the induction with respect to the number of terms. For n = k + 2, the kth difference sequence is constant and consists of exactly two terms equal to the difference of an . By (1) we get k r = ak1 = (−1)i k ak−i+1 , i (−1)i k ak−i+2 . i i=0 k r = ak2 = i=0 Renumbering (3) we obtain k+1 r = ak1 = (−1)i+1 i=1 k ak−i+2 i −1 (3) k-Arithmetic Sequences—Theory and Applications 41 and then subtracting the terms we have k+1 0= k ak−i+2 − i −1 (−1) i+1 i=1 k = (−1)k+2 a1 − i=1 k (−1)i i=0 k ak−i+2 i k k + i −1 i (−1)i = (−1)k+2 − k ak−i+2 − ak+2 k+1 (−1)i i=1 k+1 k+1 ak−i+2 − ak+2 = − ak−i+2 .

### Challenging Problems and Solutions in Intelligent Systems by Guy de Trė, Przemyslaw Grzegorzewski, Janusz Kacprzyk, Jan W. Owsiński, Wojciech Penczek, Slawomir Zadrożny

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