2) Zeige (a− b) | (a − b) für alle a, b ∈ Z und alle n ∈ N. 3) Zeige: Wenn m | n dann (a − b) | (a − b) (mit a, b ∈ Z, m,n ∈ N). 4) Zeige: Wenn 2 ∤ n für ein n ∈ N dann 8 | (n + 23). 5) Zeige: Wenn 3 ∤ n für ein n ∈ N dann 3 | (n + 23). 6) Zeige: Wenn 2 ∤ a und 2 ∤ b (mit a, b ∈ Z) dann 2 | (a + b) aber 4 ∤ (a + b). 7) Zeige: Wenn 7 | (a + b) (mit a, b ∈ Z) dann 7 | a und 7 | b. 8) Finde alle n...

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The notion of a Γ-symmetric space is a generalization of the classical notion of a symmetric space, where a general finite abelian group Γ replaces the group Z2. The case Γ = Z k has also been studied, from the algebraic point of view by V.Kac [14] and from the point of view of the differential geometry by Ledger, Obata [17], Kowalski [16] or Wolf-Gray [20] in terms of k-symmetric spaces. In th...

This study aims to evaluate the fermentative profile, losses, microbial populations, aerobic stability, chemical composition, and in situ degradability of total mixed ration silages based on forage cactus associated with xerophytic legumes. The treatments consisted four concentrate (TMRC), legumes such as Gliricidia sepium (TMRG), Leucaena (TMRL), Senna obtusifolia (TMRS). There was a significa...

Since q<3n~*, the expression in the parenthesis is positive for n large enough, and ]C*-i|/(**)| < 2 n , unless <z = 0. But g = 0 means that every s„= — 1, and f(z) = (z+l). The theorem has thus been proved for large values of n. (5) For small values of n the theorem is incorrect. For example, if/(s) = (2+1) • ( 2 1 / 2 ) , then \f(e) | has relative extrema for 0 = 0, TT, arc cos (1/8), and the...

The undirected power graph of a finite group $G$, $P(G)$, is a graph with the group elements of $G$ as vertices and two vertices are adjacent if and only if one of them is a power of the other. Let $A$ be an adjacency matrix of $P(G)$. An eigenvalue $lambda$ of $A$ is a main eigenvalue if the eigenspace $epsilon(lambda)$ has an eigenvector $X$ such that $X^{t}jjneq 0$, where $jj$ is the all-one...

The notation and terminology used in this paper are introduced in the following articles: [16], [13], [2], [3], [5], [1], [7], [9], [12], [10], [8], [18], [14], [11], [6], [15], and [17]. For simplicity, we use the following convention: x, r, a, x0, p are real numbers, n, i, m are elements of N, Z is an open subset of R, and f , f1, f2 are partial functions from R to R. Next we state a number o...