Abstract. For every finite-to-one map λ : Γ → Γ and for every abelian group K, the generalized shift σ λ of the direct sum Γ K is the endomorphism defined by (x i ) i∈Γ → (x λ(i) ) i∈Γ [3]. In this paper we analyze and compute the algebraic entropy of a generalized shift, which turns out to depend on the cardinality of K, but mainly on the function λ. We give many examples showing that the generalized shifts provide a very useful universal tool for producing counter-examples.We denote by Z, P, and N respectively the set of integers, the set of primes, and the set of natural numbers; moreover N 0 = N ∪ {0}. For a set Γ, P fin (Γ) denotes the family of all finite subsets of Γ. For a set Λ and an abelian group G we denote by G Λ the direct product i∈Λ G i , and by G (Λ) the direct sum i∈Λ G i , where all G i = G. For a set X, n ∈ N, and a function f : X → X let Per(f ) be the set of all periodic points and Per n (f ) the set of all periodic points of order at most n of f in X.
Internet of Things (IoT) paradigm with strong impact on future life will be interconnected through Cognitive Radio Networks (CRNs). CRNs with Ubiquitous trait are highly promising to achieve interference-free and on-demand services. CRs are able to sense the spectral environment, to detect unoccupied bands, and to use them for signal transmissions. This opportunity encourages malicious Users to surpass CRs by Primary User Emulation (PUE) attack and use vacant spectrums. This paper proposes an unsupervised algorithm to distinguish CRs from PUs regardless of static and mobile user. Employing K-means and graph theory are coincident in our algorithm to improve detection outcomes. The edge of graph corresponding to the relation between signals is used and the result of comparison the signal properties is exposed to different clusters. The Receiver Operating Characteristic (ROC) and Detection Error Tradeoff (DET) of our proposed algorithm prove our claim.
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