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...Chapter III is of a similar nature to that of chapter II. It deals with the alternating iteration of the Smarandache function and the sum of divisor... ... Preface 3 I. An Integer as a Sum of Consecutive Integers 9 II. Alternating Iterations of the Euler φ-Function and the Pseudo-Smarandach... ...f the Euler φ-Function and the Pseudo-Smarandache Function 16 III. Alternating Iterations of the Sum of Divisors Function and the Pseudo-Sm... ...f Numbers, Erhus University Press, 1997. 16 II. Alternating Iterations of the Euler φ-Function and the Pseudo-Smarandach... ...dache Function Abstract: This study originates from questions posed on alternating iterations involving the Pseudo-Smarandache function Z(n) an... ... 2 k 1 k 1 k k Again we note that Z(n)<n. 18 In a study of alternating iterations [3] it is stated that apart from when n=2 k (k≥0... ...to the iterations of composite integers and plays an important role in the alternating Z-φ iterations which will be subject of the next section. ... ...ing Z-φ iterations which will be subject of the next section. IV. The alternating iteration of the Euler φ function followed by the Smarandach... ...wed by the Smarandache Z function. Charles Ashbacher [3] found that the alternating iteration Z(…(φ(Z(φ(n)))…) ends in 2-cycles of which he foun...

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...tedness and disconnectedness. DEFINITION 2.2.1: A walk of a graph G is an alternating sequence of points and lines v 0 , e 1 , v 1 ,…, v n-1 e n ... ...D'). A directed walk joining the vertex u 0 to the vertex u k in D is an alternating sequence w = u 0 a 1 u 1 a 2 u 2 … a k u k , 1 ≤ i ≤ k,... ... alternate between E \ M and M and its end vertices are M-unsaturated. An M-alternating path in G is a path whose edges alternate between E \ M and ... ... A neutrosophic walk of a neutrosophic graph is a walk of the graph with an alternating sequence of points and lines where at least one of the point... ...EX 1-factor, 25 A Acyclic graph, 37-8 Adjacent vertices, 20-1 Alternating sequence, 26-7, 32 Automorphism of a graph, 28 B Binar... ...Loopless bipartite graph, 54 Loopless graph, 28, 40 Loopless, 36 M M-alternating path, 40-1 Markov chain, 16 Matching, 40 Maximal clique, 2...

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... Consider the group S 4 , the symmetric group of degree 4 which has A 4 the alternating subgroup of order 12. Clearly 6/12 but A 4 has no subgro... ... Find all the elements in G which have Smarandache inverses where G = A 5 , the alternating group of degree 5. PROBLEM 6: Find the Smarandache inv... ...tead of simple as S(n) is pseudo simple for S(n) in the classical sense has the alternating group A n ⊂ S n ⊂ S(n) to be the normal subgroup. So i...

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.... Timi soara, 1987, pp.1-18. 7. J. S andor, On the irrationality of some alternating series, Studia Univ. Babe s-Bolyai 33(1988), 8-12. 166 14 On ... ... in the above result. 2. We now prove a result on the irrationality of the alternating series 1 X n=1 ( 1) n 1 S(n) n! . We can formulate our result... ... k a 1 :::a n b n + ( 1) k b k+1 a k+1 b k+2 a k+1 a k+2 +::: : The alternating series on the right side is convergent and must have an integer... ... din Timi soara, 1987, 1-18. 3. J. S andor, On the irrationality of some alternating series, Studia Univ. Babe s- Bolyai, Mathematica, XXXIII, 4, 1...

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...deliberate seclusions for feverish and tormenting seeking at the desk, always alternating with plunges in the immediate reality and in the middle of...

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... spaces are a particular case of Riemannian spaces, where the metric is sign-alternating. So a four- dimensional pseudo-Riemannian space Einstein put ...

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...) ≥ . n! (n+1)! The alternating series theorem can then be used to conclude that the series is ...

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...to do so with angles which are multiples of the initial angle, each time alternating the end about which the rotation is executed , we get fascinat...

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... the geometry of a four-dimensional pseudo- Riemannian space with the sign-alternating signature (−+++) or (+−−−) he assigned to the basic space-time ... ....N) is symmetric and non- degenerate. The metric g ab , given for the sign-alternating g αβ , can be sign-alternating, having a signature dependent on... ...tein spaces had been solved by Petrov [40]. If the space signature is sign-alternating, generally speaking, the station- ary curvatures are complex as... ...bda- matrix to its canonical form along a real path in a space of the sign-alternating metric. Although this solution is obtained only at given point,... ...eneral Relativity — the four-dimensional pseudo-Riemannian space with sign-alternating label (+−−−) or (−+++) . Its metric is strictly non-degenerated... ...ced — the four- dimensional space which, having General Relativity’s sign- alternating label (+−−−), permits the space-time metric to be fully degener... ...be a combination of both positive and negative forces in the same time, or alternating positive and negative, etc. Unenergy would similarly be a combi... ... similarly be a combination between positive and negative energies (as the alternating current, a. c., which periodically reverses its direction in a ... ...ependent of the circuit’s constants). Would it be possible to construct an alternating-energy gen- erator? To conclusion: According to the Universal D...

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...es prime: 2, 3, 5, 7, 23, 37, 53, ... [6] (ID=A019546) 14) Smarandache alternating series A series of the form - - k k k a 1 ) ( ) 1 ( where a...

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...paces are a particular case of Riemannian spaces, where the metric is sign-alternating. So the four-dimensional pseudo-Riemannian space of the signatu... ...ting contour (drawn in the same scale as that in Fig. 2a). In this case an alternating signal frequency at a constant voltage was applied to the input...

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...versation in Chinese for an ignorant of this language' a "leftist poem", 'alternating discussions of travelers, in a train station, on different the...