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...ON DIET AND A HEALTHY LIFE by Dr. Johannes Liu Table of Contents Chapter 1 ON DIET AND A HEALTHY LIFE................................................1-1 Physical Characteristics of Human Being ............................................1-1 Diet .....................

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... a positive integer 5 Zhang Tianping , Ma Yuankui Smarandache “Chopped” NN and N + 1N¡1 9 Jason Earls The 57 -th Smarandache’s problem II 13 Liu Huaning , Gao Jing Perfect Powers in Smarandache n - Expressions 15 Muneer Jebreel Karama On the m -th power residue of n 25 Li Junzhuang and Zhao Jian Generalization of the divisor products and proper divisor p...

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...ZHANG WENPENG, LI JUNZHUANG, LIU DUANSEN (EDITORS) RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER TH... ...ATICS NORTHWEST UNIVERSITY XI’AN, SHAANXI, P.R.CHINA LI JUNZHUANG, LIU DUANSEN INSTITUTE OF MATHEMATICS SHANGLUO TEACHER’S COLLEGE SHANGL... .... China. Copyright 2005 by Hexis and Zhang Wenpeng, Li Junzhuang, Liu Duansen, and all authors for their own articles Many books can be... ...ur successful conference. August 10, 2005 Zhang Wenpeng, Li Junzhuang, Liu Duansen ONTHESMARANDACHEM THPOWERRESIDUES ∗ ZhangWenpeng Department... ...ower free number se quence. SmarandacheNotionsJournal,2004,14: 66 69. [5] Liu Hongyan and Lou Yuanbing. A Note on the 29 th Smarandache’s problem. Sm... ... power free number sequence. SmarandacheNotionsJournal,2004,14: 66 69. [5] Liu Hongyan and Lou Yuanbing. A note on the 29 th Smarandache’s problem. Sm... ...ower free number se quence. SmarandacheNotionsJournal,2004,14: 66 69. [5] Liu Hongyan and Lou Yuanbing. A Note on the 29 th Smarandache’s problem. Sm... ...an Xu Zhefeng Xue Xifeng Ren Ganglian Ding Liping Li Jie Liu Yanni Ma Jinping Li Zhanhu Wang Ting Yang Hai Ren Dongmei... ...a Jinping Li Zhanhu Wang Ting Yang Hai Ren Dongmei Lu Yaming Liu Duansen Li Junzhuang Li Chao Yang Cundian Zhao Jian Wang Nianliang Gao ...

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...On the Cultural Contrast: East-West, preface by F. Liu .……………………………….4 Abstract by the Translator to the Chinese Translation of Neutrosophics………………12 Preface by the Translator to the Chinese Translation of Neutrosophics………………13 Biography of the Translator………………………………………………...

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...X. Pan and B. Liu : On the irrational root sieve sequence 1 H. Liu : Sub-self-conformal sets 4 K. Liu : On mean values of an arithmetic function 12 L. Wang and Y. Liu : On an infinite series related to Hexagon-numbers 16 X. Ren, etc. :...

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...On the Cultural Contrast: East-West, preface by F. Liu. 4 -- Abstract by the Translator to the Chinese Translation of Neutrosophics. 12 -- Preface by the Translator to the Chinese Translation of Neutrosophics. 13 -- Biography of the Translator. 18 -- Publication Review by F. ...

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...Florentin Smarandache Feng Liu NEUTROSOPHIC DIALOGUES ... ... CONTENTS On the Cultural Contrast: East-West, preface by F. Liu .………….…..………………….4 Abstract by the Translator to the Chinese Transla... ... of the Translator………………………………………………………..…….18 Publication Review by F. Liu…………………………………………………..………..22 Annotation in the Chinese Translation of... ...o.…………………………….95 4 On the Cultural Contrast: East-West (preface) Liu Feng (surname, given name) Dept. of Economic Information, School of In... ... An Interview, 2002, http://www.amtb.org.sg/2/2_10/2_10_1/2_10_1.htm. F. Liu, F. Smarandache: Logic: a Misleading Concept － A Contradiction Study to... ...dache/NeutrosophicProceedings.pdf, also in Libertas Mathematica 2002. F. Liu: On Both A and Anti-A in Neutrosophics of Logic in Excitation-Inhibitio... ... Preface by the Translator to the Chinese Translation of Neutrosophics Liu Feng (surname, given name) 1. Science Puzzles Why was neutrosop... ...ruction that readers can provide. 18 Biography of the Translator Liu Feng (surname, given name) Born in 1964 in Xi’an, China, I enter... ...itional instruction that readers can provide. 22 Publication Review Liu Feng (surname, given name) My research related to neutrosophy orig...

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...type problems. There are a few papers which are not directly related to but should fall within the scope of Smarandache type problems. They are 1. L. Liu and W. Zhou, On conjectures about the class number of binary quadratic forms; 2. W. Liang, An identity for Stirling numbers of the second kind; 3. Y. Wang and Z. Sheng, Two formulas for x^n in terms of Chebyshev polynomia...

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...L. Mao : On Algebraic Multi-Vector Spaces 1 L. Liu and W. Zhou : On conjectures concerning class number of binary quadratic forms 7 W. Zhai and H. Liu : On square-free primitive roots mod p 15 X. Pan : A new limit theorem involving the Smarandache LCM sequence 20 M. L...

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...al Class of Graphs BY R.SRIDEVI, S.NAVANEETHAKRISHNAN AND K.NAGARAJAN. . . . . . . . 59 Surface Embeddability of Graphs via Tree-travels BY YANPEI LIU. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73 Edge Maximal C3 and C5-Edge Disjoint Free Graphs BY M.S.A.BATAINEH AND M.M.M.JARADAT . . . . . . . . . . . . . . . . . . 82 A Note on Admissible Mann...

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...J. Wang : Cube-free integers as sums of two squares 1 G. Liu and H. Li : Recurrences for generalized Euler numbers 9 H. Li and Q. Yang : Some properties of the LCM sequence 14 M. Liu : On the generalization of the primitive number function 18 Z. Lv : On the F. Smarandache LCM fu...

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...e rooted maps, pioneered by W. Tutte, and today, this kind of papers are still appeared on the journals frequently today. All of those is surveyed in Liu’s book [33]. To determine the embedding of a graph on surfaces, including coloring a map on surfaces is another trend in map theory. Its object is combinatorialization of surfaces, see Gross and Tucker [22], Mohar and Tho...

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...s and Systems, Chinese Academy of Sciences, Beijing 100080, P.R.China. Y.P.Liu and R.X.Hao, Department of Applied Mathematics, Beijing Jiaotong Uni- v... ...ill appeared on the journals frequently today. All of those is surveyed in Liu’s book [33]. To deter- mine the embedding of a graph on surfaces, inclu... ... Feng Tian for his encouraging and invaluable helps and toprofessor Yanpei Liu introduced meinto thefiled ofcombinatorial map theory. Thanks are also g... ...August, 2002, http://www.icm2002.org.cn/B/Schedule.Section04.htm. [33] Y.P.Liu, Enumerative Theory of Maps, Kluwer Academic Publishers , Dor- drecht/ ... ...s, Kluwer Academic Publishers , Dor- drecht/ Boston/ London,1999. [34] Y.P.Liu, Advances in Combinatorial Maps, Northern Jiaotong University Pub- lish... ...l Maps, Northern Jiaotong University Pub- lisher, Beijing (2003). [35] Y.P.Liu, Embeddability in Graphs, Kluwer Academic Publisher, Dordrecht / Boston... ..., Kluwer Academic Publisher, Dordrecht / Boston / London (1995). [36] Y.P. Liu, The maximum non-orientable genus of a graph (in Chinese), Scientia Sin... ...hesis in Northern Jiaotong University, Beijing, 2002. [44] L.F.Mao and Y.P.Liu, New automorphism groups identity of trees, Chinese Advance in Math.,11... ...ese Advance in Math.,113-117,5(2002). References 111 [45] L.F.Mao and Y.P.Liu, On the roots on orientable embeddings of graph, Acta Math. Scientia, 2...

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... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117 The Crossing Number of Two Cartesian Products BY LIN ZHAO, WEILI HE, YANPEI LIU and XIANG REN. . . . . . . . . . . . . . . . . . . . . . . . .120...

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...M AND S. SUDHA . . . . . . . . . . . . . . . . . . . . . . . 45 Embeddings of Circular graph C(2n + 1, 2) (n ≥ 2) on the Projective Plane BY XINQIU LIU, YUANQIU HUANG AND JING WANG. . . . . . . . . . . . .53 A Note On Jump Symmetric n-Sigraph BY H. A.MALATHI AND H. C.SAVITHRI . . . . . . . . . . . . . . . . . . . . 65 New Families of Mean Graphs BY SELVAM AVADAYAPPAN...

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...aphs-II BY E.SAMPATHKUMAR, P.SIVA KOTA REDDY AND M.S.SUBRAMANYA . . . . 89 Genus Distribution for a Graph BY LIANGXIA WAN, HONGJIAN LAI AND YANPEI LIU. . . . . . . . . . . . . . 99 ...

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...wer residues function 31 H. Gunarto and A. Majumdar : On numerical values of Z(n) 34 K. Nagarajan, etc.: M-graphoidal path covers of a graph 58 H. Liu : On the cubic Gauss sums and its fourth power mean 68 A. A. K. Majumdar : On the dual functions Z¤(n) and S¤(n) 74 S. S. Gupta : Smarandache sequence of Ulam numbers 78 L. Li : A new Smarandache multiplicative functio...

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.... About the elementary properties of this function, many scholars have studied it (see reference [2]-[7]), and got some useful results. For examples, Liu Yanni [2] studied the mean value properties of ep(bk(n)), where bk(n) denotes the k-th free part of n, and obtained an interesting mean value formula for it. That is, let p be a prime, k be any fixed positive integer, the...

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...Nonlinear Fractional Programming BY E.E.AMMAR . . . . . . . . . . . . . . . . . . . . 01 Surface Embeddability of Graphs via Joint Trees BY YANPEI LIU. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15 Plick Graphs with Crossing Number 1 BY B.BASAVANAGOUD AND V.R.KULLI . . . . . . . . . . . . . . . . . . . . . . 21 Effects of Foldings on Free Pr...

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...mplements 95 L. Ding : An arithmetical function and its mean value 99 Y. Yi : Some identities involving Bernoulli numbers and Euler numbers 102 L. Liu : An arithmetical function and its mean value 108 J. Li : On Smarandache dual fuction 111...