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A new complete translation and in-depth mythological index.

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Book I (Chaos, Four Ages, Flood, Daphne, Io, Syrinx, Phaethon) Book II (Phaethon’s fall, Callisto, Coronis, Aglauros, Europa) Book III (Cadmus, Actaeon, Semele, Tiresias, Narcissus, Pentheus) Book IV (Pyramus, Leucothoe, Salmacis, Ino, Cadmus, Perseus) Book V (Perseus, Calliope, Proserpine, Arethusa, The Pierides) Book VI (Arachne, Niobe, Marsyas, Procne, Philomela, Boreas) Book VII (Jason, Medea, Minos, The Plague, Myrmidons, Procris) Book VIII (Scylla, Daedalus, Icarus, Meleager, Philemon and Baucis) Book IX (Hercules, Alcmene, Iole, Galanthis, Dryope,Byblis, Iphis) Book X (Orpheus, Pygmalion, Myrrha, Venus and Adonis, Atalanta) Book XI (Death of Orpheus, Midas, Peleus, Ceyx, Alcyone, Aesacus) Book XII (Rumour, Cycnus, Caeneus, Lapiths and Centaurs, Achilles) Book XIII (Ajax, Ulysses, Polyxena, Hecuba, Memnon, Galatea, Glaucus) Book XIV (Scylla, Sibyl, Polyphemus, Circe, Picus, Pomona, Romulus) Book XV (Pythagoras, Hippolytus, Cipus, Aesculapius, The Caesars)...

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The Metal Monster is an Abraham Merritt fantasy novel. Dr. Goodwin is on a botanical expedition in the Himalayas. There he meets Dick Drake, the son of one of his old science acquaintances. They are witnesses of a strange aurora-like effect, but seemingly a deliberate one. As they go out to investigate, they meet Goodwin's old friends Martin and Ruth Ventnor, brother and sister scientists. The two are besieged by Persians as Darius III led when Alexander of Macedon conquered them more than two thousand years ago. The group is saved by a magnificent woman they get to know as Norhala. She commands the power of lightning and controls strange metal animate Things, living, metallic, geometric forms; an entire city of sentient cubes, globes and tetrahedrons, capable of joining together and forming colossal shapes, and wielding death rays and other armaments of destruction. They are led to a hidden valley occupied by what they name The Metal Monster, a strange metal city occupied by the metal animate Things Norhala commands This city is governed by what they call the Metal Emperor, assisted by the Keeper of the Cones. Ruth is slowly being ...

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Fantasy, Fiction

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This book contains short notes or articles, as well as studies on several topics of Geometry and Number theory. The material is divided into ve chapters: Geometric theorems; Diophantine equations; Arithmetic functions; Divisibility properties of numbers and functions; and Some irrationality results. Chapter 1 deals essentially with geometric inequalities for the remarkable elements of triangles or tetrahedrons. Other themes have an arithmetic character (as 9-12) on number theoretic problems in Geometry. Chapter 2 includes various diophantine equations, some of which are treatable by elementary methods; others are partial solutions of certain unsolved problems. An important method is based on the famous Euler-Bell-Kalm ar lemma, with many applications. Article 20 may be considered also as an introduction to Chapter 3 on Arithmetic functions. Here many papers study the famous Smarandache function, the source of inspiration of so many mathematicians or scientists working in other elds. The author has discovered various generalizations, extensions, or analogues functions. Other topics are connected to the composition of arithmetic funct...

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A complete understanding of what something is must include an understanding of what it is not. In his paper, “Paradoxist Mathematics” [19], Florentin Smarandache proposed a number of ways in which we could explore “new math concepts and theories, especially if they run counter to the classical ones.” In a manner consistent with his unique point of view, he defined several types of geometry that are purposefully not Euclidean and that focus on structures that the rest of us can use to enhance our understanding of geometry in general....

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A collection of short poems in Romanian.

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Desfac o sticlă plină de doruri posomorâte şi vinul picură-n pahare amintiri. De obediente gânduri eu atârn senin (în aer spânzurat) şi-o fIacără-de-cântec izbucneşte-n gât. I open a bottle full of longing dull and wine glasses dripping-in memory. I thought hang clear of obedience (in air hung) and the-song neck breaks....

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Cifrele au început să vibreze ............................................................... 5 A scrie scriere .................................................................................... 6 i'v1iorita ................................................................................................. 7 Marele zid românesc ........................................................................... 8 Mama latină şi tata cel dac ................................................................. 9 Am trăi fiecare tablou ....................................................................... 10 Femeie care plângi în versul meu ..................................................... II Eminescieni lci În palidc hemoragii .................................................. 12 Susur de ape ..................................................................................... 13 Naştere ............................................................................................. 14 Noi făntâni săpăm pentru lumină ...................................................... 15 Alergare in noapte ..........................................

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1. O ke Anahonua ka mea e i ike ai ke ano o na mea i hoopalahaiahaia, oia na kaha, a me na ili, a me na paa. Ekolu mau ano o na mea i hoopalahaiahaia, he loa, he laula, a he manoanoa. 2. O ke kaha ; he loa wale no ko ke kaha; aole laula, aole manoanoa. O na welau o ke kaha he mau kiko ia: nolaila, o ke kiko, aole ona loa, aole laula, aole manoanoa, aka he wahi e ku wale ai no. 3. O ke kaha pololei ka loa pokole mai kekahi kiko a i kekahi kiko. 4. O ke kaha pololei ole, a i hui ole ia na kaha pololei, he kaha poai ia Penei, AE he kaha pololei ia; AIOE he kaha hakiia oia, i huiia na kaha pololei AI, IO, OE; AUE he kahapoai ia. Ina ma keia palapala, ua loaa ka olelo kaha wale no, ua manaoia he kaha pololei....

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The aim of this current article is to illustrate in detail Armenian relativistic formulas and compare them with Lorentz relativistic formulas so that readers can easily differentiate these two theories and visualize how general and rich our Armenian Theory of Special Relativity really is with a spectacular build in asymmetry. Then we are going behind this comparison and illustrating that build in asymmetry inside Armenian Theory of Special Relativity is reincarnating the aether as a universal reference medium, which is not contrary to relativity theory. We mathematically prove the existence of aether and we show how to extract infinite energy from the time-space or sub-atomic aether medium. Our theory explains all these facts and peacefully brings together followers of absolute aether theory, relativistic aether theory or followers of dark matter theory. We also mention that the absolute aether medium has a very complex geometric character, which has never been seen before. We are explaining why NASA’s earlier "BPP" and DARPA’s "Casimir Effect Enhancement" programs failed. We are also stating that the time is right to reopen NA...

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Abstract 1. Introduction 2. Legend of the Used Symbols 3. Comparison Armenian and Lorentz Relativistic Formulas 4. Conclusions

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In a lecture that Einstein gave to the Prussian Academy of Sciences in 1921, he said the following: "Geometry predicates nothing about relations of real things, but only geometry together with the purport of physical laws can do so... The idea of the measuring rod and the idea of the clock contained with it in the theory of relativity do not find their exact correspondence in the real world. It is also clear that the solid body and the clock do not in the conceptual edifice of physics play the part of irreducible elements, but that of composite structures, which may not play any independent part in theoretical physics." / Einstein then went on to say that, in spite of the foregoing comment, we should temporarily support the use of the length and time transformations as though they were physically real....

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Preface. 5 -- About the cover 9 -- Welcome from Campus President, DrRaul Ramirez. 10 -- Registrants/Attendees. 11 -- Keynotes. 12 -- The light barrier: its construction and demolition. 14 -- Jim Malmberg. 17 -- DrDavid Iadevaia. 25 -- DrLarissa BBorrisova & Dmitri DRabounski. 29 -- DrFlorentin Smarandache. 36 -- Greg Holmberg AKA BBash. 43 -- Homer BTilton. 83 -- Credits . 100 -- Photos. 101 --...

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This is an amazing little volume and a paradox all of its own. It is published in the United States, in a French translation of a compilation of essays from the original Romanian made jointly by Paul Georgelin and the man it is all about: Smarandache....

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POEMATIO LUDENS A1. En bas d’en haut Rien n’est plus beau En haut d’en bas… Quel bla bla bla : Règnent Ne et Pas !

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C.M.Popa – Le mouvement littéraire paradoxiste -------- 7 F.Smarandache – Neutrosophic degree of a paradoxicity 9 L.Kuciuk, M.Antholy – Paradoxism in geometry -------- 13 Un paradox aritmetic ----------------------------------------- 18 Lediana Paja – Paradoksizmi – rryma më e fundit… ---- 19 A.Balog – Viaţa şi opera scriitorului F. Smarandache --- 24 Paradoxurile bogăţiei ----------------------------------------- 33 Azulejo colgado en muro de Sevilla ------------------------ 34 Quantum Smarandache quasi paradoxes and quantum Smarandache sorites paradoxes --------- 34 T.Negoescu – În miezul paradoxismului ------------------- 39 Folclor românesc din America – Balada lui… ------------ 40 Taifasuri – chat: J.Nică – M. Monu – F.Smarandache --- 43 Folclor paradoxist România – Decalog -------------------- 62 Alt paradox aritmetic ----------------------------------------- 62 Mirifica Românie în imagini -------------------------------- 63 Folklore English ----------------------------------------------- 65 Auzite din popor ---------------------------------------------- 65 Constituţia României revizuită în 2010 (pamflet) ---...

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Florentin Smarandache, an American mathematician of Romanian descent has generated a vast variety of mathematical problems. Some problems are easy, others medium, but many are interesting or unsolved and this is the reason why the present book appears. Here, of course, there are problems from various types. Solving these problems is addictive like eating pumpkin seed: having once started, one cannot help doing it over and over again....

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Preface 5 Definitions, Problems, Theorems 7 References 76

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Apollonius of Tyana (ca. 40-120 AD) was a Greek Pythagorean philosopher and teacher. He hailed from the town of Tyana in the Roman province of Cappadocia in Asia Minor. His date of birth is a matter of conjecture as some say he was roughly a contemporary of Jesus. After Apollonius' death his name remained famous among philosophers and occultists. In a novelistic invention inserted in the Historia Augusta, Aurelian, at the siege of Tyana in 272, was said to have experienced a visionary dream in which Aurelian claimed to have seen Apollonius speak to him, beseeching him to spare the city of his birth. In part, Aurelian said that Apollonius told him Aurelian, if you desire to rule, abstain from the blood of the innocent! Aurelian, if you will conquer, be merciful! By far the most detailed source is the Life of Apollonius of Tyana, a lengthy, novelistic biography written by the sophist Philostratus at the request of empress Julia Domna. Philostratus’ account shaped the image of Apollonius for posterity and still dominates discussions about him in our times. To some extent it is a valuable source because it contains data from older writi...

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Biography, Memoirs, Philosophy, Religion

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Sidelights on Relativity contains ETHER AND THE THEORY OF RELATIVITY, an address delivered on May 5th, 1920, in the University of Leyden; and GEOMETRY AND EXPERIENCE, an expanded form of an address to the Prussian Academy of Sciences in Berlin on January 27th, 1921. (Intro from Project Gutenberg)...

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Nature, Science

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Taking an unusual approach to its topic, “Principles of Anarchism” applies classical logic and deductive reasoning to the subject of human interactions and rights. Beginning with the assumption that all human beings are equal, a concept which is termed the human equality axiom, the author goes on to draw a number of conclusions from this postulate in the manner of a Euclidean proof. These include the surprising contention that no such thing as human rights can exist, the assertion that anarchism is essentially intellectual rather than activist, and the uncomfortable suggestion that historical developments such as cities and industries are harmful to the human race rather than helpful. More centrally, though, the pamphlet argues very simply that any form of authority or government is logically incompatible with the existence of humanity. And it is the implications of this theory, rejection of which requires rejection of the equality axiom as well, which make the argument offered both novel and exceptionally relevant....

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Taking an unusual approach to its topic, “Principles of Anarchism” applies classical logic and deductive reasoning to the subject of human interactions and rights. Beginning with the assumption that all human beings are equal, a concept which is termed the human equality axiom, the author goes on to draw a number of conclusions from this postulate in the manner of a Euclidean proof. These include the surprising contention that no such thing as human rights can exist, the assertion that anarchism is essentially intellectual rather than activist, and the uncomfortable suggestion that historical developments such as cities and industries are harmful to the human race rather than helpful. More centrally, though, the pamphlet argues very simply that any form of authority or government is logically incompatible with the existence of humanity. And it is the implications of this theory, rejection of which requires rejection of the equality axiom as well, which make the argument offered both novel and exceptionally relevant....

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A collection of poetry relating to the author himself based on personal experiences.

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FATE IRONY before may I have been ”me” the fate was done from geometric signs Lobacevsky’s ones after my face but not looking like me in dropping of time...

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Meeting of minds: Florentin Smarandache, preface by Cezar Ivãnescu: 5 1) IN THE VICINITY OF VICINITY – cycle: 6 PIT OF WORDS: 7 FATE IRONY: 8 DEFEATED TREE: 9 THE YELLOW ROSE: 10 WHEN I LOOK AT YOU: 11 BROKEN MIRROR: 12 LEFT FLOWER AND GREEN: 13 GOOD MORNING: 14 NIGHT SPIRIT: 15 SOUL WITHOUT SOUL: 16 2) TOO LONG ROUTE AMONG THE FOREIGNERS (and I cannot turn back my sight anymore) – cycle: 17 PASSING THOUGHTS: 18 THE DEAF AND THE DUMB: 19 HEALING: 20 THE MIRROR: 21 A YOKE: 22 WHAT DO WE WANT?: 23 THE GREAT ARMY: 24 TEST AT GEOGRAPHY: 25 MYSELF LIKE A CONTRADICTION: 26 FIRST CLASS PLANE TICKET: 28 NOBODY: 33 HUNTING: 34 HOSPITALITY: 35 DESIRE: 36 WITHOUT CEREMONIES: 37 YES AND NO: 39 STRIP-TEASE: 40 OH, YES: 41 COURAGE: 42 RECIPROCITY: 43 PRAYER: 44 CONTENTMENT: 48 WISH: 49 ALONE IN THE CHRISTMAS NIGHT: 51 THE MATCHMAKERS: 57 THE END: 65 ADIEU: 66 3) POEMS FROM MY SOUL’S EXILE – cycle: 68 THE SHADOW OF AUROCHS: 69 EMIGRANT TO INFINITY: 70 THE COUNTRY AND ME: 71 THE SAP OF THE LIFE: 72 THORNY ROSE OF LOVE: 73 YOU ARE SO SINGLE ON THE EARTH…: 74 SYMPHONY IN WALNUT WOOD: 75 DON’T SET YOUR ...

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This book includes part of the research results about the Smarandache problems written by Chinese scholars at present, and its main purpose is to introduce various results about the Smarandache problems, such as Smarandache function and its asymptotic properties, series convergence, solutions about special equations. At the same time, we put forward to some new interesting problems either in order to research further. We hope this booklet will guide and inspire readers to these fields....

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前言 数论这门学科最初是从研究整数开始的, 所以叫整数数论. 后来整数 数论又进一步发展, 就叫做数论了. 确切地说, 数论就是一门研究整数性 质的学科. 数论和几何学一样, 是古老的数学分支. 数论在数学中的地位是特殊的, 高斯曾经说过:“数学是科学的皇后, 数论是数学中的皇冠”. 虽然数论中的许多问题在很早就开始了研究, 并得到了丰硕的成果, 但是至今仍有许多被数学家称之为“皇冠上的明 珠”的悬而未解的问题等待人们去解决. 正因如此, 数论才能不断地充 实和发展, 才能既古老又年轻, 才能始终活跃在数学领域的前沿. Foreword Number theory, this discipline was originally started from the study integer, so called integer number theory. Later integer further development of number theory, number theory called up. Rather, number theory is an integer of study qualitative disciplines. Number theory and geometry, is an ancient branch of mathematics. Number theory in mathematics position is special, Gauss once said: "Mathematics is the queen of sciences, number theory is the mathematics of the crown. "Although many of the problems in number theory began very early in the research, And has been fruitful, but there are still many of the mathematicians call "crown Ming Pearl "of unsolved problems waiting to be solved for this reason, number theory can continue to charge Real and development in order to both old and young, can always active in the forefront of the field of mathematics....

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目录 第一章Smarandache 函数1 1.1 引言. . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 S(n) 函数和d(n) 函数的混合均值. . . . . . . . . . . . 4 1.3 关于F.Smarandache 函数S(mn) 的渐近性质. . . . . . . . 6 1.4 复合函数S(Z(n)) 的均值. . . . . . . . . . . . . . . . 7 1.5 是否为整数的问题. . . . . . . . . . . . . . . 10 1.6 关于函数S(n) 的一个方程. . . . . . . . . . . . . . . 13 1.7 关于函数S(nk) 的一个方程. . . . . . . . . . . . . . . 15 1.8 关于Smarandache 函数值的分布. . . . . . . . . . . . . 17 1.9 S(ak(n)) 函数的值分布. . . . . . . . . . . . . . . . . 21 1.10 两个包含Smarandache 函数的方程. . . . . . . . . . . . 25 1.11 S(n) 函数及其均值. . . . . . . . . . . . . . . . . . 27 第二章Smarandache 对偶函数 . . . . . . . . . . . . . . . . .30 2.1 引言. . . . . . . . . . . . . . . . . . . . . . . . 30 2.2 Smarandache 对偶函数的渐近公式. . . . . . . . . . . . 30 2.3 关于Smarandache 对偶函数的一个方程. . . . . . . . . . 33 2.4 关于Smarandache 对偶函数S¤¤(n) . . . . . . . . . . . 37 2.5 一个包含SM(n) 函数的方程. . . . . . . . . . . . . . 40 2.6 一个包含Smarandache 对偶函数的方程. . . . . . . . . . 44 第三章关于SL(n) 函数及其对偶函数的性质 . . . . . . . . . . . . . . . . .48 3.1 引言. . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 SL(n) 函数的渐近公式. . . . . . . . . . . . . . ....

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This study investigates Europe’s motives to develop the independent satellite navigation system known as Galileo despite the existence of America’s successful global positioning system (GPS). The study begins by analyzing both systems to familiarize the reader with global navigation satellite systems (GNSS) and to provide an understanding of the strengths and weaknesses of GPS and Galileo, as well as the systems’ similarities and differences. Although the two systems have different founding principles, they employ similar infrastructures and operational concepts. In the short term, Galileo will provide better accuracy for civilian users until GPS upgrades take effect. But performance is only part of the rationale. The author contends that Europe’s pursuit of Galileo is driven by a combination of reasons, including performance, independence, and economic incentive. With Galileo, Europe hopes to achieve political, security, and technological independence from the United States....

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DISCLAIMER . . . . . . . . . . . . . . . . . . . . . . .ii FOREWORD . . . . . . . . . . . . . . . . . . . . . . .vii ABOUT THE AUTHOR . . . . . . . . . . . . . . . . .ix ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . .xi 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . 1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 GPS VERSUS GALILEO . . . . . . . . . . . . . . .11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . .27 3 WHY GALILEO? . . . . . . . . . . . . . . . . . . .31 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . .45 4 IMPLICATIONS AND RECOMMENDATIONS . .51 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ABBREVIATIONS . . . . . . . . . . . . . . . . . . . .69 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . 71 Illustrations 1 Satellite geometry 9 2 Global positioning system (GPS) satellite 14 3 Galileo satellite 16 Table 1 Number of visible satellites for various masking angles . . . . . . . . . . . . . . . . . . . .13...

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Do you think "math = awesome" is a true statement? After reading this book, you might change your answer to a yes. With "jargon avoidance" in mind, this recreational math book gives you the lowdown on why math is fun, interesting and relevant in today's society. Intended for anyone who is curious about math and where it is circa 2010. This book is less concerned with exploring the mathematical details than it is with exploring the overall impact of various discoveries and insights, and aims to be insightful, cutting edge-y and mathematically rigorous....

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Book Directory: 1. Introduction 2. Picking a Winner is as Easy as 1, 2, 3. 3. That's my Birthday! 4. Sizing up Infinity 5. I am a Liar 6. Gratuitous Mathematical Hype 7. LOL Math, Math LOL 8. In Addition to High School Geometry 9. Abstraction is for the Birds 10. NKS: Anti-Establishment as Establishment 11. 42% of Statistics are Made up 12. Undercover Mathematicians 13. I Will Never Use This 14. Gaussian Copula: $ Implications 15. A Proven Savant 16. History of the TOE and E8 17. One Heck of a Ratio 18. A Real Mathematical Hero 19. Casinos Heart Math 20. The Man who was Sure About Uncertainty 21. We Eat This Stuff Up 22. Do I Have a Question for you! 23. When Nothing is Something 24. Think Binary 25. Your Order Will Take Forever 26. When you Need Randomness in Life 27. e=mc^2 Redux 28. Quipu to Mathematica 29. Through the Eyes of Escher 30. Origami is Realized Geometry 31. Quantifying the Physical 32. Geometric Progression Sure Adds up 33. Nature = a + bi and Other Infinite Details 34. Mundane Implications of Tim...

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A combinatorial map is a connected topological graph cellularly embedded in a surface. As a linking of combinatorial configuration with the classical mathematics, it fascinates more and more mathematician’s interesting. Its function and role in mathematics are widely accepted by mathematicians today. On the last century, many works are concentrated on the combinatorial properties of maps. The main trend is the enumeration of maps, particularly the 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 Thomassen [53] and White [70], especially the [53] for detail. The construction of regular maps on surfaces, related maps with groups and geometry is a glimmer of the map theory with other mathematics....

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Aims and Scope: The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, and published quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces,. . . , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology; Applications of Smarandache multi-spaces to theoretical physics; Applications of Combinatorics to mathematics and theoretical physics; Mathematical theory on gravitatio...

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Duality Theorems of Multiobjective Generalized Disjunctive Fuzzy Nonlinear Fractional Programming Abstract: This paper is concerned with the study of duality conditions to convex-concave generalized multiobjective fuzzy nonlinear fractional disjunctive programming problems for which the decision set is the union of a family of convex sets. The Lagrangian function for such problems is defined and the Kuhn-Tucker Saddle and Stationary points are characterized. In addition, some important theorems related to the Kuhn-Tucker problem for saddle and stationary points are established. Moreover, a general dual problem is formulated together with weak; strong and converse duality theorems are proved. Key Words: Generalized multiobjective fractional programming; Disjunctive programming; Convexity; Concavity; fuzzy parameters Duality. ...

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Contents Duality Theorems of Multiobjective Generalized Disjunctive Fuzzy 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 Product of Fundamental Groups BY M.El-GHOUL, A. E.El-AHMADY, H.RAFAT AND M.ABU-SALEEM. . . . . . . 29 Absolutely Harmonious Labeling of Graphs BY M.SEENIVASAN AND A.LOURDUSAMY . . . . . . . . . . . . . . . . . . . . 40 The Toroidal Crossing Number of K4n BY SHENGXIANG LV, TANG LING AND YUANGQIU HUANG . . . . . . . . . . 52 On Pathos Semitotal and Total Block Graph of a Tree BY MUDDEBIHAL M. H. AND SYED BABAJAN. . . . . . . . . . . . . . . . . . 64 Varieties of Groupoids and Quasigroups Generated by Linear-Bivariate Polynomials Over Ring Zn BY E.ILOJIDE, T.G.JAIYEOLA AND O.O.OWOJORI . . . . . 79 New Characterizations for Bertrand Curves in Minkowski 3-Space BY BAHADDIN BUKCU, MURAT K...