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...ure 1b.] S’ S a 1 θ r a 2 v Figure 1. GDT Projective and Elliptic Geometry Theorems a.) Right conic projection in G determining S’ ellipse... ... ,b 2 } are the corresponding minor axial radii as determined from elliptic geometry. We only require reference to one of the ellipse's two focal po... ... to be determined: the elliptic length (σ e ) defined as: the length of the line from any point P on the elliptic surface to the positive focus. Th... ... Graph 2. Relative Doppler Frequency (ν ν ν ν) Equations (7.2): solid line, and (7.3): dashed line : ( v = 0.9c). (ν ν ν ν) Freq 16 The d... ...cs obey the laws of special relativity. [44] VSE was seen to be a dividing line separating general relativity (GRT) from gravitational models, whic... ... be heterogeneous with electron well loci determined by intranuclear proton geometry. It is obvious that proton charge repulsion, interior to a spher... ... (cubic) surface. Coupled with proton repulsion, the logical result of this geometry is eight proton potential wells, (octet sites) with lowest pot... ... however, the eight ionizations of the 2p and 2s orbitals proceed in a near linear progression, so that there is no clear way to differentiate betwe... ...t progression supports the primary hypothesis of charge projection. Thus, a line drawn from the nucleo through a filled outer site defines a line of...

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... As a new type of poetry with fixed from: The Paradoxist distich is a two-line poem which contains an antithesis. The second line contradicts the f... ...ontains an antithesis. The second line contradicts the first one, but both lines form together a unitary meaning defining the title. Here there are... ...y with fixed from: The Tautological Distich is an apparently redundant two-line poem, but the “redundant” lines together give a deeper meaning to th... ...tion Of the scientific AXIOMATIZATION (II) Geometrization Of the geometry SCHOPENHAUER: “I know That I know” IMITATOR Discovers ... ...s As a new form of poetry with fixed form: The Dualistic Distich is a two-line poem such that the second line is the dualistic of the first and tog... ... reviewed it, something inside of the envelope was opposing. Gerald`s four-line proseful poetry, whose each stanza starts with “and”, was eagerly wa...

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... As a new type of poetry with fixed from: The Paradoxist distich is a two-line poem which contains an antithesis. The second line contradicts the f... ...ontains an antithesis. The second line contradicts the first one, but both lines form together a unitary meaning defining the title. Here there are... ...y with fixed from: The Tautological Distich is an apparently redundant two-line poem, but the “redundant” lines together give a deeper meaning to th... ...tion Of the scientific AXIOMATIZATION (II) Geometrization Of the geometry SCHOPENHAUER: “I know That I know” IMITATOR Discovers ... ...s As a new form of poetry with fixed form: The Dualistic Distich is a two-line poem such that the second line is the dualistic of the first and tog... ... reviewed it, something inside of the envelope was opposing. Gerald`s four-line proseful poetry, whose each stanza starts with “and”, was eagerly wa...

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...ess, he actually assumes Greenleaf's identity. Disembarking from a luxurious Cunard liner in his destination, Italy, he "confesses" to a gullible te... ...fortunate state of things and its moral and practical implications. The blurring line between life and its representation in the arts is arguably... ...m "Titanic" is riddled with moral dilemmas. In one of the scenes, the owner of Star Line, the shipping company that owned the now-sinking Unsinkable... ...physical riddles and presses the viewers' intellectual stimulation buttons. A two line bit of dialogue, though, forms the axis of this nightmarish... ...ar questions arise in time travel)? This, precisely, is where the film crosses the line between the intriguing and the macabre. The master puppetee... ...e sense organs that we do (eyes) and may not be acquainted with our mathematics and geometry. Reality can be successfully described and captured by ...

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...i, ness, too. I deal in love, and so must you. Since, for me, the bottom line in life and in business is people, it is important that they are nou... ..., it is gone for, ever. "Yesterday is a page turned; you cannot add one line to it or erase one word from it. It is closed forever and can affect ... ...first begin." The fastest runner in the world will never cross the finish line unless he or she first begins the race. The great inventions of the ... ...f a great mind that has gone before. In every case, people are the bottom line. Governments, educa~ tional institutions and corporations all exist t... ...d Game, the Dymaxion Airocean World Map, his develop, ment of Synergetic Geometry and his most famous invention, the Geodesic Dome. I was stationed... ...e are one in truth Q orne people live life as if it were a o horizontal line with all of us stand, ing in it single file, fighting desperately to ...

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...quantization. Thus adaptive vector quantization embeds learning in a dynamic geometry, vector quantization may be optimal according to different cr... ...l function S j (y j ). However the competitive learning law can be reduced to linear competitive learning law ij m = S j (y j ) (x – m j ), ... ...3-linked FRMs, 4-linked FRMs and so on in general, an n-linked FRM on similar lines and obtain the hidden relation, which are not explicitly calcul... ...ns membership value in the interval [0, 1]. But very few try to visualize the geometry of fuzzy sets. It is not only of interest but is meaningful t... ...ometry of fuzzy sets. It is not only of interest but is meaningful to see the geometry of fuzzy sets when we discuss fuzziness. Till date researcher... ...sional graphs with the domain X represented as a one-dimensional axis. The geometry of fuzzy sets involves both domain X = (x 1 ,…, x n ) and the... ...main X = (x 1 ,…, x n ) and the range [0, 1] of mappings µ : X → [0, 1]. The geometry of fuzzy sets aids us when we describe fuzziness, define fuzz... ...ple will not make repeated mistakes and become HIV/AIDS victim. This was her line of thinking. Additional Observation: Her husband infected her...

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... the work of feelings combined with intellect. It uses, for beginning students, geometry, matrices and diagrams. When studying Kabbalah, they recogn... ...s the case here. Kabbalah utilizes many mathematical concepts such as matrices, geometry, numbers, graphs, characters and letters, etc. These approa... ...s called Mouth (Peh). The part where the Light enters is called Body (Guf). The line exerting a restriction, inside the Guf , is called Navel (Tabu... ...t’s pressure by twenty percent below the Peh into the Guf, until it reaches the line of the Tabur. When the Light is expelled from the Guf, the Scree... ...hen the turn of the last Partzuf arrives, its lower part crosses the separating line, the Barrier (Machsom) between the spiritual world and our worl... ...making comparisons between spiritual and material laws. He even wrote a book on geometry. Perceiving one of the highest spiritual worlds, he was abl... ...riction. A new border for the spreading of the Light is being formed along this line: the Parsa. This border was located before in the Sium of Galga... ...elfish properties; immediately Malchut, ascending upward, covers it and forms a line, called Parsa, to limit the downward spreading of the Light. Th...

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...s s ds, Anumbertheoreticfunctionanditsmeanvalue 1 21 we move the integral line from s = 3 2 ±iT to s = 1 2 ±iT. This time, the function g(s) = ζ(s) (... ...ar number, because there are close relations between these numbers and the geometry. Now for any positive integern, we definearithmeticalfunctionc(n)as... ...perfectk thpowernumbers 93 To estimate the main term, we move the integral line fromb = 1 k + 1 logx to a = 1 2k + 1 logx Therefore, 1 2πi Z b+iT b−... ...ainterm 1 2πi Z 5 2 +iT 5 2 −iT ζ 2 (s)R(s) x s s ds, we move the integral line from s = 5 2 ±iT to s = 1 2 ±iT. This time, the function ζ 2 (s)R(s) x... ...s s ds, Onthemeanvalueofanewarithmeticalfunction 147 we move the integral line from s = 2±iT to s = 1 2 ±iT. This time, the function f(s) = ζ(s)x s ζ...

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...nd 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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...cover is a representation of an s-manifold illustrating some of the behavior of lines in an s-manifold. This book has been peer reviewed and re... ... a manner consistent with his unique point of view, he defined several types of geometry that are purposefully not Euclidean and that focus on struc... ... focus on structures that the rest of us can use to enhance our understanding of geometry in general. To most of us, Euclidean geometry seems self... ... and continued with Riemann, when they found counterexamples to the notion that geometry is precisely Euclidean geometry. This opened a whole univer... ...sely Euclidean geometry. This opened a whole universe of possibilities for what geometry could be, and many years later, Smarandache’s imagination h... ...s now generally called hyperbolic geometry. Compared to Euclidean geometry, the lines in hyperbolic geometry are less prone to intersecting one anot... ...n the slightest change upsets the delicate balance of parallelism for Euclidean lines, parallelism of hyperbolic lines is distinctly more robust. On... ...rbolic lines is distinctly more robust. On the other hand, it is impossible for lines to be parallel in Riemann’s geometry. It is not clear which Ri... ...otes uniformity. If we require that through any two points there is exactly one line, for example, then all points share this property. Each axiom o...

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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 ...

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...n gave to the Prussian Academy of Sciences in 1921, he said the following: "Geometry predicates nothing about relations of real things, but only ge... ...tor is α x = (t,x,y,z) and it defines a path in space/time called a world line. The 4-velocity is α u = ∂ x dt d and the 4-acceleration is . α ... ...) 00 2 1 g c w − = is gravitational potential, 00 0 g g c v i i − = is the linear velocity of the space rotation, k i ik dx dx h d = 2 σ is the squ... ...ong imaginary world-trajectories σ τ d cd < , having imaginary relativistic\linebreak masses 1 v 2 2 0 − = c im m (tachyons). World- lines of the bo... ...ectories represented in the terms of observable quantities are pure spatial lines of imaginary three- dimensional lengths σ d , although being taken ... ...ther, is not invariant ∗ Any space of Riemannian geometry has the strictly non-degenerated metric by definition of such me... ... 2 ≠ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = = c c U U g U U k i ik k i so that looking Riemannian geometry for a regular observer, the real geometry of zero-space within ... ...nsolved Problems, Conjectures, and Theorems in Number Theory and 41 Geometry", edited by M. L. Perez, 86 p., Xiquan Publishing House, Phoeni... ... turn destroys local geometry. Furthermore, quantum field theory (QFT) is a linear theory which is being used to describe non- linear data. Namely, ...

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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 ...

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... contains short notes or articles, as well as studies on several topics of Geometry and Number theory. The material is divided into ve chapters: Geom... ...mes have an arithmetic character (as 9-12) on number theoretic problems in Geometry. Chapter 2 includes various diophantine equations, some of which a... .... . . . . . . . . . . . . . . . . . . . . 27 9 On certain constants in the geometry of equilateral triangle . . . . . . . . . 35 10 The area of a Pyth... .... . . . . . . . . . . . . . . . . . . . . . 42 12 An arithmetic problem in geometry . . . . . . . . . . . . . . . . . . . . . . 53 Chapter 2. Diophant... ...ctly convex on h 0; 4 i , 96 therefore it’s graph is belong the segment line y = y passing on the points (0,0) and 4 ; 1 . Therefore tg ... ... equality for a = 1, 4, and a = prime (see [3]), we have the following one-line proof of this result: S(2 k 1 (2 k 1)) = maxfS(2 k 1 );S(2 k 1)g... ...2p and 3p there is at least a prime, implying P (p 2 ) 3p 1. On the same lines, for su ciently large p, there is a prime between mp and (m + 1)p. Th... ...h prime pjA we have also pjB. The left-side inequality follows by the same lines. Write now b =qa, q 1 and apply (22): (b) = (qa) q (a) = b a (a) ... ... (k + 1) (k + 1) (2k + 2) +O(x k ); k> 1 (55) The proof runs on the same lines, based on ’ k (n) = X d 2 l=n (d) k (l) 209 and X n x k (n) = ...

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...ughout the history, starting from the Near East Bronze Age until our days. The reader is expected to be familiar with basics of the Flower of Life geometry. Elementary knowledge of the ancient Near East history and history of mathematics is helpful. Knowledge of visual arts, geometry, and comparative religion studies are also useful. I hope that my independent resear...

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...Introduction Foreword Preface Part 1: Introspections From a point to a line From a line to a plane From a plane to the Flower of Life Part 2: Meaning of the symbol Previous work Pondering the name Hittite sun discs Fleur-de-lis as a part of the Tree of Life ...

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..., 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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...The construction of regular maps on surfaces, related maps with groups and geometry is a glimmer of the map theory with other mathematics. In fact, ma... ...zation or making combinatorial refinement for the Riemann surfaces, Riemann geometry and finally, the Smarandache geometries by maps is a very interesti... ...ntribution of combinatorics to classical mathematics, such as, the Riemann geometry and Smarandache geometries is realized. Those are the wish of mine... ...able, 2−g(S)−k(S). if Sis non−orientable. 1.1.2 Definition of a Smarandache geometry By the history, we know that classical geometries include the Eucl... ...Each of the later two is proposed by denial the 5th postulate for parallel lines in the Euclid postu- lates of geometry. The Smarandache geometries is... ...the embedding sequence of the graph Γ. Define a functionF(x ¯ ,y ¯ ) to bey-linear if it can be represented as the following form F(x ¯ ,y ¯ ) =f(x 1 ,... ...x 2 ,···) Λ∈O Λ(y ¯ ), where, I denotes a subset of index and O a set of linear operators. Notice that f(x 1 ,x 2 ,···) =F(x ¯ ,0 ¯ ), where 0 ¯ = (... ...(Γ) |Aut1 2 Γ| g ¯ (Γ)) =0. Even more, assume the functionF(x ¯ ,y ¯ ) isy-linear and 2ε(Γ) |Aut 1 2 Γ| ,Γ∈H is a constant. If the embedding sequences... ...x ¯ , 2ε(Γ) |Aut1 2 Γ| g ¯ (Γ))= 0. Now assume that F H (x ¯ ,y ¯ ) is a y-linear function and has the following form F H (x ¯ ,y ¯ ) =f(x 1 ,x 2 ,···...

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...nal 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,...

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... . 22 Special Smarandache Curves in the Euclidean Space BY AHMAD T. ALI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 The H-Line Signed Graph of a Signed Graph BY R.RANGARAJAN, M. S. SUBRAMANYA AND P. SIVA KOTA REDDY. . . .37 Min-Max Dom-Saturation Number of a Tree BY S. ARUMUGAM AND S. SUDHA . . . . . . . . . . . . . . . . . . . . . . . 45 Em...

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...nal 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,...

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...s On (∈ vq)- Fuzzy Bigroup BY AKINOLA L.S. and AGBOOLA A.A.A. . . . . . . . . . . . . . . . . . . . . . . . . . . 01 Connectivity of Smarandachely Line Splitting Graphs BY B.BASAVANAGOUD and VEENA MATHAD . . . . . . . . . . . . . . . . . . . . . . 08 Separation for Triple-Harmonic Di?erential Operator in Hilbert Space BY E.M.E.ZAYED and S.A.OMRAN . . . . . . . . . . ...

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...nal 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,...

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.... 107 Group Connectivity of 1-Edge Deletable IM-Extendable Graphs BY KEKE WANG, RONGXIA HAO AND JIANBING LIU . . . . . . . . . . . . 113 A Note On Line Graphs BY P.SIVA KOTA REDDY, KAVITA. S. PERMI AND B.PRASHANTH. . . . . . . 119 One-Mother Vertex Graphs BY F.SALAMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123...

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...nal 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,...

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... . . . 85 Weak and Strong Reinforcement Number For a Graph BY PINAR DUNDAR, TUFAN TURACI and DERYA DOGAN. . . . . . . . . . . . . . 91 Tulgeity of Line, Middle and Total Graph of Wheel Graph Families BY AKBAR ALI.M.M, S.PANAYAPPAN and VERNOLD VIVIN.J . . . . . . . . . . . . 98 Labeling, Covering and Decomposing of Graphs – Smarandache’s Notion in Graph Theory BY LINF...

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...ortance that our government protect the right of every citizen to have access to knowledge without regard to an individual’s ability to pay.? -- Time Line of Information Technology Revolutions -- Epilogue -- Acknowledgments -- About the Author -- About Michael Hart -- Bibliography -- Index --...

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...ss to knowledge without regard to an individual’s ability to pay.‖ Time Line of Information Technology Revolutions Epilogue Acknowledgments ... ...od, heads back to the hive, then dances on the wall in circles or straight lines to point others to where it‘s been. Other birds and beasts make s... ...emorized lists of spaces with visual details to bring to mind thousands of lines of poetry. Church ritual and liturgy were passed down in the same ma... ...from one aboriginal ancestral tongue and (2) split geographically along a line that runs roughly from Scandinavia to Greece and almost touches the B... ... and seed the tongues of today’s speakers of Indo-European languages? A line in map connecting the Baltic Sea and India seems to demark halfway be... ...oved on to the studies of:  arithmetic—math applied to basic numbers  geometry—math applied to space  music—math applied to the time and sound ... ... is zero. Similarly, famed Greek mathematicians, who specialized in the geometry of areas and volumes, thought it a silly exercise to try to subtr... ... any void spooked the early Greeks. All modern mathematics— math beyond geometry— fixes the zero as the starting linchpin in plus and minus sequen...

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...book Frate cu meridianele _i paralelele, Vol. IV, OffsetColor, Rm. Vâlcea, Romania, pp. 201-202, 2008 ...........................................175 GEOMETRY ..................................178 21. Nine Solved and Nine Open Problems in Elementary Geometry, by Florentin Smarandache, extended version of some ideas from author’s books Problèmes avec et sans.. problèmes!, ...

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...book Frate cu meridianele _i paralelele, Vol. IV, OffsetColor, Rm. Vâlcea, Romania, pp. 201-202, 2008 ...........................................175 GEOMETRY ..................................178 21. Nine Solved and Nine Open Problems in Elementary Geometry, by Florentin Smarandache, extended version of some ideas from author’s books Problèmes avec et sans.. problèmes!, ...