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...Geometry, Orthological Triangles, Tangential Triangle, Contact Triangle, Cotangent Triangle, Anti-Supplementary Triangle, Podal Triangle, Three Images Triangle, Cyclocevian Triangle, Fuhrmann Triangle, Degenerate Triangles, O...

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... Ion Pătrașcu Florentin SmarandacheTHE GEOMETRY OF THE ORTHOLOGICAL TRIANGLESPeer Reviewers:Prof. dr. Bencz... ...ania Ion Pătrașcu Florentin Smarandache THE GEOMETRY OF THEORTHOLOGICAL TRIANGL... ...7 Ion Pătrașcu, Florentin Smarandache 5 THE GEOMETRY OF THE ORTHOLOGICAL TRIANGLESTABLE OF CONTENTS2 ORTHOLOGICA... ....................... 496 Ion Pătrașcu, Florentin Smarandache THE GEOMETRY OF THE ORTHOLOGICAL TRIANGLES ... ...0 Ion Pătrașcu, Florentin Smarandache 7 THE GEOMETRY OF THE ORTHOLOGICAL TRIANGLESTABLE OF CONTENTSOBSERVATION 1... ... two authors have designed this work that seeks to provide an integrativeimage of elementary geometry by the prism of this "filter". Basically, the p... ...to graft on the central theme - the orthology of triangles-, manyresults from the elementary geometry. In particular, we approach theconnection betwee... ...o graft on the central theme, of theorthological triangles, many results from the elementarygeometry. In particular, we approach the connection betwee...

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...of New Planetoids in the Solar System and Quantization of Celestial System, by V. Christianto, F. Smarandache..................28 3. Open and Solved Elementary Questions in Astronomy, by Florentin Smarandache.. 36 BIOLOGY......................................40 4. Statistical Modeling of Primary Ewing Tumors of the Bone, by Sreepurna Malakar, Florentin Smarandache, Suka...

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...onomics and business and politics, education and administration, game theory,geometry, graph theory, information fusion, neutrosophic logic and set, n... ... graph theory, information fusion, neutrosophic logic and set, non-Euclidean geometry,number theory, paradoxes, philosophy of science, psychology, qua... ...iplinary method of study and research, notbeing ashamed to ask and seek even elementary or impossible questions.Thus, Neutrosophic Transdisciplinarity... ...f the operator of S-denying a proposition, or an axiomatic system, from geometry to respectively any theory in a given domain of knowledge. By S-... ...em, by V. Christianto, F. Smarandache…………………………………28 3. Open and Solved Elementary Questions in Astronomy, by Florentin Smarandache… 36 ... ...75 7 GEOMETRY …………………………………………………………………178 21. Nine Solved and Nine Open ... ... …………………………………………………………………178 21. Nine Solved and Nine Open Problems in Elementary Geometry, by Florentin Smarandache, extended version of... ...……694 87. Unleashing the Quark within: LENR, Klein-Gordon Equation, and Elementary Particle Physics, preliminary report by Florentin Smara... ... 35 OPEN AND SOLVED ELEMENTARY QUESTIONS IN ASTRONOMY Florentin Smarand...

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...of New Planetoids in the Solar System and Quantization of Celestial System, by V. Christianto, F. Smarandache..................28 3. Open and Solved Elementary Questions in Astronomy, by Florentin Smarandache.. 36 BIOLOGY......................................40 4. Statistical Modeling of Primary Ewing Tumors of the Bone, by Sreepurna Malakar, Florentin Smarandache, Suka...

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...onomics and business and politics, education and administration, game theory,geometry, graph theory, information fusion, neutrosophic logic and set, n... ... graph theory, information fusion, neutrosophic logic and set, non-Euclidean geometry,number theory, paradoxes, philosophy of science, psychology, qua... ...iplinary method of study and research, notbeing ashamed to ask and seek even elementary or impossible questions.Thus, Neutrosophic Transdisciplinarity... ...f the operator of S-denying a proposition, or an axiomatic system, from geometry to respectively any theory in a given domain of knowledge. By S-... ...em, by V. Christianto, F. Smarandache…………………………………28 3. Open and Solved Elementary Questions in Astronomy, by Florentin Smarandache… 36 ... ...75 7 GEOMETRY …………………………………………………………………178 21. Nine Solved and Nine Open ... ... …………………………………………………………………178 21. Nine Solved and Nine Open Problems in Elementary Geometry, by Florentin Smarandache, extended version of... ...……694 87. Unleashing the Quark within: LENR, Klein-Gordon Equation, and Elementary Particle Physics, preliminary report by Florentin Smara... ... 35 OPEN AND SOLVED ELEMENTARY QUESTIONS IN ASTRONOMY Florentin Smarand...

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... . . . . . . . . . . . . . . . . 85 2.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Chapter 3. Topology with Smarandache Geometry . . . . . . . . . . . . . . 91 3.1 Algebraic topology . . . . . . . . . . . . . . . . . . . . 92 3.1.1 Topological space. . . . . . . . . . . . . . . . . . . . . . . .92 3.1.2 Metric space . . . . . . . . . . . . ...

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... LINFAN MAO COMBINATORIAL GEOMETRY WITH APPLICATIONS TO FIELD ... ...il: maolinfan@163.com Combinatorial Geometry with Applications to Field ... ...ushroomed in lastcentury. Its related with algebra, probability theory and geometry has made it to animportant subject in mathematics and interesting ... ...includes the development of a fewii Combinatorial Geometry with Applications to Fieldtopics discussed in the first edition. ... ...tional fields and how to constructcombinatorial gauge basis and fields. Elementary applications of combinatorial fieldsto many-body mechanics, ... ...≤ m}. f e For the case of multi-groups (A ; O), some elementary properties are presented f ein [Mao3], particular... ...bers. We have knows these openintervals (a, b) for a ≤ b, a, b ∈ R in elementary mathematics. Define open sets in Rto be a union of finite open... ...d neighborhoods Ux that satisfies thecondition just stated is called an elementary neighborhood and p is often called a eprojection from... ...is example, each opened subinterval on S1 serves as an elementaryneighborhood.Definition 3.1.5 Let S, T be topological spaces, x...

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...The authors approach several themes of classical geometry of the circle and complete them with some original results, showing that not everything in traditional math is revealed, and that it still has an open character. The topics were chosen according to authors’ aspiratio...

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...n Patrascu | Florentin Smarandache Complements to Classic Topics of Circles Geometry Pons Editions ... ... Brussels | 2016Complements to Classic Topics of Circles Geometry Ion Patrascu | Florentin Smarandache Complements to... ...orentin Smarandache Complements to Classic Topics of Circles Geometry 1Ion Patrascu, Florentin Smarandache In the me... ... 2 Complements to Classic Topics of Circles Geometry Ion Patrascu | Florentin SmarandacheComplements to Classic Topic... ...ascu | Florentin SmarandacheComplements to Classic Topics of Circles Geometry Pons Editions Brussels | 2016 ... ...ing the first theorem, C. Cosnita suggests an elementary proof by employing the concept of polar. In the followi... ...he twocircles, the points 𝐾, 𝐾, 𝐾 are collinear.Remarks. 1. Another elementary proof of this theorem is to be found in [3] . 2. If...

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... mathematical generalizations to the complex behavior of helix, rectifying, hyperquadrical, and hyperelastic curves in Galilean 3-space and Minkowski geometry, this collection serves as a testament to the dynamic interplay between theory and practice. Each chapter exemplifies the creativity and rigor of its contributors, providing both foundational knowledge and new perspe...

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... ALGORITHMS AND VARIATIONS: MODERN MATHEMATICAL THEORIES PROF. DR. AHMET YÜCESAN GEOMETRY, ALGORITHMS ... ...t his academic career, he has undertaken many important researches in the fields of differential geometry, theory of curves and surfaces, man... ... www.bztturanpublishinghouse.com /bztturanpublishinghouse GEOMETRY, ALGORITHMS AND VARIATIONS: MODERN ... ... ALGORITHMS AND VARIATIONS: MODERN MATHEMATICAL THEORIES PROF. DR. AHMET YÜCESAN GEOMETRY, ALGORITHMS ... ...t his academic career, he has undertaken many important researches in the fields of differential geometry, theory of curves and surfaces, man... ...strate their theoretical and geometric features.3 Süleyman Demirel University, Department of Division of Elementary Mathematics Edu- cation, Facult...

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...er to make the book self contained. Section two gives the properties of fuzzy matrices. Since the data need to be transformed into fuzzy models, some elementary properties of graphs are given. Further, this section provides details of how to prepare a linguistic questionnaire to make use of in these fuzzy models when the data related with the problem is unsupervised....

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...atrix addition follows theassociative law. We next show the subtraction of elementarymatrices. Just like matrix addition we see matrix subtraction isa... ...smembership value in the interval [0, 1]. But very few try tovisualize the geometry of fuzzy sets. It is not only of interest butis meaningful to see ... ...etry of fuzzy sets. It is not only of interest butis meaningful to see the geometry of fuzzy sets when we discussfuzziness. Till date researchers over... ...ional graphs with the domain X represented as a one-dimensional axis. The geometry of fuzzy sets involves both domain X =(x1,…, xn) and the range [0,... ... fuzziness,define fuzzy concepts and prove fuzzy theorems. Visualizingthis geometry may by itself provide the most powerful argumentfor fuzziness. An... ...vide the most powerful argumentfor fuzziness. An odd question reveals the geometry of fuzzy sets. Whatdoes the fuzzy power set F(2X), the set of all ...

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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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... until our days.The reader is expected to be familiar with basics of the Flower of Life geometry 2 .Elementary knowledge of the ancient... ...r is expected to be familiar with basics of the Flower of Life geometry 2 .Elementary knowledge of the ancient Near East history and his... ...ncient Near East history and history of mathematics ishelpful. Knowledge of visual arts, geometry, and comparative religion studies are also useful.I ... ... piscis, fleur-de-lis, lily symbolism, tree of life, menorah history, sacred geometry, ancient mathematics, hittite sun disc L... ...e material. Instead, it simply offers a clear,complete survey of the Flower of Life: the geometry, its origins, and history. Such a survey islong over... ...n, but wecan be sure that it was an ideal place for a conversation of the mastermind and geometrylover, Plato. In one sense Plato further developed th...

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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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... manner consistent with his unique point ofview, he defined several types of geometry that are purposefully not Euclidean and thatfocus on structures ... ...us on structures that the rest of us can use to enhance our understanding of geometryin general.To most of us, Euclidean geometry seems self-evident a... ...nd continued with Riemann, when they foundcounterexamples to the notion that geometry is precisely Euclidean geometry. Thisopened a whole universe of ... ...ly Euclidean geometry. Thisopened a whole universe of possibilities for what geometry could be, and many yearslater, Smarandache’s imagination has wan... ...earslater, Smarandache’s imagination has wandered off into this universe.The geometry associated with Gauss, Bolyai, and Lobachevski is now generally ... ... New York: Dover Publications, Inc.5. Boyce, W.E. and DiPrima, R.C. (1992). Elementary Differential Equations and Boundary Value Problems (5th ed.),...

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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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...ns. Other themes havean arithmetic character (as 9-12) on number theoretic problems in Geometry. Chapter 2includes various diophantine equations, some... ...metry. Chapter 2includes various diophantine equations, some of which are treatable by elementary meth-ods; others are partial solutions ... ... . . . . . . . . . . . . 27 9 On certain constants in the geometry of equilateral triangle . ... .... . . . . . . . . . . . . 42 12 An arithmetic problem in geometry . . . . . ... ...e triangle). For questions of priority(which is very difficult to decide in this field of Elementary Geometry - Algebra - Analysis)we note that (1) appea... .... For questions of priority(which is very difficult to decide in this field of Elementary Geometry - Algebra - Analysis)we note that (1) appears in [R. S... ...s impossible, by assumption. Certain direct results follow from the elementary connections existing between theel... ...anyi, Combinatorial and geometrical investigations in elementary number theory, Periodica Mat. Hunarica 6(3... ...ctiong a = m, b = n,c = m + p, d = n + q). For other connections of this equation to Elementary Number Theory, see [3].Bibliography 1. G.H. Hardy a...

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...inal research papers and survey articles in all aspects of Smarandachemulti-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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... generating the group or hsgg in short. In this paper we prove some results on hsggs of a group. Order class of a group are special halfsubgroupoids. Elementary abelian groups are characterized as groups with maximum special halfsubgroupoids. Order class of a group with unity forms a typical halfsubgroup. Key words: halfsubgroup, hsgg, order class of an element. AMS(20...

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...all aspects of Smarandache multi-spaces,Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topologyand their applications ... ...s those of algebraicmulti-systems, multi-metric spaces,· · · , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topologica... ...is paper we prove some results on hsggs of a group. Order class of a group are special halfsubgroupoids. Elementary abelian groups are characterized... ... agroup of odd order. Proof It is easy to verify. Theorem 1.4 An elementary Abelian p−group, p > 3 is a di... ...each of which is a cyclic p−group which is the union of distinct pure hsggs. Proof By the definition of elementary Abelian p−group, ... ...run S. MuktibodhExample 2.2 All groups with exponent p a prime are such groups.Example 2.3 All elementary Abelian groups are such groups.Ex... ...re of order p. Thus, G is a p-group. Now we prove a theorem which gives the characterization of an elementary Abelian groups.Theorem 2.2 ... ... ein a local chart (U, [ϕ]), then D is called torsion-free on (U, [ϕ]). Similar to that of Riemannian geometry, metrics on a smoothly combinatoria... ...at of Riemannian geometry, metrics on a smoothly combinatorial manifoldand the combinatorially Riemannian geometry are defined in next definition. ...

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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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...was first intiated by Vasantha Kandasamy. The objective of this paper is to present the concept of bicoset of a bivector space and obtain some of its elementary properties. Key Words: bigroup, bivector space, bicoset, bisum, direct bisum, inner biproduct space, biprojection. ...

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...i-spaces,Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topologyand their applications to other sciences. Topics in de... ...ms, multi-metric spaces,· · · , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Ran... ...present the concept of bicoset of a bivector space and obtain some of its elementary properties. Key Words: bigroup, bivector space, bicoset, bisum,... ... however, we look at the bicoset of a bivectorspace and obtain some of its elementary properties.Definition 1.1([2]) A set (G, +, ·) with two binary op... ...rong: Basic topology, McGrow-Hill(1979). [3] M. P. Docarmo:Riemannian geometry, Boston,Birkhauser(1992). [4] A. E. El-Ahmady: Fuzzy folding of ... ...of the first author in thefollowing style:Books[4]Linfan Mao, Combinatorial Geometry with Applications to Field Theory, InfoQuest Press,2009.Research p...

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... the usage of the symbol roughly from 1600 BC to 1600 AD. The reader of this document is expected to be familiar with basics of the Flower of Life geometry1. 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 my independent research p...

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...to 1600 AD.The reader of this document is expected to be familiar with basics of the Flower of Lifegeometry 1 . Elementary knowledge of the ancient N... ...ntary knowledge of the ancient Near East history and history ofmathematics is helpful. Knowledge of visual arts, geometry, and comparative religion st... ...Life, six-petal rosette, intersecting circles, equilateral triangle, concave square, apsamikku, sacred geometry Language English Publis... ...called Apsamikku is discussed inMesopotamian mathematics 1 by Eleanor Robson. See also: Some old Babylonian geometry2 by Duncan J. Melville. A s... ...t)2.1.1. Eleanor Robson: Mesopotamian mathematics 1999 Chapter 22.1.2. Duncan J. Melville: Some Old Babylonian geometry 20052.1.3. wikipedia.org: C... ...parts, which corresponds to 360 equal angles or degrees', isn't a choice that is completely independent of the geometry because this division can be...

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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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...osition of Graphs into Internally Disjoint Trees BY S.SOMASUNDARAM, A.NAGARAJAN AND G.MAHADEVAN . . . . . . . . 90 An Introduction to Combinatorial Geometry with Application to Field Theory . . 104 ...

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...all aspects of Smarandache multi-spaces,Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topologyand their applications ... ...s those of algebraicmulti-systems, multi-metric spaces,· · · , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topologica... ... two variables functions. So our intention is touse several new concepts, which are defined in differential geometry [7]. 1Received Feb.25, 2009. Accept... ...es- An Introduction with Ap- plications, Boston, Birkhauser, 2001. [6] B.O’Neill. Semi-Riemannian Geometry With Application to Relativity,... ...vings, they have appearedin human activities since prehistoric times. Their mathematical theory is mostly elementary,but nevertheless it contains a ri... ...t.104 International Journal on Mathematical Combinatorics Combinatorial Geometry with Application to Field...

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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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... . . . . . . . . . . 20 Supermagic Coverings of Some Simple Graphs BY P.JEYANTHI AND P.SELVAGOPAL . . . . . . . . . . . . . . . . . . . . . . . 33 Elementary Abelian Regular Coverings of Cube BY FURONG WANG AND LIN ZHANG. . . . . . . . . . . . . . . . . . . . . .49 Super Fibonacci Graceful Labeling of Some Special Class of Graphs BY R.SRIDEVI, S.NAVANEETHAKRISHNAN AN...

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...all aspects of Smarandache multi-spaces,Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topologyand their applications ... ...s those of algebraicmulti-systems, multi-metric spaces,· · · , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topologica... ...(1960). [6] Ekmekci, N. ve Ilarslan K., On Bertrand curves and their characterization, Differential Geometry Dynamical Systems, 3, No.2, 17-24... ...alaysian Mathematical Sciences Society, 31 (1). pp. 1-10, 2008. [8] Mellish A.P., Notes on differential geometry, Annals of Mathematics, Second ... ...ls of Mathematics, Second Series, Vol. 32, No. 1, pp. 181-190, Jan. 1931. [9] Oprea J., Differential Geometry and its Applications, Prentica Hal... ...m. AMS(2010): 05A10, 05A19§1. IntroductionBernoulli numbers are the signs of a very strong bond between elementary number theory,complex analytic nu...

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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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...he Direct Product of Theta Graphs with Cycles BY M.M.M. JARADAT and K.M.A.AL-SHORMAN . . . . . . . . . . . . . . . . . . 87 A Note on Differential Geometry of the Curves in E4 BY SUHA YILMAZ, SUUR NIZAMOGLU AND MELIH TURGUT . . . . . . . . . . . 104 The Upper and Forcing Vertex Detour Numbers of a Graph BY A.P.SANTHAKUMARAN AND P.TITUS . . . . . . . . . . . . . . . ...

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...all aspects of Smarandache multi-spaces,Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topologyand their applications ... ...s those of algebraicmulti-systems, multi-metric spaces,· · · , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topologica... ... and his special equations which play important role in mechanics andkinematics as well as in differential geometry (for more details see [2]). At the ... ...ory opened a door of use of new geometries. One ofthem, Minkowski space-time, which is simultaneously the geometry of special relativity and thegeomet... ...anifold - a specialcase of Smarandache geometries ([4]), was introduced and some of classical differential geometrytopics have been treated by the rese... ..., the basic elements of the theory of curves 4in the space E1 are briefly presented (a more complete elementary treatment can be found in[1]). ... ...0, (12) dt2 τ τwhere u(t) is a real valued function. (12) has an elementary solution. It follows that ... ...ns, here, the basic elements of the theory of curvesin the space E4 are briefly presented (a more complete elementary treatment can be found in[4]). ...

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...cal empirics have emphasized the philosophical methodological significance of the theory. Reichenbach approached the GR through the "relativity of geometry" thesis, trying to build a "constructive axiomatization" (Rendall 2005) of relativity based on "elementary matters of fact" (Elementartatbestande) for the observable behavior of light rays, rods and clocks. The ma...

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...eral relativity (GR). The inertial movement in classicalmechanics is related to the geometry of space and time, practically along geodesics in which t... ...vity. This allows the definition of a new class of bodies in afree fall, defining a geometry of space and time by a geodetic motion that depends on th... ...ic, at least up to a positive scalar factor, resultingin a conforming structure (or geometry). If gravity is taken into account, the temporal strai... ...ing an inertial framewithout gravity are curved, resulting in a change in spacetime geometry. (Schutz and Schutz 1985) Proper time measured with cl... ...ented the fieldequations 2 that include gravity, which specifies how space and time geometry is influenced bymatter and radiation.General Relativity (... ...ying to builda "constructive axiomatization" (Rendall 2005) of relativity based on "elementary matters of fact"(Elementartatbestande) for the observab... ...ard): "The supreme task of the physicist is … the search for those most general, elementary laws from which the world picture is to be obtained ... ...picture is to be obtained through pure deduction. No logical path leads to these elementary laws; it is instead just the intuition that rests on an...

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...bability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc....

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... started in 1995 and their applications in any field,such as the neutrosophic structures developed in algebra, geometry, topology, etc. The submitted... ...olombia,Kemale Veliyeva & Sadi Bayramov, Department of Algebra Email: rafael2188797@correo.uis.edu.co.and Geometry, Baku State University, 23 Z. ... ...egree of negation in (0, 1] of this axiom }:http://fs.unm.edu/Geometries.htm.36. Definition of S-geometries A geometry is called S-geometry [5] if it... ...entific orhumanistic proposition, theory, etc.). Many geometries, such as pseudo-manifold geometries, Finsler geometry, combinatorial Finslergeometri... ...Finsler geometry, combinatorial Finslergeometries, Riemann geometry, combinatorial Riemannian geometries, Weyl geometry, Kahlergeometry are particular... ...dthe neutrosophic quadruple algebraic structures, also studied neutrosophic quadruple rings andpresented their elementary properties. Basset et al. [4...

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...bability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc....

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...ation of the classicalas the neutrosophic structures developed in algebra, geometry, statistics.topology, etc. What ... ...example, Neutrosophy is proposed by Prof. Florentinfrom elementary mathematical equations develop and ex- Smarandache in 1... ...nature, and scope of neutralities, as well as theirdevelop and expand into geometry equations, trigonomet- interactions with different ideatio... ...and <Anti-A> ideas together are re-the concepts of domain of function, the geometry elements ferred to as <Non-A>.included in domain of functio... ...ke; as well as the concepts of pointquadratic equation, multiple equation, geometry equation, solution, line solution, plane solution, solid soluti... ...ble for the entire inner surface of theconcepts of domain of function, the geometry elements sphere.included in domain of function, and the like.... ...h to select the best F~ (x) : X  [0, 1] for each point x X and we have,elementary school for children using neutrosophic MADM Nwith interval ...

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...bability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc....

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...ation of the classicalas the neutrosophic structures developed in algebra, geometry, statistics.topology, etc. What ... ...ever, all weighted average operators and all geometry operators also lead to the sameApparently, there is no need to com... ...oped inwisdom] means knowledge of neutral thought. While algebra, geometry, topology etc.“neutrosophic” (adjective), means having the nature ... ...we havestudied refined neutrosophic group and we have presentedsome of its elementary properties.A.A.A. Agboola, On Refined Neutrosophic Algebraic Str...