Real Examples of NeutroGeometry & AntiGeometry

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Publisher : Infinite Study
ISBN 13 :
Total Pages : 8 pages
Book Rating : 4./5 ( download)

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Book Synopsis Real Examples of NeutroGeometry & AntiGeometry by : Florentin Smarandache

Download or read book Real Examples of NeutroGeometry & AntiGeometry written by Florentin Smarandache and published by Infinite Study. This book was released on 2023-01-01 with total page 8 pages. Available in PDF, EPUB and Kindle. Book excerpt: For the classical Geometry, in a geometrical space, all items (concepts, axioms, theorems, etc.) are totally (100%) true. But, in the real world, many items are not totally true. The NeutroGeometry is a geometrical space that has some items that are only partially true (and partially indeterminate, and partially false), and no item that is totally false. The AntiGeometry is a geometrical space that has some item that are totally (100%) false. While the Non-Euclidean Geometries [hyperbolic and elliptic geometries] resulted from the total negation of only one specific axiom (Euclid’s Fifth Postulate), the AntiGeometry results from the total negation of any axiom [and in general: theorem, concept, idea etc.] and even of more axioms [theorem, concept, idea, etc.] and in general from any geometric axiomatic system (Euclid’s five postulates, Hilbert’s 20 axioms, etc.), and the NeutroAxiom results from the partial negation of any axiom (or concept, theorem, idea, etc.). Clearly, the AntiGeometry is a generalization of Non-Euclidean Geometries.

Applications of NeutroGeometry and AntiGeometry in Real World

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Publisher : Infinite Study
ISBN 13 :
Total Pages : 19 pages
Book Rating : 4./5 ( download)

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Book Synopsis Applications of NeutroGeometry and AntiGeometry in Real World by : Erick González-Caballero

Download or read book Applications of NeutroGeometry and AntiGeometry in Real World written by Erick González-Caballero and published by Infinite Study. This book was released on 2023-01-01 with total page 19 pages. Available in PDF, EPUB and Kindle. Book excerpt: NeutroGeometries are those geometric structures where at least one definition, axiom, property, theorem, among others, is only partially satisfied. In AntiGeometries at least one of these concepts is never satisfied. Smarandache Geometry is a geometric structure where at least one axiom or theorem behaves differently in the same space, either partially true and partially false, or totally false but its negation done in many ways. This paper offers examples in images of nature, everyday objects, and celestial bodies where the existence of Smarandechean or NeutroGeometric structures in our universe is revealed. On the other hand, a practical study of surfaces with characteristics of NeutroGeometry is carried out, based on the properties or more specifically NeutroProperties of the famous quadrilaterals of Saccheri and Lambert on these surfaces. The article contributes to demonstrating the importance of building a theory such as NeutroGeometries or Smarandache Geometries because it would allow us to study geometric structures where the well-known Euclidean, Hyperbolic or Elliptic geometries are not enough to capture properties of elements that are part of the universe, but they have sense only within a NeutroGeometric framework. It also offers an axiomatic option to the Riemannian idea of Two-Dimensional Manifolds. In turn, we prove some properties of the NeutroGeometries and the materialization of the symmetric triad , , and .

Applications of NeutroGeometry and AntiGeometry in Real World

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Publisher : Infinite Study
ISBN 13 :
Total Pages : 19 pages
Book Rating : 4./5 ( download)

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Book Synopsis Applications of NeutroGeometry and AntiGeometry in Real World by : Erick Gonzalez-Caballero

Download or read book Applications of NeutroGeometry and AntiGeometry in Real World written by Erick Gonzalez-Caballero and published by Infinite Study. This book was released on 2023-01-01 with total page 19 pages. Available in PDF, EPUB and Kindle. Book excerpt: NeutroGeometries are those geometric structures where at least one definition, axiom, property, theorem, among others, is only partially satisfied. In AntiGeometries at least one of these concepts is never satisfied. Smarandache Geometry is a geometric structure where at least one axiom or theorem behaves differently in the same space, either partially true and partially false, or totally false but its negation done in many ways. This paper offers examples in images of nature, everyday objects, and celestial bodies where the existence of Smarandechean or NeutroGeometric structures in our universe is revealed. On the other hand, a practical study of surfaces with characteristics of NeutroGeometry is carried out, based on the properties or more specifically NeutroProperties of the famous quadrilaterals of Saccheri and Lambert on these surfaces. The article contributes to demonstrating the importance of building a theory such as NeutroGeometries or Smarandache Geometries because it would allow us to study geometric structures where the well-known Euclidean, Hyperbolic or Elliptic geometries are not enough to capture properties of elements that are part of the universe, but they have sense only within a NeutroGeometric framework. It also offers an axiomatic option to the Riemannian idea of Two-Dimensional Manifolds. In turn, we prove some properties of the NeutroGeometries and the materialization of the symmetric triad ,

A note on AntiGeometry and NeutroGeometry and their application to real life

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Publisher : Infinite Study
ISBN 13 :
Total Pages : 15 pages
Book Rating : 4./5 ( download)

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Book Synopsis A note on AntiGeometry and NeutroGeometry and their application to real life by : Carlos Granados

Download or read book A note on AntiGeometry and NeutroGeometry and their application to real life written by Carlos Granados and published by Infinite Study. This book was released on 2022-04-27 with total page 15 pages. Available in PDF, EPUB and Kindle. Book excerpt: Dealing with NeutroGeometry in true, false, and uncertain regions is becoming of great interested for researchers. Not too many studies have been done on this topic, for that reason, aim of this work is to define a new method to deal with NeutroGeometry in true, false, and neutrogeometry (T,C,I,F). Furthermore, some real-life application examples in 3D computer graphics, Astrophysics, nanostructure, neutrolaw, neutrogender, neutrocitation, neutrohealth-food, neutroenvironment and quantum space are presented.

Theory of Distances in NeutroGeometry

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Publisher : Infinite Study
ISBN 13 :
Total Pages : 11 pages
Book Rating : 4./5 ( download)

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Book Synopsis Theory of Distances in NeutroGeometry by :

Download or read book Theory of Distances in NeutroGeometry written by and published by Infinite Study. This book was released on 2024-01-01 with total page 11 pages. Available in PDF, EPUB and Kindle. Book excerpt: NeutroGeometry is one of the most recent approaches to geometry. In NeutroGeometry mod-els, the main condition is to satisfy an axiom, definition, property, operator and so on, that is neither entirely true nor entirely false. When one of these concepts is not satisfied at all it is called AntiGeometry. One of the problems that this new theory has had is the scarcity of models. Another open problem is the definition of angle and distance measurements within the framework of NeutroGeometry. This paper aims to introduce a general theory of distance measures in any NeutroGeometry. We also present an algorithm for distance measurement in real-life problems.

NeutroGeometry & AntiGeometry are alternatives and generalizations of the Non-Euclidean Geometries (revisited)

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Publisher : Infinite Study
ISBN 13 :
Total Pages : 22 pages
Book Rating : 4./5 ( download)

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Book Synopsis NeutroGeometry & AntiGeometry are alternatives and generalizations of the Non-Euclidean Geometries (revisited) by : Florentin Smarandache

Download or read book NeutroGeometry & AntiGeometry are alternatives and generalizations of the Non-Euclidean Geometries (revisited) written by Florentin Smarandache and published by Infinite Study. This book was released on 2021-10-01 with total page 22 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper we extend the NeutroAlgebra & AntiAlgebra to the geometric spaces, by founding the NeutroGeometry & AntiGeometry. While the Non-Euclidean Geometries resulted from the total negation of one specific axiom (Euclid’s Fifth Postulate), the AntiGeometry results from the total negation of any axiom or even of more axioms from any geometric axiomatic system (Euclid’s, Hilbert’s, etc.) and from any type of geometry such as (Euclidean, Projective, Finite, Affine, Differential, Algebraic, Complex, Discrete, Computational, Molecular, Convex, etc.) Geometry, and the NeutroGeometry results from the partial negation of one or more axioms [and no total negation of no axiom] from any geometric axiomatic system and from any type of geometry. Generally, instead of a classical geometric Axiom, one may take any classical geometric Theorem from any axiomatic system and from any type of geometry, and transform it by NeutroSophication or AntiSophication into a NeutroTheorem or AntiTheorem respectively in order to construct a NeutroGeometry or AntiGeometry. Therefore, the NeutroGeometry and AntiGeometry are respectively alternatives and generalizations of the Non-Euclidean Geometries. In the second part, we recall the evolution from Paradoxism to Neutrosophy, then to NeutroAlgebra & AntiAlgebra, afterwards to NeutroGeometry & AntiGeometry, and in general to NeutroStructure & AntiStructure that naturally arise in any field of knowledge. At the end, we present applications of many NeutroStructures in our real world.

Research on the topics of neutrosophic operations research

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Publisher : Infinite Study
ISBN 13 :
Total Pages : 366 pages
Book Rating : 4./5 ( download)

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Book Synopsis Research on the topics of neutrosophic operations research by : Florentin Smarandache

Download or read book Research on the topics of neutrosophic operations research written by Florentin Smarandache and published by Infinite Study. This book was released on 2023-08-10 with total page 366 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this volume, we present a set of research that was published in cooperation with a number of researchers and those interested in keeping pace with the great scientific development that our contemporary world is witnessing, and one of its products was neutrosophic science, which was founded by the American scientist and mathematical philosopher Florentin Smarandache in 1995. Through it, we present a new vision for some research methods. Operations research to the concepts of this science.

NeutroGeometry, NeutroAlgebra, and SuperHyperAlgebra in Today's World

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Author :
Publisher : IGI Global
ISBN 13 : 1668447428
Total Pages : 280 pages
Book Rating : 4.6/5 (684 download)

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Book Synopsis NeutroGeometry, NeutroAlgebra, and SuperHyperAlgebra in Today's World by : Smarandache, Florentin

Download or read book NeutroGeometry, NeutroAlgebra, and SuperHyperAlgebra in Today's World written by Smarandache, Florentin and published by IGI Global. This book was released on 2023-05-15 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt: NeutroAlgebra and AntiAlgebra were extended to NeutroGeometry and AntiGeometry in order to catch the Non-Euclidean Geometries. In the real world, the spaces and the elements that populate them and the rules that apply to them are not perfect, uniform, homogeneous, or complete. They are fragmentary and disparate, with unclear and conflicting information, and they do not apply in the same degree to each element. Therefore, these partial, hybrid, and mixed structures are necessary. NeutroGeometry, NeutroAlgebra, and SuperHyperAlgebra in Today's World presents applications of many NeutroStructures in our real world and considers NeutroGeometry and AntiGeometry as new fields of research that resemble everyday life. Covering key topics such as hyperbolic geometry, elliptic geometry, and AntiGeometry, this reference work is ideal for mathematicians, industry professionals, researchers, scholars, academicians, practitioners, instructors, and students.

Theory and Applications of NeutroAlgebras as Generalizations of Classical Algebras

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Publisher : IGI Global
ISBN 13 : 1668434970
Total Pages : 333 pages
Book Rating : 4.6/5 (684 download)

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Book Synopsis Theory and Applications of NeutroAlgebras as Generalizations of Classical Algebras by : Smarandache, Florentin

Download or read book Theory and Applications of NeutroAlgebras as Generalizations of Classical Algebras written by Smarandache, Florentin and published by IGI Global. This book was released on 2022-04-15 with total page 333 pages. Available in PDF, EPUB and Kindle. Book excerpt: Neutrosophy is a new branch of philosophy that studies the origin, nature, and scope of neutralities as well as their interactions with different ideational spectra. In all classical algebraic structures, the law of compositions on a given set are well-defined, but this is a restrictive case because there are situations in science where a law of composition defined on a set may be only partially defined and partially undefined, which we call NeutroDefined, or totally undefined, which we call AntiDefined. Theory and Applications of NeutroAlgebras as Generalizations of Classical Algebra introduces NeutroAlgebra, an emerging field of research. This book provides a comprehensive collection of original work related to NeutroAlgebra and covers topics such as image retrieval, mathematical morphology, and NeutroAlgebraic structure. It is an essential resource for philosophers, mathematicians, researchers, educators and students of higher education, and academicians.

Neutrosophic Sets and Systems, vol. 49/2022

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Publisher : Infinite Study
ISBN 13 :
Total Pages : 611 pages
Book Rating : 4./5 ( download)

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Book Synopsis Neutrosophic Sets and Systems, vol. 49/2022 by : Florentin Smarandache

Download or read book Neutrosophic Sets and Systems, vol. 49/2022 written by Florentin Smarandache and published by Infinite Study. This book was released on 2022-04-01 with total page 611 pages. Available in PDF, EPUB and Kindle. Book excerpt: “Neutrosophic Sets and Systems” has been created for publications on advanced studies in neutrosophy, neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc. Neutrosophy is a new branch of philosophy that studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. This theory considers every notion or idea together with its opposite or negation and with their spectrum of neutralities in between them (i.e. notions or ideas supporting neither nor ). The and ideas together are referred to as . Neutrosophy is a generalization of Hegel's dialectics (the last one is based on and only). According to this theory every idea tends to be neutralized and balanced by and ideas - as a state of equilibrium. In a classical way , , are disjoint two by two. But, since in many cases the borders between notions are vague, imprecise, Sorites, it is possible that , , (and of course) have common parts two by two, or even all three of them as well. Neutrosophic Set and Neutrosophic Logic are generalizations of the fuzzy set and respectively fuzzy logic (especially of intuitionistic fuzzy set and respectively intuitionistic fuzzy logic).

Nidus Idearum. Scilogs, X: via neutrosophica

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Publisher : Infinite Study
ISBN 13 :
Total Pages : 99 pages
Book Rating : 4./5 ( download)

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Book Synopsis Nidus Idearum. Scilogs, X: via neutrosophica by : Florentin Smarandache

Download or read book Nidus Idearum. Scilogs, X: via neutrosophica written by Florentin Smarandache and published by Infinite Study. This book was released on 2022-11-01 with total page 99 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this tenth book of scilogs – called via neutrosophica (the neutrosophic way) –, one may find new and old questions and solutions, referring mostly to topics on NEUTROSOPHY, but also MULTISPACE, with miscellaneous addition of topics on Physics, Mathematics, or Sociology – email messages to research colleagues, or replies, notes about authors, articles, or books, spontaneous ideas, and so on. Exchanging ideas with A. Elhassouny, Junhui Kim, Jeong Gon Lee, Kul Hur, Hojjatollah Farahani, W. B. Vasantha Kandasamy, Said Broumi, Mumtaz Ali, Mohamed Abdel-Basset, Ozen Ozer, Madad Khan, Gheorghe Săvoiu, John Mordeson, Adesina Agboola, Waldyr Rodrigues, Ajay Sharma, Stephen Crothers, Vlad, Dmitri Rabounski, Victor Christianto, Trung Duyên, Mirela Teodorescu, Ioan Aurel Pop.

Neutrosophic Sets and Systems, Vol. 46, 2021

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Publisher : Infinite Study
ISBN 13 :
Total Pages : 487 pages
Book Rating : 4./5 ( download)

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Book Synopsis Neutrosophic Sets and Systems, Vol. 46, 2021 by : Florentin Smarandache

Download or read book Neutrosophic Sets and Systems, Vol. 46, 2021 written by Florentin Smarandache and published by Infinite Study. This book was released on 2021-10-19 with total page 487 pages. Available in PDF, EPUB and Kindle. Book excerpt: Papers on neutrosophic programming, neutrosophic hypersoft set, neutrosophic topological spaces, NeutroAlgebra, NeutroGeometry, AntiGeometry, NeutroNearRings, neutrosophic differential equations, etc.

Collected Papers. Volume IX

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Publisher : Infinite Study
ISBN 13 :
Total Pages : 1008 pages
Book Rating : 4./5 ( download)

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Book Synopsis Collected Papers. Volume IX by : Florentin Smarandache

Download or read book Collected Papers. Volume IX written by Florentin Smarandache and published by Infinite Study. This book was released on 2022-05-10 with total page 1008 pages. Available in PDF, EPUB and Kindle. Book excerpt: This ninth volume of Collected Papers includes 87 papers comprising 982 pages on Neutrosophic Theory and its applications in Algebra, written between 2014-2022 by the author alone or in collaboration with the following 81 co-authors (alphabetically ordered) from 19 countries: E.O. Adeleke, A.A.A. Agboola, Ahmed B. Al-Nafee, Ahmed Mostafa Khalil, Akbar Rezaei, S.A. Akinleye, Ali Hassan, Mumtaz Ali, Rajab Ali Borzooei , Assia Bakali, Cenap Özel, Victor Christianto, Chunxin Bo, Rakhal Das, Bijan Davvaz, R. Dhavaseelan, B. Elavarasan, Fahad Alsharari, T. Gharibah, Hina Gulzar, Hashem Bordbar, Le Hoang Son, Emmanuel Ilojide, Tèmítópé Gbóláhàn Jaíyéolá, M. Karthika, Ilanthenral Kandasamy, W.B. Vasantha Kandasamy, Huma Khan, Madad Khan, Mohsin Khan, Hee Sik Kim, Seon Jeong Kim, Valeri Kromov, R. M. Latif, Madeleine Al-Tahan, Mehmat Ali Ozturk, Minghao Hu, S. Mirvakili, Mohammad Abobala, Mohammad Hamidi, Mohammed Abdel-Sattar, Mohammed A. Al Shumrani, Mohamed Talea, Muhammad Akram, Muhammad Aslam, Muhammad Aslam Malik, Muhammad Gulistan, Muhammad Shabir, G. Muhiuddin, Memudu Olaposi Olatinwo, Osman Anis, Choonkil Park, M. Parimala, Ping Li, K. Porselvi, D. Preethi, S. Rajareega, N. Rajesh, Udhayakumar Ramalingam, Riad K. Al-Hamido, Yaser Saber, Arsham Borumand Saeid, Saeid Jafari, Said Broumi, A.A. Salama, Ganeshsree Selvachandran, Songtao Shao, Seok-Zun Song, Tahsin Oner, M. Mohseni Takallo, Binod Chandra Tripathy, Tugce Katican, J. Vimala, Xiaohong Zhang, Xiaoyan Mao, Xiaoying Wu, Xingliang Liang, Xin Zhou, Yingcang Ma, Young Bae Jun, Juanjuan Zhang.

NeutroGeometry, NeutroAlgebra, and SuperHyperAlgebra in Today's World

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Publisher :
ISBN 13 : 9781668447413
Total Pages : 0 pages
Book Rating : 4.4/5 (474 download)

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Book Synopsis NeutroGeometry, NeutroAlgebra, and SuperHyperAlgebra in Today's World by : Madeline Al Tahan

Download or read book NeutroGeometry, NeutroAlgebra, and SuperHyperAlgebra in Today's World written by Madeline Al Tahan and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: NeutroAlgebra and AntiAlgebra were extended to NeutroGeometry and AntiGeometry in order to catch the Non-Euclidean Geometries. In the real world, the spaces and the elements that populate them and the rules that apply to them are not perfect, uniform, homogeneous, or complete. They are fragmentary and disparate, with unclear and conflicting information, and they do not apply in the same degree to each element. Therefore, these partial, hybrid, and mixed structures are necessary. NeutroGeometry, NeutroAlgebra, and SuperHyperAlgebra in Today's World presents applications of many NeutroStructures in our real world and considers NeutroGeometry and AntiGeometry as new fields of research that resemble everyday life. Covering key topics such as hyperbolic geometry, elliptic geometry, and AntiGeometry, this reference work is ideal for mathematicians, industry professionals, researchers, scholars, academicians, practitioners, instructors, and students.

NeutroGeometry & AntiGeometry are alternatives and generalizations of the Non-Euclidean Geometries

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Publisher : Infinite Study
ISBN 13 :
Total Pages : pages
Book Rating : 4./5 ( download)

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Book Synopsis NeutroGeometry & AntiGeometry are alternatives and generalizations of the Non-Euclidean Geometries by : Florentin Smarandache

Download or read book NeutroGeometry & AntiGeometry are alternatives and generalizations of the Non-Euclidean Geometries written by Florentin Smarandache and published by Infinite Study. This book was released on with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper we extend the NeutroAlgebra & AntiAlgebra to the geometric space, by founding the NeutroGeometry & AntiGeometry. While the Non-Euclidean Geometries resulted from the total negation of only one specific axiom (Euclid’s Fifth Postulate), the AntiGeometry results from the total negation of any axiom and even of more axioms from any geometric axiomatic system (Euclid’s, Hilbert’s, etc.), and the NeutroAxiom results from the partial negation of one or more axioms [and no total negation of no axiom] from any geometric axiomatic system. Therefore, the NeutroGeometry and AntiGeometry are respectively alternatives and generalizations of the Non-Euclidean Geometries. In the second part, we recall the evolution from Paradoxism to Neutrosophy, then to NeutroAlgebra & AntiAlgebra, afterwards to NeutroGeometry & AntiGeometry, and in general to NeutroStructure & AntiStructure that naturally arise in any field of knowledge. At the end, we present applications of many NeutroStructures in our real world.

Nidus Idearum. Scilogs, XI: in-turns and out-turns

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Publisher : Infinite Study
ISBN 13 :
Total Pages : 133 pages
Book Rating : 4./5 ( download)

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Book Synopsis Nidus Idearum. Scilogs, XI: in-turns and out-turns by : Florentin Smarandache

Download or read book Nidus Idearum. Scilogs, XI: in-turns and out-turns written by Florentin Smarandache and published by Infinite Study. This book was released on 2023-05-31 with total page 133 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this eleventh book of scilogs – called in-turns and out-turns –, one may find new and old questions and solutions, referring mostly to topics on Neutrosophy, but also Multispace, with miscellaneous addition of topics on Physics, Mathematics, or Sociology – email messages to research colleagues, or replies, notes about authors, articles, or books, spontaneous ideas, and so on. Exchanging ideas with Prem Kumar Singh, Feng Liu, Nicolae Bălașa, Jimmy Quellet, Minodora Rușchița, Frank Gelli, A. R. Vătuiu, Victor Christianto, Vladimir I. Rogozhin, Robert Neil Boyd, Akira Kanda, Stefan Spaarmann, Oliver Consa, Yabin Shao, Junle Zhuo, Nivetha Martin, Said Broumi, Jean Dezert, Erick González, Tomasz Witczak, B. De Baets, I. Couso, D. Dubois, L. Good, Mihaela Colhon, Atiqe Ur Rahman, Muhammad Saeed, Muhammad Ihsan, Edeline Nancy, Shawkat Alkhazaleh, Shazia Rana, Vasantha Kandasamy, Santanu Acharjee, Mohamed Al-Shmrani, Parimala Manie, Mehmet Unver, Murat Olgun, Adel Aleidhri, Akira Kanda, Ilanthenral Kandasamy, Mircea Zărnescu, R. Tayebi Khorami, Arsham Borumand Saeid, Maisam Jdid, M. Abobala, Riad Hamido, Rafael Rojas, Esmaeil Zarei, Faisal Khan, Rouzbeh Abbassi, Adel Al-Odhari, Henry Garrett, Ackbar Rezaei, Saeid Jafari, Karthika Muthusamy, Harish Garg, Huda E. Khalid, Gonca D. Güngör, Muslim A. Noah Zainal, Fakhry Asad Agusfrianto, Madeleine Al-Tahan, Mariam Hariri, Yudi Mahatma, Masoud Ghods, Marco Brigliadori, Luis Enrique Aponte Pérez, Amr Mohammed, M. Aslam (in order of reference in the book).

Nidus Idearum. Scilogs, XII: seed & heed

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Publisher : Infinite Study
ISBN 13 :
Total Pages : 99 pages
Book Rating : 4./5 ( download)

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Book Synopsis Nidus Idearum. Scilogs, XII: seed & heed by : Florentin Smarandache

Download or read book Nidus Idearum. Scilogs, XII: seed & heed written by Florentin Smarandache and published by Infinite Study. This book was released on 2023-06-01 with total page 99 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this twelfth book of scilogs – called seed & heed –, one may find topics on Neutrosophy, Superluminal Physics, Mathematics, Information Fusion, Philosophy, or Sociology – email messages to research colleagues, or replies, notes, comments, remarks about authors, articles, or books, spontaneous ideas, and so on. Exchanging ideas with Pritpal Singh, Mohamed Abobala, Muhammad Aslam, Ervin Goldfain, Dmitri Rabounski, Victor Christianto, Steven Crothers, Jean Dezert, Tomasz Witczak (in order of reference in the book).