Proceedings of the First International Conference on Smarandache Multispace & Multistructure Edited by Linfan Mao Beijing University of Civil Engineering and Architecture The Education Publisher Inc. July, 2013 Proceedings of the First International Conference On Smarandache Multispace & Multistructures (28-30 June 2013, Beijing, China) Edited by Linfan Mao Beijing University of Civil Engineering and Architecture The Education Publisher Inc. July, 2013 This proceedings can be ordered from: The Educational Publisher Inc. 1313 Chesapeake Ave. Columbus, Ohio 43212,USA Toll Free: 1-866-880-5373 E-mail: [email protected] Website: www.EduPublisher.com Peer Reviewers: Said Broumi, University of Hassan II Mohammedia, Hay El Baraka Ben M’sik, Casablanca, Morocco. Linfan Mao, Academy of Mathematics and Systems, Chinese Academy of Sciences, Beijing 100190,P.R.China. Y.P.Liu, Department of Applied Mathematics, Beijing Jiaotong University, Beijing 100044, P.R.China. Jun Zhang, School of Economic and Management Engineering, Beijing University of Civil En- gineering and Architecture, P.R.China. Ovidiu Ilie Sandru, Mathematics Department, Polytechnic University of Bucharest, Romania. W. B. Vasantha Kandasamy, Indian Institute of Technology, Chennai, Tamil Nadu, India. Copyright 2013 Beijing University of Civil Engineering and Architecture, The Education Publisher Inc., The Editor, and The Authors for their papers. Many books can be downloaded from the following Digital Library of Science: http://fs.gallup.unm.edu/eBooks-otherformats.htm ISBN: 978-1-59973-229-9 Printed in America Contents TheFirstInternationalConferenceonSmarandacheMultispaceandMultistructure was held in China......................................................................04 TheFirstInternationalConferenceonSmarandacheMultispaceandMultistructure was held in China (In Chinese).......................................................06 S-Denying a Theory By Florentin Smarandache................................................................08 Smarandache Geometry A geometry with philosophical notion (cid:4) By Linfan Mao ...........................................................................15 Neutrosophic Transdisciplinarity — Multi-Space & Multi-Structure By Florentin Smarandache................................................................22 Non-Solvable Equation Systems with Graphs Embedded in Rn By Linfan Mao ...........................................................................25 Some Properties of Birings By A.A.A.Agboola and B.Davvaz . . . . . . . . . . ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41 Surface Embeddability of Graphs via Tree-travels By Yanpei Liu............................................................................51 Surface Embeddability of Graphs via Joint Trees By Yanpei Liu............................................................................60 Surface Embeddability of Graphs via Reductions By Yanpei Liu............................................................................66 Recent Developments in Regular Maps By Shaofei Du ...........................................................................73 Smarandache Directionally n-Signed Graphs A Survey (cid:4) By P.Siva Kota Reddy ...................................................................82 Neutrosophic Diagram and Classes of Neutrosophic Paradoxes, Or To The Outer- Limits of Science By Florentin Smarandache ..............................................................92 Neutrosophic Groups and Subgroups By Agboola A.A.A., Akwu A.D. and Oyebo Y.T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Neutrosophic Rings I By Agboola A.A.A., Akinola A.D. and Oyebola O.Y......................................114 Neutrosophic Degree of a Paradoxicity By Florentin Smarandache ..............................................................128 Appendix: The Story of Mine with That of Multispaces (In Chinese) By Linfan Mao...........................................................................132 Proceedings of Conference on Multispace & Multistructure June 28-30, 2013, Beijing The First International Conference on Smarandache Multispace and Multistructure was held in China In recent decades, Smarandache’s notions of multispace and multistructure were widely spreadandhaveshownmuchimportanceinsciencesaroundtheworld. OrganizedbyProf.Linfan Mao, a professional conference on multispaces and multistructures, named the First Interna- tionalConferenceonSmarandacheMultispaceandMultistructurewasheldinBeijingUniversity ofCivilEngineeringandArchitectureofP.R.ChinaonJune28-30,2013,whichwasannounced by American Mathematical Society in advance. The Smarandache multispace and multistructure are qualitative notions, but both can be applied to metric and non-metric systems. There were 46 researchers haven taken part in this conference with 14 papers on Smarandache multispaces and geometry, birings, neutrosophy, neutrosophic groups, regular maps and topological graphs with applications to non-solvable equation systems. Prof.Yanpei Liu reports on topological graphs 1http://fs.gallup.unm.edu/Multispace.htm NewsonConference 5 Prof.Linfan Mao reports on non-solvable systems of differential equations with graphs in Rn Prof.Shaofei Du reports on regular maps with developments Applications of Smarandache multispaces and multistructures underline a combinatorial mathematicalstructureandinterchangeabilitywithothersciences,includinggravitationalfields, weak and strong interactions, traffic network, etc. All participants have showed a genuine interest on topics discussed in this conference and would like to carry these notions forward in their scientific works. 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Key Words: operator of S-denying,axiomatic system AMS(2010): 51M15, 53B15, 53B40, 57N16 §1. Introduction Let T be a theory in any domain of knowledge, endowed with an ensemble of sentences E, on a given space M. E can be for example an axiomatic system of this theory, or a set of primary propositions of this theory, or all valid logical formulas of this theory, etc. E should be closed under the logical implications, i.e. given any subset of propositions P ,P , in this theory, if Q is a 1 2 ··· logical consequence of them then Q must also belong to this theory. A sentence is a logic formula whose each variable is quantified i.e. inside the scope of a quantifier such as: (exist), (for all), modal logic quantifiers, and other various modern ∃ ∀ logics’ quantifiers. With respect to this theory, let P be a proposition, or a sentence, or an axiom, or a theorem, or a lemma, or a logical formula, or a statement, etc. of E. It is said that P is S-denied on the space M if P is valid for some elements of M and invalid for other elements of M, or P is only invalid on M but in at least two different ways. An ensemble of sentences E is considered S-denied if at least one of its propositions is S- denied. And a theoryT is S-denied if its ensemble of sentences is S-denied, which is equivalent to at least one of its propositions being S-denied. The proposition P is partially or totally denied/negated on M. The proposition P can be simultaneouslyvalidated inone wayand invalidatedin(finitely orinfinitely) many different ways on the same space M, or only invalidated in (finitely or infinitely) many different ways. The invalidation can be done in many different ways. For example the statement A =: x = 5 can be invalidated as x = 5 (total negation), but x 5,6 (partial negation). (Use 6 ∈ { } a notation for S-denying, for invalidating in a way, for invalidating in another way a different 1ThemultispaceoperatorS-denied(Smarandachely-denied)hasbeeninheritedfromthepreviouslypublished scientificliterature(seeforexampleRef. [1]and[2]). S-DenyingaTheory 9 notation; consider it as an operator: neutrosophic operator? A notation for invalidation as well.) But the statement B =: x>3 can be invalidated in many ways, such as x 3, or x=3, ≤ orx<3,orx= 7,orx=2,etc. Anegationisaninvalidation,butnotreciprocally-sincean − invalidation signifies a (partialor total) degree of negation,so invalidationmay not necessarily be a complete negation. The negation of B is B =: x 3, while x= 7 is a partial negation ≤ − (therefore an invalidation) of B. Also, the statement C =: John’s car is blue and Steve’s car is red can be invalidated in many ways,as: John’s car is yellow and Steve’s car is red, orJohn’s car is blue and Steve’s car is black, or John’s car is white and Steve’s car is orange, or John’s car is not blue and Steve’s car is not red, or John’s car is not blue and Steve’s car is red, etc. Therefore, we can S-deny a theory in finitely or infinitely many ways, giving birth to many partially or totally denied versions/deviations/alternatives theories: T ,T , . These 1 2 ··· new theories represent degrees of negations of the original theory T. Some of them could be useful in future development of sciences. Whydo westudysuchS-denyingoperator? Becauseourrealityisheterogeneous,composed of a multitude of spaces, each space with different structures. Therefore, in one space a state- ment may be valid, in another space it may be invalid, and invalidationcan be done in various ways. Or a proposition may be false in one space and true in another space or we may have a degree of truth and a degree of falsehood and a degree of indeterminacy. Yet, we live in this mosaic of distinct (even opposite structured) spaces put together. S-denying involved the creation of the multi-space in geometry and of the S-geometries (1969). It was spelt multi-space, or multispace, of S-multispace, or mu-space, and similarly for its: multi-structure,or multistructure, or S-multistructure, or mu-structure. §2. Notations Let <A> be a statement (or proposition, axiom, theorem, etc.). a) For the classical Boolean logic negation we use the same notation. The negation of <A> is noted by A and A=<nonA>. An invalidation of <A> is noted by i(A), while ¬ ¬ a validation of <A> is noted by v(A): i(A) 2<nonA> and v(A) 2<A> , ⊂ \{∅} ⊂ \{∅} where 2X means the power-set of X, or all subsets of X. All possible invalidations of <A> form a set of invalidations, notated by I(A). Similarly for all possible validations of <A> that form a set of validations, and noted by V(A). b) S-denying of <A> is noted by S (A). S-denying of <A> means some validations of ¬ < A > together with some invalidations of < A > in the same space, or only invalidations of <A>inthesamespacebutinmanyways. Therefore,S (A) V(A) I(A)orS (A) I(A)k ¬ ¬ ⊂ ⊂ for k 2. ≥ S