Full Text Search Details
... American Research Press Rehoboth, NM 2002 A 11 L A 7 A 8 A 6 A 10 A 1 A 2 A 3 A 4 A 5 2 The picture on the co... ...ace 5 1. General Fundamentals 1.1 Basic Concepts 7 1.2 A few properties of groups and graphs 8 1.3 Lattices and its pro... ... of loops 29 2.6 On a new class of loops and its properties 31 2.7 The new class of loops and its application to proper edge colourin... ...nd associator subloops 67 3.6 Smarandache identities in loops 71 3.7 Some special structures in S-loops 74 3.8 Smarandache mixed direct p... ...parents, Worathur Balasubramanian and Krishnaveni for their love. 7 Chapter one GENERAL FUNDAMENTALS In this chapter we shall recall... ...generally, if a, b, c are elements in a Moufang loop G such that (ab)c = a(bc) then a, b, c generate an associative loop. The proof is left for t... ...= b, b < a. (ii) if a < b and b < c then a < c. (iii) if a < b then ac < bc and ca < c b. The relation a > b is interpreted as usual, to mean b <... ... triple a, b, c ∈ L is said to be pseudo associative if (ab) (xc) = (ax) (bc) for all x ∈ L. If (ab) (xc) = (ax) (bc) for some x ∈ L we say the trip... ...ciative loop L if for a non-associative triple a, b, c in L we have (ax) (bc) = (ab) (xc) then for all x ∈ L, we say the triple is strongly pseudo a...

Full Text Search Details
... Preface 5 1. Preliminary Notions 1.1 Integers 7 1.2 Groupoids 8 1.3 Definition of Semigroup with Examples 11 1.4 ... ...Smarandache Automaton 96 6.3 Direct Product of Smarandache Automaton 102 7. Research Problems 105 Index 109 ... ...s, Vol. III, Abaddaba, Oradea, 78-81, (2000). 7 CHAPTER ONE In this chapter, w... ...(mod 2). Z 9 = {0, 1, 2, ... , 8}, 3 + 6 ≡ 0 (mod 9), 3.3 ≡ 0 (mod 9), 2.8 ≡ 7 (mod 9), 6.2 ≡ 3 (mod 9). This notation will be used and Z n will... ... H = {a 1 , a 3 } and K = {a 2 , a 4 } to be subgroupoids of G. Example 1.2.7: G is a groupoid given by the following table: ∗ a 0 a 1 a 2 a ... ...ay or may not have the associative law to be true. Remark: Even (ab) c ≠ a (bc) in general for all a, b, c ∈ <S>. Thus unlike a free semigroup whe...

Full Text Search Details
...es and other number problems Henry Ibstedt Glimminge 2036 7, rue du Sergent Blandan 280 60 Broby 92130 Issy les Moulineaux Sweden... ... my world of numbers. March 2003 Henry Ibstedt 6 7 Contents Preface 3 I. An Integer as a Sum o... ... w h e n M = L . Case 4. s≠0, k≠0. Equation (2) takes the form (7) ) 1 2 m 2 m m m ( 2 1 Q k 1 k s 2 1 + ⋅ − = + − Q is an intege... ...mples are given in table 1. Table 1. The number of sequences for L=105 is 7 and is independent of s in N=L⋅2 s . N= 105 s=0 N= 210 s=1 N= ... ...1 Q M Q M Q M Q M 34 3 69 3 1119 3 2239 3 19 5 40 5 670 5 1342 5 12 7 27 7 477 7 957 7 1 14 7 15 217 15 441 15 6 10 1 20 150 21 310 21 ... ...cd. B= abcdabcdabcdabcdabcd…… i C I Period # F(i) M(i) L(i) 1 a a 2 bc bc 3 dab 1 d ab 4 cdab 1 cd ab 5 cdabc 1 cd abc 6 dabcda 1 d...

Description Details
...specially to the sequence of prime numbers. The author has solved various conjectures by Smarandache, Bencze, Russo etc.; see especially articles 4,5,7,8,9,10. Finally, Chapter 5 studies certain irrationality criteria; some of them giving interesting results on series involving the Smarandache function. Article 3.13 (i.e. article 13 in Chapter 3) is concluded also with a t...

Full Text Search Details
...jectures by Smarandache, Bencze, Russo etc.; see espe- cially articles 4,5,7,8,9,10. Finally, Chapter 5 studies certain irrationality criteria; some o... ...ic inequality of Arslanagi c and Milo sevi c . . . . . . . . . . . . 23 7 A note on the Erd os-Mordell inequality for tetrahedrons . . . . . . . .... ...phantine equation x 1 ! +x 2 ! +::: +x n ! =x n+1 ! . . . . . . . . . . 68 7 The diophantine equation xy =z 2 + 1 . . . . . . . . . . . . . . . . . . ... ...n convexity of S . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7 A note on S(n), where n is an even perfect nunber . . . . . . . . . . . .... ...ype inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 7 On certain inequalities for primes . . . . . . . . . . . . . . . . . . . ... ...0 ;C 0 be the feet of the altitudes of an acute-angled triangle ABC (A 0 2 BC, B 0 2 AC, C 0 2 AB). Let a 0 ;b 0 ;b 0 denote the sides of the podaire ... ...lds true (see Theorem 25 in page 112): Let AD and AE be two cevians (D;E2 (BC)) forming angles ; with the sides AB;AC, respectively. If b A 90 an... ... b p By 2 + p z a p Ax p x c p Cz 2 0; or equivalently, y +z x bc aA + z +x y ca bB + x +y z ab cC 2 a p BC + b p CA + c p AB : ... ...CA + c p AB : (3) By using again the A.M.-G.M. inequality, we obtain a p BC + b p CA + c p AB 3 abc ABC 1 3 : Then, on base of (iii), one gets...

Full Text Search Details
... Preface 5 1. Preliminary notions 1.1 Binary Relation 7 1.2 Mappings 9 1.3 Semigroup and Smarandache Semigroups 10 ... ... Smarandache Hyper Subsemigroup 37 4.6 Smarandache Lagrange Semigroup 39 4.7 Smarandache p-Sylow Subgroup 41 4.8 Smarandache Cauchy Element i... ... Direct Product in Smarandache Semigroups 80 7. Open Problems 85 Index 89 ... ... Abaddaba, Oradea, 78-81, (2000). 7 CHAPTER ONE One of the essential and out... ...n explicit we illustrate them by the following examples: 12 Example 1.3.7: Let Z 8 = {0, 1, 2, ... , 7} be the semigroup under multiplicatio... ...         d c b a where a, b, c, d are real numbers such that ad − − − − bc ≠ 0. G is a group under matrix multiplication with        ...

Full Text Search Details
...us. As we will define them, s-manifolds are a very restricted subclass of the 7 polyhedral surfaces. The relationship between polyhedral surfaces ... ...les depends on the number of triangular disks around that vertex. Figure 7. A deformed hyperbolic star: both angles ∠AOB have measure 210º. ... ...rtex for a total of 420º, and an s-line will form two 210º angles, as in Figure 7. Here, both angles designated as ∠AOB have measure 210º. This is h... ...d the path ABC. The tangent vector on the path formed by the segments AB and BC has the same total change in direction θ, but all of this change ... ...ir of points. 32 Figure 6. The points P and Q are 5-aligned. Figure 7. The points P and Q are ∞-aligned. We can also get multiple alig... ...igned. We can also get multiple alignments as a result of topology. In Figure 7, we have a cylindrical s-manifold. The points on the top and botto... ...lso passes through a point of the segment AC, or through a point of the segment BC. Probably the most intuitive notion of a point C lying between ... ...lso passes through a point of the segment AC, or through a point of the segment BC. 40 This is Pasch’s axiom, and it states roughly that an s-l... ...′B′ is also congruent to the segment A′′B′′. III-3. On the line a, let AB and BC be two segments which except for B have no point in common. Furth...

Full Text Search Details
...american research press 2002 a 13 a 18 a 2 a 8 a 5 a 3 a 16 a 7 a 14 a 6 a 10 a 17 a 11 a 9 a 1 a 4 a 15 a 12 a 0 1 Sma... ...-ideals of the Smarandache mixed direct product ring R = Z 3 × Z 12 × Z 7 . This is a major difference between a ring and a Smarandache ring. For,... ... Preface 5 1. Preliminary notions 1.1 Groups 7 1.2 Semigroups 11 1.3 Lattices 14 1.4 Smarandache semi... ...otient rings 28 2.5 Special rings 30 2.6 Modules 34 2.7 Rings with chain conditions 35 3. Smarandache rings and its pr... ...51 3.5 Substructures in S-rings 56 3.6 Smarandache modules 63 3.7 Rings satisfying S. A.C.C. and S. D.C.C. 67 3.8 Some special types of... ...s are satisfied. 1. The set G is non-empty. 2. If a, b, c ∈ G then a(bc) = (ab) c. 3. There are exists in G an element e such that (a) For an... ...) + (c + id) = (a + c, i(b + d)) and (a + ib) (c + id) = (ac – bd, i(ad + bc)). The unity of Z[i] is 1. DEFINITION 2.5.5: Consider the set Q[i] =... ...DB ⊂ B and BD ⊂ B; i.e., bd(and db) are in B. with 65 b(d + c) = bd + bc ∀ d, c ∈ D and b ∈ B ((d + c) b = db + cb). If B is simultaneously a S-... ... collection of all right ideals of R. If (A + B) (A + C) (A + D) = A + BC (A + D) + BD (A + C) + DC (A + B) ∗ for all right ideals A, B, C, D in...

Full Text Search Details
...5 1. BASIC CONCEPTS 1.1 Basics of vector space and bilinear forms 7 1.2 Smarandache vector spaces 8 1.3 Basic definitions of other algebrai... ...algebras and their Smarandache analogue 123 7. SUGGESTED PROBLEMS 127 References 139 Index 145 5 ... ...omination which still continues and requires our intensive struggle. 7 Chapter 1 BASIC CONCEPTS This chapter on basic concepts has thre... ...roper subset, which is a group under the operations of L. DEFINTION 1.3.7: Let R be a non empty set on which is defined two binary operations '+' ... ... These identities will be termed as Moufang identities. DEFINITION 2.1.7: A loop L is called a Bruck loop if (x (yx)) z = x (y (xz)) and (xy) -1 ... ...bloop in L, if we have an associative triple a, b, c in A such that (ax) (bc) = (ab) (xc) for some x ∈ A, we say L is a Smarandache pseudo associati... ...a S-pseudo associative loop, then PA (L s ) = 〈{t ∈ A / (ab) (tc) = (at) (bc) where (ab) c = a (bc) for a, b, c ∈ A} 〉 denotes the Smarandache pseud... ... a strongly modular element of L if (a + b) (a + c) (a + d) (a + e) = a + bc (a + d) (a + e) + bd (a + c) (a + e) + be (a + b) (a + c) + cd ( a + b)...

Full Text Search Details
... two limits and it has, on either of its sides, the spiritual reality. 7 6. We associate, as a representation of the rational entity, two concu... ...process becomes perfectly explainable if we put forward a new hypothesis. 7. Material reality, as a whole, has a retrograde movement, from the right ... ... under the control of the divine will. Critical notes of the reader: 7. The biblical episode of waters parting ... ...nominator of its levels – rationality. Critical notes of the reader: 7. On death “Dasein is factual Being , power of Being, as... ...nce of the divine revelation. Critical notes of the reader: 81 7. Hell and the original sin According to the MESER concept, the monadi...

Full Text Search Details
...e Chapter One: INTRODUCTORY CONCEPTS 1.1 Groups, Loops and S-loops 7 1.2 Semigroups and S-semigroups 11 1.3 Groupoids and S-groupoids 13 1... ...mirings and S-semivector spaces 35 1.6 Near-rings and S-near-rings 43 1.7 Vector spaces and S-vector spaces 62 Chapter Two: BIGROUPS AND SMARAND... ...MIRINGS, S-BISEMIRINGS, BISEMIVECTOR SPACES AND S-BISEMIVECTOR SPACES 7.1 Bisemirings and its properties 175 7.2 Non associative bisemirings an... ... properties 175 7.2 Non associative bisemirings and its properties 180 7.3 S-bisemirings and its properties 183 7.4 Bisemivector spaces and S-bi... ...and motivation to rally on and carry forward the collective struggles. 7 Chapter 1 INTRODUCTORY CONCEPTS This chapter has seven sections.... ...in) then DB ⊂ B and BD ⊂ B i.e. bd (and db) are in B with b(d + c) = bd + bc for all d, c ∈ D and b ∈ B. On similar lines we can define S-left mod... ...p homomorphism which is well defined φ/G 2 : G 2 → H 2 φ(A) → (ad – bc) where A =       d c b a ∈ G 2 . Thus φ/G 2 : G 2 → H 2... ...2 . be groups with G 1 = ()       ≠ − ∈         2 mod 0 bc ad with Z d , c , b , a d c b a 2 and G 2 =       ≠ − ∈ ... ...d c b a 2 and G 2 =       ≠ − ∈         ) 5 (mod 0 bc ad with Z d , c , b , a d c b a 5 . Clearly both G 1 and G 2 are no...

Full Text Search Details
... on T, I, F, nor on their sum n=t+i+f. The neutrosophic set generalizes: 7 - the intuitionistic set, which supports incomplete set theories (for 0... ...viewpoints, neutrosophy combines with the truth of neutral ones as well. 7. Fallibilism attributes uncertainty to every class of beliefs or proposit... ...aissance Humanism", American Historical Association, Washington, 1973. [ 7] Burnet, John, "Greek Philosophy: Thales to Plato", St. Martin's Press, I... ... Model M 6 (which is 5% stable, and 95% unstable) M o d e l M 7 (which is 100% unstable) He gives Orientation Tables for Physical ... ..., or from new arising conditions. This resulted from practice. 104 2.7. Neutrosophical Modal Logic: In modal logic, the primitive operato...

Full Text Search Details
...ties 39 1.5 Fuzzy birings 64 1.6 Fuzzy fields and their properties 79 1.7 Fuzzy semirings and their generalizations 84 1.8 Fuzzy near-rings and th... ...r-ring 381 6.3 Smarandache fuzzy binear-rings 389 6.4 Problems 398 7. APPLICATIONS OF SMARANDACHE FUZZY ALGEBRAIC STRUCTURES 7.1 Applic... ... STRUCTURES 7.1 Applications of Smarandache algebraic structures 401 7.2 Some applications of fuzzy algebraic structures and Smarandache alge... ...of fuzzy algebraic structures and Smarandache algebraic structures 426 7.3 Problems 431 References 433 Index 443 5 PREFACE In... ... me overwhelmingly, and his thought drives me to dissent and to dare. 7 PART ONE 8 ... ..., then DB ⊂ B and BD ⊂ B; i.e., bd(and db) are in B. with b(d + c) = bd + bc ∀ d, c ∈ D and b ∈ B ((d + c) b = db + cb). If B is simultaneously a S-r... ... or may not have the associative law to be true. Remark: Even (ab)c ≠ a(bc) in general for all a, b, c ∈ 〈S 〉. Thus unlike a free semigroup where ...

Full Text Search Details
... Smarandache is either exasperating/ exasperated or he is nothing! 7 From paradox to paradoxism In aesthetics, the paradox means the appare... ...ter’s Choice, USA; http:/ members.spree.com/writer/htm3/newman.htm, 1998; 7. Nonnovel, Aius, Craiova, 1993; introductions by Alexandru Cioranescu and... ...sm’s main roots, Xiquan Publishing House, Phoenix, USA, 1994, 64p., pp. 3-7. Soare, Ion, A writer of the paradoxes: Florentin Smarandache, Almarom,... ... Castleman, Dave, Nonpoems, in “Dusty Dog Reviews”, Gallup, NM 1992, no. 6-7. Skemer, Arnold, Nonpoems, in “ZXY”, Bayside, NY, 1992, no.4, pp.3-4. ... ...em Cadernos”, Blumenau, Santa Catarina, Brazil, Tomo XXXIV, July 1993, no.7, p.224. Editor, Poesia International em Joinville, in ”A Ilha”, Supleme...