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...andache substructures in loops 51 3.3 Some new classical S-loops 56 3.4 Smarandache commutative and commutator subloops 61 3.5 Smaranda... ...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... ... 2 2 2 3 e 1 5 4 3 3 5 4 e 2 1 4 4 2 1 5 e 3 5 5 4 3 2 1 e THEOREM [56]: Let L n be the class of loops for any n > 3, if k 2 1 k 2 1 p p p n ... ... . The proof is left for the reader as an exercise. 34 THEOREM [56]: L n contains one and only one commutative loop. This happens when m ... ... easily verified by using simple number theoretic techniques. THEOREM [56]: Let L n be the class of loops. If k 2 1 k 2 1 p p p n α α α = K , th... ...ative loop and the left alternative loop is not commutative. THEOREM [56]: Let L n be the class of loops: 1. L n does not contain any Moufan...

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...etter to not want, than to want and not have, or in the words of Lao-tzu (604 BC - 531 BC), “Manifest plainness; embrace simplicity; reduce selfishn... ...g Light; upward arrows mark the crea- ture’s desire to delight the Creator. 56 KABBALAH REVEALED pleasure of being identical to the Creator. If you...

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...rance had to be where Tuthmosis’ granite Dream Stela was. During the period of 1427-1401 BC, Tuthmosis had placed the Dream Stela in front of the Sp... ... to explain this?” “You will not,” Khamir stated. “I will.” CHAPTER 56 QUESTIONS TO ANSWER BY the time Mitch and Alex... ...e light- house was built by Ptolemy Soter, Alexander the Great’s commander, at around…290 BC, I believe. The lighthouse was completely destroyed by e...

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...min jUTY qin jUiT hY .............. 55 jpu qpu sMjmu swDIAY ............. 56 gur qy inrmlu jwxIAY ............ 57 suix mn BUly bwvry ................. ...drsnu Bytq pwp ............... 915 isKhu sbid ipAwrho ........... 916 mn bc /im rwm nwmu ........... 916 mhlw 3 AnMd AnMdu BieAw myrI ............... ... kvn kwj isrjy ................ 970 ijh ismrin hoie ............... 971 56 bMDic bMDnu pwieAw ............ 971 cMdu sUrju duie joiq ................ ...AwveI mnmuiK bUJ n pwie ] ijsu vyKwly soeI vyKY nwnk gurmuiK pwie ]4]23]56] sRIrwgu mhlw 3 ] ibnu gur rogu n quteI haumY pIV n jwie ] gur prswdI ... ...Cu nwih ]8]4] isrIrwgu mhlw 1 ] min jUTY qin jUiT hY ijhvw jUTI hoie ] 56 muiK JUTY JUTu bolxw ikau kir sUcw hoie ] ibnu AB sbd n mWjIAY swcy q... ...syv lgwny ] jn nwnk hir hir hir jip pRgty miq gurmiq nwim smwny ]4]4]18]56] gauVI pUrbI mhlw 4 ] myry mn so pRBu sdw nwil hY suAwmI khu ikQY hir ... ...vK nsy ]1] swDsMig kIrqn Plu pwieAw ] jm kw mwrgu idRsit n AwieAw ]2] mn bc /m goivMd ADwru ] qw qy CuitE ibKu sMswru ]3] kir ikrpw pRiB kIno Apnw... ...m hY insqwr ]2] inrml gux gwvhu inq nIq ] geI ibAwiD aubry jn mIq ]3] mn bc /m pRBu Apnw iDAweI ] nwnk dws qyrI srxweI ]4]102]171] gauVI mhlw 5 ]... ...y ibnu aubr n hoaU ] XwhU qrn qwrn smrwQw ] rwiK lyhu inrgun nrnwQw ] mn bc /m ijh Awip jnweI ] nwnk iqh miq pRgtI AweI ]43] sloku ] rosu n kwhU ...

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... 6 LW 31, 83–252. 7 PE 1, 56–71. 8 Ibid., 81–101. 9 Ibid., 294–326. 10 PE 2, 9–31. 11 Smith... ... desert did not receive the sacrament in any form for many years at a time. 56 Therefore I do not urge that both kinds be seized upon by force, as ... ...e “Bohemians”) the privilege of administering the sacrament in both kinds. 56 Cf. A Treatise Concerning the Ban, PE 2, 40. 57 Cf. above, p. 13 n.... ... they are ignorant of what they ought to know and do. They fulfil what Isa. 56[:10, 11] says: “His watchmen are blind, they are all without knowledg... ... 125 Cf. p. 64, above, and PE 1, 56. 126 res; cf. p. 23 n. 45, above. that which you were baptized to ...

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...55 min jUTY qin jUiT hY ..............55 jpu qpu sMjmu swDIAY..............56 gur qy inrmlu jwxIAY.............57 suix mn BUly bwvry................. ...5 drsnu Bytq pwp.................915 isKhu sbid ipAwrho............916 mn bc k®im rwm nwmu...........916 mhlw 3 AnMd Anµdu BieAw myrI............... ... ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔ 56 ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ CMq mhlw 5 pRIqm pRym Bgiq ky dwqy.... ...n AwveI mnmuiK bUJ n pwie ] ijsu vyKwly soeI vyKY nwnk gurmuiK pwie ]4]23]56] sRIrwgu mhlw 3 ] ibnu gur rogu n quteI haumY pIV n jwie ] gur prswdI ... ... ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔ 56 ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ jUTI hoie ] muiK JUTY JUTu bolxw ik... ...r syv lgwny ] jn nwnk hir hir hir jip pRgty miq gurmiq nwim smwny ]4]4]18]56] gauVI pUrbI mhlw 4 ] myry mn so pRBu sdw nwil hY suAwmI khu ikQY h... ...ivK nsy ]1] swDsMig kIrqn Plu pwieAw ] jm kw mwrgu idRsit n AwieAw ]2] mn bc k®m goivMd ADwru ] qw qy CuitE ibKu sMswru ]3] kir ikrpw pRiB kIno ... ...im hY insqwr ]2] inrml gux gwvhu inq nIq ] geI ibAwiD aubry jn mIq ]3] mn bc k®m pRBu Apnw iDAweI ] nwnk dws qyrI srxweI ]4]102] 171] gauVI mhlw... ...py ibnu aubr n hoaU ] XwhU qrn qwrn smrwQw ] rwiK lyhu inrgun nrnwQw ] mn bc k®m ijh Awip jnweI ] nwnk iqh miq pRgtI AweI ]43] sloku ] rosu n kw...

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...I can't" in one area even triggers "I can't" in other THE POWER IN YOU 56 areas of your life. I can't get a job, I can't get a girlfriend or boyf...

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... ă se a şteaptă şi la terori şti anti-americani... cu bombe, crime... 56 Eu cred c ă va fi o noapte obi şnuit ă. Anul 2000 e o conven ţie între ... ...eun ă cu George P. Coatmundi publicaser ă o carte la Trafford (Victoria – BC, Canada), „A New Copernican Revolution”, pe care Greg a prezentat-o. ...

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...field of information. We are born; T H E K A B B A L A H E X P E R I E N C E 56 we feel ourselves in the biological, “protein” body for some time, a... ...main completely abstruse to us. Q: The Zohar was written in the second century BC, but was first discovered in approximately the 13th century. Why di...

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...The Republic is a Socratic dialogue written by Plato around 380 BC concerning the definition of justice and the order and character of the just city-state and the just man....

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...da tempo, kaj uzu la aliajn tri partojn de la tempo por provizi domon por 56 370 Parto Du si, kaj vestaĵojn, kaj ŝuojn, tiel ke oni ne komune kunlabo... ...o, li trovus ke Hesiodo tre saĝe diris ke iel ‘duono estas pli ol la tuto’ 56 .” “Se li konsultus min,” li komentis, “li kontentiĝus pro sia vivo.” “D... ...oj, kiun ni ĵus diskutis, rilate al la edukado kaj prizorgo de infanoj kaj 56 Hesiodo, Laboroj kaj Tagoj 40. 155 Parto Kvin 467 protektado de la ali...

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...NTS IN THE SCIENCES 7 ABSTRACT ON THE ATOM AND ITS ENERGY 11 GREAT IDEAS THAT REVOLUTIONIZED OUR SCIENTIFIC WORLD 16 DEMOCRITUS (470 - 380 BC) Greek Philosopher 16 NICHOLAS COPERNICUS (1473 - 1543) Polish Astronomer 17 GALILEO GALILEI (1564 - 1642) Italian Mathematician and Astronomer 18 RENE DESCARTES (1596 - 1650) French Mathematician and Philosopher. 19 I...

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... . . . . . . . . . . . . . . . . . . . 53 Chapter 2. Diophantine equations 56 1 On the equation 1 x + 1 y = 1 z in integers . . . . . . . . . . . . . ... ...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... ...yi, Arithmetic problem in geometry (Hungarian), Math. Phys. Lapok 1(1891), 56-57. 4. T. Weszely, The life and work of Gy. V alyi , Ed. Kriterion, 198... ...Fermat (1601-1665)..." (H. Davenport, The higher arithmetic, London, 1962) 56 1 On the equation 1 x + 1 y = 1 z in integers It is su cient to conside... .... As a corollary, X n x k (n) (k + 1) (k + 1) (2k + 2) (x!1); k 1 (56) 2. Recall the de nition of "normal order of magnitude" ([7]). Let P be ... ...rrational numbers (Romanian), Gazeta Mat. A (Bucure sti), 1-2/1983, pp.54-56. 12. J. S andor, Irrational numbers (Romanian), Univ. of Timi soara, 1...

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...lation (the contact of matter with anti-matter) will never take place. 56 - Owing to the divine will, the space, time and cosmos are eternal. ...

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...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 ... ...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... ...r B′ also have no point in common. In that case, if AB is congruent to A′B′ and BC is congruent to B′C′, then AC is congruent to A′C′. III-4. Ever... ...c has an angle of 120° relative to the s-line a, and since the region between 56 them is Euclidean outside of the diagram, this angle will be pres...

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...entem ab alio Quovis curvae pun&o g agatur ZR’ .difiantk S P parallela , Bc aemittatur .$& T perpendicularis ad diltantiam illam SP : vis centripe... ...~6 ut 'T'zt.fk sequalis ipfi Tv ; deinde cape z~lv, quz fit ad vG ut eit ‘BC q@ad. ad T C pad. Et quoniam ex conicis efi $9-v q~d. ad T v G ut ZICg... ...antur tria puntia fit folvitur expeditius. Dentur pun&a B, C, I), Juntias BC, C2) produc ad E, F, ut fit EB ad E Cut SB ad SC, 8~ FC ad FB ut SC ad ... ...ifim EC-CA ad CA-CCL, fei. 23 A ad A,!&~& Fompof& EA ad EA=-kAL feu EL ut BC ad E C+CA feu EB; ideoque, ob knilitudinem triangularurn EA.& EL I, E C... ... fuit. 2lcf@~$vprimus d 2 4 8 16 32 64 Afcenf?s ultimus ; -; 33 7 14 28 56 A?umerus OfSlat. 226 228 ‘193 140 90; 53 30 1 QzJhfqs primus 1 2 4 8... ...in eadem ’ jovis a terra diltantia ex temporibus periodicis prodeunt 2’. 56”. 47”‘, & 11. 5 If?, 6”?. Diameter jovis micrometro in telefcopio pede... ...2. prop. ~xvx. lib, x. Jupiter utique refpeAu fixarum revolvitur horis 9. 56’, mars horis 24. 39’. Venus ‘horis 23 circiter,’ terra horis 23. 56’, ... ... spedurn Ig6I58Go. a centro, firrgcri,.- lis diebus fidereis horarum 23. .56’. 4” uniformiter revolvens tempo-. re minuti unius i‘ecund.i defcribet ... ...s diminutz Wnde cum terra refpe&u fixarum revblvatur ,SYSTEMAJE horis 23. 56’, jupiter autem horis 9 ~6’~ fintque temporum quadrata .ut 29 ad 5, & r...

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...aucracy under King David and his successors in the general period 800- 700 BC. Hebrew religious culture did not exist in isolation, but our chart does... ...quired. On the first level we find Egyptian religion of approximately 1400 BC which became codified and highly organized under King Akhenaton. 11 It w... ...eaucracy under King David and his successors in the general period 800-700 BC. Hebrew religious culture did not exist in isolation, but our chart does... ...ery pervasive influence of Mesopotamian culture. During the period 800-500 BC, a creedal statement of vast 11. Details in Cyril Aldred, Akhenaten, Tha... ...ned and regulated religious organization. 13 In the general period 500-200 BC three major new influences impinged upon the Hebrew system in Palestine ... ...n original 1923: “Das Ich und Das Es”) in vol. 13. THE RELIGIOUS DIMENSION 56 Jung dethroned the intellect as supreme arbiter of reality. Ulti- mate ... ...ious, does not ap- pear to include any concept of that person’s own death. 56 The fear of your own death is a function of conscious mental activity an... ... Science which is a world-wide movement. Our funeral ceremonies also para- 56. Sigmund Freud, The Ego and the Id (1923), Hogarth Press, London, 1952. ... ...ympathetic, less dangerous than the more powerful Spirit Beings (Swain, p. 56). Also, unlike Totemic Beings, Spirit Beings are most often a composite ...

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...2 0 293 83 3 202 634 36 30 111 80 9 16 358 760 4 20 35 2 6 39 12 3 61 56 5 2 2 1 2 0 3 0 2 2 11 5 20 B s 872 2573 297 779 422 887 452 786 12... ... CS1 16 -5⋅10 -5 0.6419 06100846 8766935832 66017539 5627913126 56 Table 3. Some general continued fractions converted to simple... ...he function values, which up n=32000 is displayed in numerical form on page 56 of the Smarandache Function Journal, vol. 2-3 [1]. This study has bee... ...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... ...7 139 140 141 43 j 1 6 0 4 6 6 6 1 p=31 k 3 9 12 13 14 17 d=3 i 1 30 56 59 60 61 64 j 1 1 2 2 2 2 2 p=37 k 2 3 4 12 13 14 d=3 i 1 9 30... ....9091.333667.F174 5 3.7.11.13.37.41.271 55 67.F200 6 3.7.11.13.37.239.4649 56 3.11.F204 7 11.73.101.137.239.4649 57 3.31.37.F206 8 3 2 .11.37.73.1...

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...Smarandache idempotents in rings 51 3.5 Substructures in S-rings 56 3.6 Smarandache modules 63 3.7 Rings satisfying S. A.C.C. and S. D... ...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] =... ...2}. 2. Can Z 49 = {0, 1, 2, ..., 48} have nontrivial S-idempotents? 56 3. Find all S-idempotents in the group ring Z 7 G where G = S 7 the sy... ...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... ...(0) and S 1 = (1 + g n , 0) is such that S 1 2 =(0). DEFINITION 3.10.56: Let R be a ring, we say R has a Smarandache trivial subring (S-trivial... ...e can find a subset M of R which is a tight subset of R. Example 3.10.56: Let G = 〈g / g 2 =1〉 and Z 2 ={0, 1} the ring of integers modulo 2. Th... ...t a locally semiunitary ring, which is not a S-locally semiunitary ring? 56. Give an example of a weakly CN-ring, which is not a CN-ring. 57. Prov...

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...d that. And yet we are in an era where this is something that is happening.” 56 In one notable experiment conducted in the fall of 2003, a researcher ... ... traffic, much of which is unidentifi- The Rise and Stall of the Generative Net 56 able even to the user of the machine, and much of which happens even ... ... by the public at large than one in which big firms help sand off rough edges. 56 GNU/Linux has become user-friendly thanks to firms that package and sel... ... BBC N, June 14, 2007, http://news.bbc.co.uk/2/hi/technology/6752853.stm. 56. Posting of Bob Sullivan to The Red Tape Chronicles, Is Your Computer ... ...pt_rooted/. 67. See Karasaridis, supra note 55. 68. See Sullivan, supra note 56. 69. Id. 70. CERT has also noted another threat, evidenced by the expl... ...tp://news.cnet.co.uk/software/0,39029694,39190155,00.htm. 18. Hiawatha Bray, BC Warns Its Alumni of Possible ID Theft After Computer Is Hacked, B...

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...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... ...ing, the interested reader is requested to refer [120]. DEFINITION 1.4.56: Let R be a ring. R is called a Smarandache F-ring (S-F-ring) if we hav... ...che Marot near-ring (S-Marot near-ring) if each regular S-ideal of N is 56 generated by regular elements where by regular S-ideal we mean an ideal... ...ng) if for every S-subnear-ring of N is a S-ideal of N. DEFINITION 1.6.56: Let N be a near-ring {I k } be the collection of all S-ideals of N. N ... ...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... ...give the definition of non-associative biseminear-rings. DEFINITION 8.1.56: Let (N, +, ) be a non-empty set. N is said to be a non-associative bis... ...ar-ring NL = N 1 L 1 ∪ N 1 L 2 ∪ N 2 L 1 ∪ N 2 L 2 . DEFINITION 8.2.56: Let (N, +, ) be a non-empty set. We call N a Smarandache na- quad semin...