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...s played. Any city that has had a great impact on learning has had a great library behind it, fueling that learning. Libraries date back to 1900 BC in the Sumerian city of Nippur. Nippur’s library held tens of thousands of inscribed clay tablets. These tablets themselves dated as far back as 2600 BC and included mathematical tables and almanacs. The first great lib...

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... George V was outraged. Upon his death, the Prince of Wales became King on January 20, 1936. Ernest Simpson - who was having a long-term affair of ... ...their digestive juices. Ants share digested food with each other. They can carry 15-20 times their body weight. Only the colony's queen breeds. U... ...first world war broke, Turkey allied itself with the Germans. All Armenian men aged 20-45 were conscripted to the army as soldiers, soon to be disar... ... in the dialogs Timaeus and Critias written by the Greek philosopher Plato (428-347 BC). An Egyptian priest was supposed to have described it to the... ...tian priest was supposed to have described it to the Greek statesman Solon (638-559 BC). The priest insisted that Atlantis was enormous - bigger ... ...echnologically advanced civilization, recounted the priest, in the 10th millennium BC (c. 12,000 years ago). Curiously, he also said that the A... ...an island in the Aegean Sea (Thera, or Santorini it suffered an earthquake in 1640 BC and housed the flourishing Cycladic civilization), through th... ...nois, noticed in the early 1960s that though the bacterial population doubles every 20 minutes - the quantity of luciferase (the light producing enz... ...s of the Confederacy seceded at once. Seven - the "core" - seceded between December 20, 1860 and February 1, 1861. They were: South Carolina, Missi...

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...g loop is central small Frattini. The proof is left to the reader. 20 DEFINITION [40]: Let L be a loop. The commutant of L is the set (L) =... ...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... ...very ARIF loop is a S-loop. Hence every RIF loop is a S-loop. Fenyves [20](1969) has introduced the concept of C-loops. By Corollary 2.6 of Ki... ... proposed in Chapter 5. For more about C-loops please refer (F. Fenyves) [20]. 51 Thus we have seen the definition of S-loops and the classes... ...xample 3.4.5: Let L 19 (10) be a loop which is a commutative loop of order 20. This loop has no S-subloops but L 19 (10) is a S-loop. Then for this l... ...9. Characterize those loops L which have only one S-commutator subloop. 20. Characterize those loops L that has always the S-commutator subloop to ...

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... partial solutions of certain unsolved problems. An important method is based on the famous Euler-Bell-Kalm ar lemma, with many applications. Article 20 may be considered also as an introduction to Chapter 3 on Arithmetic functions. Here many papers study the famous Smarandache function, the source of inspiration of so many mathematicians or scientists working in other eld...

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...d on the famous Euler-Bell-Kalm ar lemma, with many applications. Article 20 may be considered also as an introduction to Chapter 3 on Arithmetic fun... ...n equations involving n! . . . . . . . . . . . . . . . . . . . . . . . 103 20 On certain diophantine equations for particular arithmetic functions . .... ... of arithmetical functions at factorials I . . . . . . . . . . . . . . 182 20 On certain inequalities for k . . . . . . . . . . . . . . . . . . . . ... ...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... ...en b h b , c h c , and the remaining inequalities of (6) follow at once. 20 Remark. For another proof ofR +r minfh b ;h c g, see [4], pp.70-72. Wh... ... immediate application is for P O, in which case R 2 abc a +b +c = 2Rr (20) giving again Euler’s inequality. We note that forP I there is equality...

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...s would allegedly have found a first period of time lapse of 23 hours and 20 minutes. - The second book of the Emperors in the Old Testament 20,11 r... ...t the Bible is, as it is known to be, the most accurate history book.) 20 The presence of the forms of substantial existence of the spirit does n... ...he only God, your God …you shall not have Gods others than Me> Exodus 20.2,3. This divine commandment refers to existence, as a coming into Bein... ...s, or down on earth or in the waters below the ground” V.T. “ Exodus” 20.4 Far from the naïve interpretation which sustains the idea that... ... 3)< Do not take in vain the Name of God, your God…> V.T. “Exodus” 20.7 The rational entity must hold the belief that it is a creation of div...

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...         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        ... ...l direct product of N 1 , …, N n . Then prove G and T are isomorphic. 20 Supplementary Reading: 1. Birkhoff. G and Maclane. S, A Brief S... ...11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24} that is Z 25 \ {0, 5, 10, 15, 20} = B is also a subgroup under multiplication modulo 25. Thus o(A)... ...= B is also a subgroup under multiplication modulo 25. Thus o(A) = 2 and o(B) = 20. Clearly, B is not a cyclic subgroup of Z 25 . Here Z 25 is not ... ...ut Z 25 is not a Smarandache 34 cyclic semigroup as Z 25 \ {0, 5, 10, 15, 20} = B is a proper subset of Z 25 which forms an abelian group und... ...che cyclic semigroup? PROBLEM 5: Find all cyclic subgroups in the semigroup Z 20 × Z 15 where Z 20 × Z 15 is the Cartesian product of the semi...

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...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... ..., 8} is a subgroup of Z 10 which is a S- normal subgroup of Z 10 . 20 PROBLEMS: 1. Show Z 15 is a S-semigroup. Can Z 15 have S-normal su... ... Clearly I = {0, 11} is an ideal of Z 22 . Also P = {0, 2, 4, 6, 8, … , 20} is an ideal of Z 22 . Example 2.3.6: Let Z 7 = {0, 1, 2, …, 6}, th... ...{0, 1, 2, … , 24} be the ring of integers modulo 25. 〈5〉 = {0, 5, 10, 15, 20} is an ideal of Z 25 (‘〈5〉’ denotes the ideal generated by 5.) which is... ...are local rings. PROBLEMS: 1. Find a ring homomorphism φ between Z 20 and Z 18 such that the ker φ≠{0}. 2. Let f: Z 25 → Z 16 be a ring h... ...ind the quotient rings Z 36 /I and Z 36 /J. 4. Let Z 21 = {0, 1, 2, …, 20}. Find an ideal I of Z 21 such that Z 21 /I is a field. 5. Prove for ... ...) + (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...

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...ir incentives largely aligned with those of independent software developers. 20 The more outside developers there were The Rise and Stall of the Gener... ...xternalities that are based on pre- The Rise and Stall of the Generative Net 20 sumptions such as how trustworthy we can expect people to be. As thos... ...tations. A PC owner named Tom Jennings wrote FIDOnet in the spring of 1984. 20 FIDOnet was BBS software that could be installed on many PCs. Each FID... ...curity standards and an- other to write a campus-wide acceptable use policy. 20 It described consensus among computer scientists that Morris’s acts wa... ...ave undesirable features—and they are not easy for third parties to improve. 20 In design terms, there is no hourglass. Carriers have forced telephone... ...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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...idempotent element in the semigroup R 2×2 given in example 1.3.4. PROBLEM 20: Does an idempotent exist in (Z + , ×)? Justify. PROBLEM 21: Does... ... ∗ a 2 )) ∗ a 1 (a 1 ∗ a 3 ) ∗ (a 2 ∗ a 3 ) = a 3 ∗ a 0 = a 3 . 20 Now (a 1 ∗ (a 3 ∗ a 2 )) a 1 = (a 1 ∗ a 3 ) ∗ a 1 = a 3 ∗ a 1... ...al groupoid? PROBLEM 5: Is Z n (t, 0) a Bol groupoid? PROBLEM 6: Will Z 20 (0, 4) be an alternative groupoid? PROBLEM 7: Can Z n (t, 0) sa... ...upoid but not a Smarandache inner commutative groupoid. PROBLEM 1: Is Z 20 (3, 8) a SG? Does Z 20 (3, 8) have semigroups of order 5? PROBL... ...ong Moufang groupoid in Z ∗ (17) \ Z (17). PROBLEM 18: Find all SGs in Z ∗ (20) \ Z (20). PROBLEM 19: Find all the SGs in Z ∗ (22) \ Z (22). Whi... ...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...

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...-semiconjugate and the converse in general is not true. DEFINITION 1.3.20: Let G be a S-groupoid. We say G is Smarandache inner commutative (S-in... ...eal of R if (0) ⊂ P ⊂ K then P = (0) or K = P. We call an ideal I to be 20 principal if I is generated by a single element. We say an ideal X of R... ...is both a S-pseudo right ideal and S- pseudo left ideal. DEFINITION 1.4.20: Let R be a ring. I a S-ideal of R; we say I is a Smarandache minimal i... ...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... ...oduce a new notion called Smarandache anti zero divisor. DEFINITION 1.5.20: Let S be a semiring. An element x ∈ S is said to be a Smarandache anti... ..., which is taken in this definition as loop near field. DEFINITION 1.6.20: Let L be a loop. N a loop near field. The loop-loop near-ring NL of th... ...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...

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...3 2.2 Properties of loop-rings and introduction to Smarandache loop-rings 20 2.3 Smarandache elements in loop rings 30 2.4 Smarandache substructure... ...bloop, Smarandache normal subloop and related concepts. DEFINITION 2.1.20: Let L be a loop a proper subset A of L is said to be a Smarandache sub... ...Smarandache pair wise associative loop (S-pairwise associative loop). 20 The Smarandache associator subloop (S-associator subloop) of L is den... ...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... ...ote the collection of S- semi-idempotents in R by SSI (R). THEOREM 2.3.20: Let R be a NA ring if α is a S-semi-idempotent then α is a semi- idempo... ...’ in general is not used in the mutually exclusive sense. THEOREM 2.4.20: Every strongly right commutative loop ring RL is commutative. Proof... ... 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)...

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... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 C. Castro, M. Pavˇ siˇ c The Extended Relativity Theory in Clifford Spac... ... web site at York University, Canada, http://at.yorku.ca/cgi-bin/amca/caft-20). 6. Smarandache F. Invisible paradox. Neutrosophy, Neutrosophic Probabi... ...rgyφ and in the non-stationary case, it reads: − ∂S R ∂t = 1 2 mV 2 +Q+φ, (20) where S R is the real part of the complex action (i. e., S R /2mD is th... ...formed Poincar´ e algebras was raised in [68]. A thorough study of Finsler 20 C. Castro. On Dual Phase-Space Relativity, the Machian Principle and Mod... ...nt. Monthly Notices of the Royal Astronomical Society, v. 327, 2001, 1208. 20. Uzan J. P. The fundamental constants and their variations: observationa... ..., defned in anal- ogous way as in eqs. (57): DA A DX B = ∂A A ∂X B + ˜ Γ A BC A C . (101) From eq.-(100) we obtain Ds Dx μν =[D μ , D ν ]s=K μν ρ ∂ ρ ...

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...l such an assembly in years: Scythian, Etruscan, VOICES FROM THE PAST 20 Turkish, Negro. Bowls of incense sent threads to the ceiling. Wisps f... ...talking to a friend; Peter’s voice has always pleased me, so deep. Elul 20 Yesterday, when I was in Naim, someone pointed out a sick man huddled i... ...them as we must help God. Heal. Lift up our eyes. Nazareth—home Kislev 20 When I picked corn in a field with my disciples I was reproved because ... ...apa’s working in his shop. CHRIST’S JOURNAL 183 Nazareth Shevat 20 Today was cool and windy. I visited Simeon. I visited Mark. I visited ... ...ttle Nazareth may have a worthwhile collection someday. Jerusalem Adar 20 My enemies come closer. Verily, I say unto you, the man who climbs t...

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... then we say that f is a fuzzy symmetric subgroup of S n . THEOREM 1.2.20: Let f be a fuzzy symmetric subgroup of the symmetric group S 3 then o... ...x) =      = = otherwise 0 ) 132 ( ), 123 ( x if 5 . 0 e x if 1 20 where e is the identity element of S 3 . It can be easily verified that ... ...t if and only if x ∈ µ t and x ∉ µ s for all s > t. 45 THEOREM 1.4.20: A fuzzy ideal µ of a ring R is fuzzy semiprime if and only if µ , t ∈ ... ... domains so R µ = 1 R µ ∪ 2 R µ is an integral bidomain . THEOREM 1.5.20: Let µ be a fuzzy bi-ideal of biring R = R 1 ∪ R 2 such that Im µ = ... ...y prime bi-ideal. Proof: Follows as in case of rings. DEFINITION 1.5.20: A fuzzy bi-ideal µ of a biring R = R 1 ∪ R 2 is called fuzzy maximal ... ..., 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 ...

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................................................................................. 20 The Third Major Split................................................. ...lly turned the Medium of The Path of Splitness Chapter One: The Universe 20 Formlessness into 2 Totalities of Everythingness and Nothingness. ... ...days instead of 365. Why? Because they divided their numbers into groups of 20 called ‘toons’… not 10, and because they regarded the number 13 and... ...alled ‘toons’… not 10, and because they regarded the number 13 and the number 20 as sacred numbers. So… 13 x 20 = 360. Since harmonically, the num... ...days instead of 365. Why? Because they divided their numbers into groups of 20 called ‘toons’… not 10, and because they regarded the number 13 and...