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..."The Roosevelt Corollary," a statement of president Theodore Roosevelt, was renewal of the Monroe Doctrine. The Monroe doctrine of 1823 informed Europe that the United States held that the rest of the Americas could only be colonized by the...

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...ucture based on the possibility of fighting two major theater wars satisfy the requirements for steady-state operations? Major Hazdra examines three corollary issues: air mobility as a form of airpower that enables the military instrument of power in two basic ways, requirements placed on mobility air forces, and the structure of mobility air forces and the effectiveness ...

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...ption for Meeting the Strategic Airlift Shortfall, Lt Col Chad T. Manske points to the growing dependency on strategic airlift as well as the abiding corollary that there will continue to be a shortfall in strategic airlift. To get to the analysis, Colonel Manske raised three crucial questions: (1) are operational requirements able to justify unmanned airlifters, (2) are ...

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...oposition is found in the realization by the Air Force that distinctions between “strategic” and “tactical” airpower are artificial and limiting. The corollary to the second proposition is that the relative effectiveness of a particular airpower role or mission is situationally dependent. ...

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...F. Two things are to be noted importantly. (1) We see that often the subsutras are not used under the main sutra for which it is the subsutra or the corollary. This is the main deviation from the usual mathematical principles of theorem (sutra) and corollaries (subsutra). (2) It cannot be easily compromised that a single sutra (a Sanskrit word) can be mathematically inte...

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...itivenumber. Takingk = 2,3inourTheorem,wemayimmediatelydeducethefollowing: Corollary. Foranyrealnumberx≥ 1,wehavetheasymptoticformulae X n≤x δ m (a 2 ... ...thefollowingCorollaries: 42 RESEARCHONSMARANDACHEPROBLEMSINNUMBERTHEORYII Corollary1. Foranyrealnumberx≥ 1,wehavetheasymptoticformula X n≤x n∈A 2 b 2... ... b 2 (n) = 4 π 2 x 3 2 +O(x) and X n≤x n∈A 2 b 3 (n) = 9 2π 2 x 4 3 +O(x). Corollary2. Foranyrealnumberx≥ 1,wehavetheasymptoticformula X n≤x n∈A 4 b 4... ...oductoverallprimenumbers. Fromthistheorem,wemayimmediatelygetthefollowing: Corollary 1. Let a(n) be the square complements sequence, then for any real... ...Y p†k p 4 −p 3 +p 2 (p 4 −1) Y p|k p 3 (p+1)(p 2 −1) ! +O x 3 2 + . Corollary 2. Let b(n) be the cubic complements sequence, then for any real ... ...HONSMARANDACHEPROBLEMSINNUMBERTHEORYII wemayimmediatelydeducethefollowing: Corollary. Foranyrealnumberx≥ 1,wehavetheasymptoticformula: X n≤x d(S 2 (n)... ...theRiemannzeta function. FromthisTheoremwemayimmediatelydeducethefollowing Corollary1. Foranyrealnumberx> 1,wehavetheasymptoticformula X n≤x d(S 2 (n)... ...symptoticformula X n≤x d(S 2 (n)) = π 2 6 x+ζ 1 2 x 1 2 +O x 1 3 . Corollary2. Foranyrealnumberx> 1,wehavetheasymptoticformula X n≤x d(S 4 (n)... ...ShannxiProvince(04JK132) 60 RESEARCHONSMARANDACHEPROBLEMSINNUMBERTHEORYII Corollary 1. Taking m = 2 and s = 3/2 or m = s = 2 in the above Theorem,the...

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...ength 6. As an immediate consequence of this we have the following: COROLLARY [65]: Let L n (m) ∈ L n . If m, k (k < n) such that ((m–1) k +(–... ... (|x – a|, n) < n. So the permutation R a does not contain a k-cycle. COROLLARY [65]: Let L n (m) ∈ L n . If there exists k ∈ N (N-natural number... ...ose into cycles of length 12, hence there exist 2 cycles of length 12. COROLLARY [65]: Let L n (m) ∈ L n . Then each permutation R a ∈ π (a ≠ e a... ...e l.c.m. of the non-zero k i 's. Thus we have the following result: COROLLARY [65]: Order of each permutation R a in π is the same for all a ∈... ...s on Moufang loops and Theorem 3.9 of Tim Hsu [63] or in the finite case, Corollary 1 of Glauberman and Wright [22]). Using the main theorem A of Ti... ...ociative or L is a Moufang loop then L is a S-loop. Proof: In view of Corollary 1 of Kinyon [42] we see for an A-loop L. The two concepts; L is ... ...loop. Fenyves [20](1969) has introduced the concept of C-loops. By Corollary 2.6 of Kinyon [41], every flexible C-loop is a ARIF-loop, we have... ...oop is a S-loop. Proof: We see every flexible C-loop is a ARIF loop by Corollary 2.6. of Kinyon [41], Hence by Theorem 3.1.8 every flexible C-loo... ...in B\{e} and it will be an S-associative triple of L. Hence the claim. COROLLARY 3.5.1: If L is a S-associative loop, then L is a S-loop. 6...

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...y condition of the nuclear surface. We see particle volume is a necessary corollary to the intrusion principle. Furthermore, the relativistic prop... ...an axiom, Identity (I.8), as yet unproven but used as a basis for following corollary theorems and hypotheses, some of which are logically and mathe... ...n rather than the MT virtual particle premise. 7. From the particle volume corollary to the intrusion principle, (Sec.4.11.A), we assume as a neces...

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...l success, they never abandoned the notion "all you need is love" and its corollary, "love is all there is." It's interesting to note that the first...

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... obtain the following 2 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY Corollary. For any real number , we have the asymptotic formula 2. One sim... ...n of the number of prime factors respectively, then we have the following: Corollary 1. For any real number , we have the asymptotic formula where is... ...xed positive number. 18 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY Corollary 2. For any real number , we have the asymptotic formula where i... ...pper bounds of , then Finally, we have This proves the Theorem. Now the Corollary 1 follows from the Theorem, the asymptotic formula (see [2]), an... ... asymptotic formula (see [2]), and the estimate (for all ). And then, the Corollary 2 follows from the Theorem, the asymptotic formula (See [3]), an... ...obtain the following 22 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY Corollary. For any real number , we have the asymptotic formula 2. Some le... ...constant. Taking in the theorem, we may immediately obtain the follow- ing Corollary. For any real number , we have the asymptotic formula 34 RESEAR... ...an easily compute the rst several : Then use the Theorem, we can get the Corollary. References [1] Smarandache F. Only problems, not Solutions. Chic... ...itive num- ber. From this theorem, we may immediately deduce the following Corollary 1. Let be the square complement sequence, then for any real numbe...

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...|P ∈E and P g =P}. ♮ From the Theorem 1.3.4, we get the following results. Corollary 1.3.1 The numbers n O (Γ),n N (Γ) and n L (Γ) of non-isomorphic u... ...Γ) and P g =P or P gα =P}, Φ L 1 (g)={P|P ∈E L (Γ) and P g =P or P gα =P}. Corollary 1.3.2The numbersη O (Γ),η N (Γ) andη L (Γ) of non-equivalentembed... ...graph Γ and the numbers of non-equivalent embeddings of the graph Γ by the Corollary 1.3.1 and 1.3.2 as follows. Theorem 1.3.5 The numbers n O (Γ), n ... ...)x i + i≥0 2ε(Γ) g i (Γ)x −i = 2ε(Γ)G[Γ](x). This completes the proof. ♮ Corollary 1.4.1 Let be Γ a graph and s ≥ 0 be an integer. If r s (Γ) and g ... ...ε(Γ)g s (Γ). 4.2 The number of rooted maps underlying a graph on genus The Corollary 1.4.1 in the previous section can be used to find the implicit rel... ... =F(x ¯ ,y ¯ )+( |Aut 1 2 Γ| 2ε(Γ) −1)F(x ¯ ,0 ¯ ). Proof According to the Corollary 1.4.1 in this chapter, for any integer s ≥ o and Γ∈H, we know tha... ... we can also get that F ⋆ H (x ¯ ,g ¯ (Γ))= 0. This completes the proof. ♮ Corollary 1.4.2 Let G be a graph family and H ⊆ G. If the embedding sequenc... ... a(i,Γ)|Aut1 2 Γ| 2ε(Γ) r i (Γ) =0, and vice via. A typical example of the Corollary 1.4.2 is the graph family bouquets B n ,n≥ 1. Notice that in [24]... ...ooted maps on an orientable surface of genusm underlying a graphB n by the Corollary 1.4.2 (n 2 −1)(n−2)r m (n) = (2n−1)(2n−3)(n−1) 2 (n−2)r m−1 (n−2)...

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... and further the symbols Y k=n ² and X k=n ² mean the product and the sum, respectively, of all divisors of n. The following assertion is obtained as ... ...also arbitrary arithmetic function. Putting there g(x)= lnx for every real positive number x, we obtain (12), since P d (k)= Y t=k t: When f(x)´1, as ... ...ive real number x and (14) is proved. From (13) and (14) we obtain Proposition 8. For n¸1theidentity n Y k=1 P d (k)= n Y k=1 [ n k ]! (15) holds. As ... ...(16), we obtain Proposition 10. For n¸2theidentity n Y k=1 p d (k)= n¡1 Y k=1 (k!) [ n k ]¡[ n k+1 ] (17) 54 On Some Smarandache's problems holds. As ... ...chisimpossible,becausethesenumbers areconsecutiveoddnumbersandhavinginmindthatv = C n +4 and C 1 =7, the smallest of them satis¯es the inequality v¸11... ...n is proved. The following statement can be proved by analogy Proposition 4. For each natural number n: P(n)=1¡ ¼(n)+P(n)¡1 Y i=1 (1¡C 2 (p i ;n)): 8... ...he Theorem is proved. Further, we use some well-known functions (see, e.g., [15]): chx´ e x +e ¡x 2 ; shx´ e x ¡e ¡x 2 ; thx´ shx chx ; cthx´ chx shx ... ...g 1 )¡¼(cthx) § j=1 ¼( chx g ¼( cthx)+j ); if ¼( chx g 1 )>¼(cthx) (30) holds. The same way, putting: a=sh x; b=c h x; where x2R + and chx ¸ g 1 , as ... ....e., g=2;3;5;7;11;13;::: Then the function ¼(x) coincides with the famous function ¼ of the prime number distribution. Thus, from our Theorem we obtai...

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...sociative. Proof: Simple number theoretic methods will yield the result. COROLLARY 2.4.3: No groupoid in this class Z / / + is a semigroup. ... ..., c ∈ Z n only if t 2 ≡ t (mod n) and u 2 ≡ u (mod n). Hence the claim. COROLLARY 3.1.2: If in the above theorem when; n is a prime Z (n) never... ...iques as every row and every column in the table have distinct n elements. COROLLARY 3.1.7: If Z p (t, u) is a groupoid such that t + u = p, (t, ... ... t 2 y if n is a prime. Hence, Z n (t, t) is not an alternative groupoid. COROLLARY 3.3.5: Let Z n (t, t) be a groupoid where n is not a prime. ... ...p (mod 2p) ≡ p (mod 2p). Thus, {p} is a semigroup so Z 2p (1, 2) is a SG. COROLLARY 5.1.3: Let Z 3p (1, 3) be a groupoid in Z (3p) (p ≠ 3 and p ... ...roper subset, which is a semigroup; hence, Z 3p (1, 3) is a SG of order 3p. COROLLARY 5.1.4: Let (1, p 2 ) and 2 1 p p Z (1, p 1 ) be groupoids i... ... (mod p 1 p 2 ). So {p 2 } is a semigroup and 2 1 p p Z (1, p 1 ) is a SG. COROLLARY 5.1.5: Let Z n (1, p) be a groupoid in Z (n) where p is a pr... ...s and they are semigroups. So       + + 2 1 p , 2 1 p Z p is a SG. COROLLARY 5.3.2:       + + 2 1 p , 2 1 p Z p is a Smarandache id... ...). Proof: Follows by the very definition of Z ∗∗ (p) and theorem 5.3.1. COROLLARY 5.3.3: Let n be odd. Then       + + 2 1 n , 2 1 n Z n ...

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... it follows that SSC(r-"-l) - SSC(r) > M contradicting the choice ofM 0 Corollary 1: There are no positive integers M and k such that ~ SSC(x) - S... ...roof: Apply the reasoning of theorem 3, replacing all squares by cubes. D Corollary 2: There are no positive integers M and k such that • SCC(x) - S... ...Lipschitz condition for any exponent k. Proof: Same reasoning as that for corollary I. D The functions SSC(n) and SCC(n) share one property. Theore... ...as 8*gk.: - 9*10 k ": and which goes to one in the limit as k -> 00. 0 Corollary 3: Let d be a decimal digit in the set { 0,1,2,3,4,5,6,7,8,9} Ifm... ...re, the digit sum of n is evenly divisible by 3 and therefore so is n. 0 Corollary 4: For any number of the form n = 1234. (k-l)k(k-l) 4321 or n ... ...her k or kk yields a number also having digit sum evenly divisible by 3 0 Corollary 5: Ifn is of the form n = 1234 k 4321 58 or n = 1234 kk 4321... ...ulo k Therefore, neither of those numbers can be evenly divisible by 3. 0 Corollary 6: If n is of the form n = 1234 k 4321 where k == 2 modulo 3, t... ...== 2 modulo 3, then n is not evenly divisible by 3 Proof: By the previous corollary, m = 1234 (k-I )(k-I) 4321 is congruent to 2 modulo 3. Insertio...

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...1] 1[2 3::: (1+b) 1] 2 3:::b =b! with equality only for a = 1, b = 1. Corollary. (x 1 +x 2 +::: +x n )!>x 1 ! +x 2 ! +::: +x n ! for n 3, x i ... ...1), x n+1 = n give a solution, since n(n 1)! =n!. Now, remark that by the Corollary, for n 3, x n+1 <x 1 +::: +x n . Let us suppose that x 1 x 2 ... ...s a consequence of this lemma, we obtain from (6) and a simple integration Corollary 1. For x> 1 we have: ln(x 1)< (x)< lnx (6) Corollary 2. For x> 1... ...1;r), i.e. when m and n are coprime. 3. As an application of (10), we get: Corollary 1. a) S(a) a S(b) b , if bja (13) b) If a has a composite divis... ...). We note that (14) o ers an improvement of inequality (6). We now prove: Corollary 2. Let m;n;r;s be positive integers. Then: S(mn) +S(rs) maxfS(m)... ...ly on (10)-(15), we can obtain extensions of these theorems to n numbers). Corollary 3. Let a;b be composite numbers, a6= 4, b6= 4. Then S(ab) ab S(... ... 7) and the stronger inequality from 6), namely S(m)S(n) [m;n] = mn (m;n) Corollary. S(mn) mn p (m;n) . 9) maxfS(m);S(n)g S(mn) (m;n) for allm;n, w... ...n result: maxfS(m);S(n)g = S([m;n]). On the other hand, since [m;n]jmn, by Corollary 1 from our paper [1] we get S(mn) mn S([m;n]) [m;n] 130 Since ... ...xtended over all divisors ofn. Since this product divides Y k n k =n!, by Corollary 1 from [1] we can write S(n!) n! S 0 @ Y kjn k 1 A Y kjn k ; wh...

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...ry finite groups. The Lagrange's theorem has some very important Corollaries. COROLLARY 3.1.2: If G is a finite group and a ∈ G, then o(a) | o(G). ... ...of: With Lagrange's theorem already in hand, it seems most natural to prove the corollary by exhibiting a subgroup of G whose order is o(a). The elem... ...generated by a has o(a) elements, whence, by Lagrange's theorem, o(a) | o(G). COROLLARY 3.1.3: If G is a finite group and a ∈ G, then a o(G) = e.... ...ROLLARY 3.1.3: If G is a finite group and a ∈ G, then a o(G) = e. Proof: By Corollary 3.1.2, o(a) | o(G); thus o(G) = mo(a). Therefore, a 0(G) ... ... only alternative, namely, that β = α. S is required subgroup of order p α . COROLLARY 3.4.2: If G is abelian of order o(G) and p α | o(G), p α+1... ... semigroup. Remark: The condition n ≥ 3 is important for S(2) is abelian. COROLLARY: S(n) is a Smarandache weakly cyclic semigroup. Proof: S... ...all the semigroup S(n) as the Smarandache symmetric semigroup of order n n . COROLLARY 4.4.4: The Smarandache symmetric semigroup S(n) has its lar... ... do not have x ∈ S n \ {1, g 2 } such that xg 2 = g 2 . Hence the theorem. COROLLARY 6.1.4: S n has elements which have Smarandache inverses. ... ...for we cannot find a p i , i ≤ 3 such that p u p i = p 5 . Hence the claim. COROLLARY: The symmetric group of degree 4 that is S 4 has Smarandac...

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... ii) The proof that \/ is associative is similar and will not be done. Corollary: If { A 1 , A 2 , A 3 , . . . , A n } is a set of elements in N... ...e principle of mathematical induction, the formula is true for all n. Corollary: If { A 1 , A 2 , A 3 , . . . , A n } is a set of elements in N... ... 1 , f 2 , f 3 , . . . , f n }. Proof: Similar to that of the previous corollary and will not be done. Theorem 4.2.9: If A, B and C are expres... ...esting is done on the first term of the triplet rather than the third. Corollary: If { A 1 , A 2 , A 3 , . . . , A n } is a set of elements in N... ...1 , i 2 , i 3 , . . . , i n }, max{ f 1 , f 2 , f 3 , . . . , f n }}. Corollary: If { A 1 , A 2 , A 3 , . . . , A n } is a set of elements in N... ... z, all results follow directly from the definitions of the operators. Corollary: If { A 1 , A 2 , A 3 , . . . , A n } is a set of elements in I... ... . . . , i n }, f = max{ f 1 , . . . , f n } and u = 1 – t – i – f. Corollary: If { A 1 , A 2 , A 3 , . . . , A n } is a set of elements in I... ... . . . , i n }, f = min{ f 1 , . . . , f n } and u = 1 – t – i – f. Corollary: If { A 1 , A 2 , A 3 , . . . , A n } is a set of elements in I... .... , f n }, u = min{ u 1 , . . . , u n } and 70 i = 1 – t – f – u. Corollary: If { A 1 , A 2 , A 3 , . . . , A n } is a set of elements in I...

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...tiple of p + 0 => a (p,r) ≡ 0(mod p) and the proof is complete. Corollary: F(1#p) ≡ 2 (mod p) Proof: a ... ...0 where r = [ α/2], α = 2r or α = 2r + 1. This completes the proof. Corollary: r F’(p 1 α p 2 2 ) = ∑ (k+2) [ P( α-2k)... ...0 In the case where α = 2r+1, the second term in the expression of the corollary can be expanded as P( α) + P( α-1) + P( α-2) + P( α-3) +. . .... ... α-2k) + P( α-2k-1)] k=0 and the proof of the corollary is complete. Section 10 Length/Extent of Smarandache Factor... ... = p a q b r c . . . ≡ 1 ( mod p n ). Which completes the proof. Corollary: For every prime p, there exists at least one prime q such that ... ... u(n) ≡ 1(mod p n ) , and every divisor of u(n) is ≡ 1 ( mod p n ) , the corollary holds. 108 Section 13 Smarandache Dual Symmetric Funct... ...d (2) we get N/d(N) = J. Therefore, J is the index of beauty of M*J. Corollary: If I is the index of beauty of M then p I-1 is the index of bea...

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...form 4k+1, we have satisfied the inductive step and the theorem is proven. Corollary : If p is an odd prime in the base B (B odd) symmetric sequence, ... ... if n is congruent to 0 or 2 modulo 3. Proof: A simple modulus 3 argument. Corollary: An element of the Reverse Sequence n(n-1) . . . 321 can be prime... ... divisible by 4, then there is more than one factor of two in the element. Corollary: Elements of odd index in the Concatenated Even Sequence cannot b...

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...and ∪ and ∩ distribute over each other. Hence (L, ∪, ∩) is a semiring. COROLLARY: All chain lattices are semirings. Remark: All lattices are... ...it. We find the following two observations as appropriate corollaries. COROLLARY 3.1.2: Every distributive lattice, which is not a chain lattice... ...c, d in S are not comparable. Thus we have semifields of this from also. COROLLARY 3.1.3: Let B be a Boolean algebra of order greater than two, B ... ...of C n . So C n is not a prime semifield. This gives the following: COROLLARY 3.1.7: C 2 is the prime semifield having no characteristic. ...