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...For any positive integer n, let (n) be the Euler function, and am(n) denotes the Smarandache m-th power residues function of n. The main purpose of this paper is using the elementary method to study the number of the solutions of the equation (n) = am(n), and give all solutions for this equation....

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...FuRuiqin,YangHai OntheSmarandachefunction 103 WangYongxing Meanvalueofthek powercomplementsequences 107 FengZhiyu Onthem thpowerfreenumbersequences 11... ...r n, let φ(n) be the Euler function, and am(n) denotes the Smarandachem th power residues function of n. The main purpose of this paperisusingtheeleme... ... n = p α 1 1 p α 2 2 ···p α k k denotes the factoriza tion ofn into prime powers. The Smarandachem th power residues function a m (n)aredefinedas a m ... ...t n = p α 1 1 p α 2 2 ···p α k k denotes the factorization of n into prime powers, then from thedefinitionsofφ(n)anda m (n)wehave a m (n) =p β 1 1 p β ... ...omplement function b k (n) of n is the smallest positive integer such that nb k (n) is a perfect k th power. In problem 29 of reference [1], Professor... ...chek thpowercomplements b k (n) is the smallest positive integer such that nb k (n) is a complete k th power, see problem 29 of [1]. Similar to the Sm... ...chek thpowercomplements b k (n) is the smallest positive integer such that nb k (n) is a perfect k th power (see problem 29 of [1]). Similar to the Sm... ...le mentssequence. Thatis,b m (n)denotesthesmallestpositiveintegersuchthat nb m (n) is a complete m th power. In problem 29 of the reference [1], Pro ... ...m th power comple ment b m (n) is the smallest positive integer such that nb m (n) is a complete m th power, see problem 29 of [1]. Now for any posit...

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...us to see the benefits of acting selflessly. We would know that we were in the power of the kind and eternal Creator. Just imagine how natural it woul... ...ualpath. This will ensure that we strive for all three objects with an equally powerful desire from the very outset, regardless of the fact that we pe... ...to rid ourselves of the ego, and will eventually realize that it is beyond our power to oust egoism, since it was the Creator who had endowed us, His ... ...ttothe Creator to save us, asking the Creator to reveal Himself to give us the power to break out from the prison of our egoism into the spiritual wor... ...e, apart from the pleasure for the food. That pleasure is measured in size and power, or in TheDiningTable -53- quantity and quality, according to th... ...rofourbeing.ThisfeelingabouttheCreatoris called "faith." T h ep r o c e s sc a nb ea c c e l e r a t e di fw em a k et h i sa n important objective. T... ... Creator. -382- ATTAININGTHEWORLDSBEYOND 31 ASeedofAltruism Ho wc a nah u m a nb e i n g — w h ow a sc r e a t e dw i t ht h e qualities of absolute ...

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...f the twenty-first century mathematics has been not only the recognition of the power of abstract approach but also its simplicity by way of illustr... ..., ; the unit element of G/N, X ≠ e 1 . By the very form of elements of G/N, X = Nb, b ∈ G, so that X p = (Nb) p = Nb p . Since e 1 = Ne, X p = e... ... (Nb) p = Nb p . Since e 1 = Ne, X p = e 1 , X ≠ e 1 translates into Nb p = N, Nb ≠ N. Thus b p ∈ N, b ∉ N. Using one of the corolla... ...eorem we must finally show that c ≠ e. However, if c = e, b o(N) = e, and so (Nb) o(N) = N. Combining this with (Nb) p = N, p / o(N), p a prime ... ... = N. Combining this with (Nb) p = N, p / o(N), p a prime number, we find that Nb = N, so b ∈ N, a contradiction. Thus c ≠ e, c p = e, and we have... ...p of G, o(ST) | o(G); thus p γ | o(G) violating the fact that α is the largest power of p which divides o(G). Thus no such subgroup T exists, and S ...

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...e . . . . . . . . . 35 10 The area of a Pythagorean triangle, as a perfect power . . . . . . . . . . . 39 11 On Heron Triangles, III . . . . . . . . .... ... equation of degree 3 . . . . . . . . . . . 80 12 On two equal sums of mth powers . . . . . . . . . . . . . . . . . . . . . . . 83 13 On the equation ... ...1991, pp.260- 263. 38 10 The area of a Pythagorean triangle, as a perfect power Let ABC be a Pythagorean triangle of legs b;c and hypothenuse a. Then... ... 6 4 = 6 5 . Thus, the area of a Pythagorean triangle may be a perfectnth power, forn 3. However, the following can be proved: Theorem 3. The area o... ...: Theorem 3. The area of a primitive Pythagorean triangle cannot be an nth power (for any n 2). Proof. Let in (1) d = 1. Then area =uv(v 2 u 2 ). L... ...rion X n m 0 b n $ X n m 0 2 n (log 2 n +a) (2 n +b) = X n m 0 2 n (nb +a) (2 n +b) = X n m 0 c n : Now, lim n!1 c n+1 c n = 1 2 1 , b... ...rion proves the theorem for 6= 1. But for = 1 we get the series X 2 n (nb +a) 2 n +b , which is clearly divergent. Bibliography 1. J. S andor, On...

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...zy subgroups of a cyclic group. THEOREM [16]: Let G be a group of prime power order. Then G is cyclic if and only if there exists a fuzzy subgroup... ... δ ⊗ ∈ . Now we proceed on to define fuzzy polynomial subring and fuzzy power series subring. DEFINITION [40]: Let S be a commutative ring with... ...t S be a commutative ring with identity. Let R = S [[x 1 , …, x n ]] be a power series ring in the indeterminates x 1 , x 2 , …, x n over S. Let A b... ... 1 K K K = ∞ = , min (t j )j = 1, 2, …, n}} then A is called a fuzzy power series subring of R where t j is defined as in the above equation. ... ...0) = A(0). If A has the sup property and B is pure in A, then n B t = (nB) t for all t ∈ (0, A(0)]. Proof: Exactly as in case of fuzzy group a... ...pose that A has the sup property. If B is S-pure in A, then for all n ∈ N, nB = B ∩ nA. ii. Suppose that A and B have sup property. If for all n ∈ ... ... = B ∩ nA. ii. Suppose that A and B have sup property. If for all n ∈ N, nB = B ∩ nA, then B is S-pure in A. iii. Suppose that A ∗ is S-torsi... ...s S-torsion free, then B is S-pure in A if and only if for all n ∈ N, nB = B ∩ nA. iv. For C ⊆ B. If C is S-pure in B and B is S-pure in A, th... ... S-semigroup. Let n ∈ N then i. n U U Ω α α Ω α α ∈ ∈ =         nB B ii.         ∩         = ∩ ∈ ∈ ∈ U U U Ω α α Ω α α Ω ...

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...nd eventful lifetime.” RUSSELL KIRK, novelist: “The scenes are drawn with power. Bartlett is an accomplished writer.” PAUL ENGLE in The Chicago Tri... ...volution of 1910 and descrip- tive of hacienda life, Forward, Children!, a powerful antiwar novel, and numerous short stories. He was also the author... ...happy surprise he did so, in a work which is at once poetic, dramatic and powerful. In Sappho’s Journal he does more than create a vague illusion of... ...must evoke beauty. I must capture the magnificence of the sea and use its power. I must trap changes and repetitions, the storm’s core and summer’s ... ...g’s arrival, the brevity of summer, the dying year. It has the shepherd’s power, the forest’s—passion tamed and sanctified. Another one I like is... ...

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...RSDAY, MARCH 91 ' 8,00 p. m.—Thompson Course enter- tainment, J. H, Leland Powers. FOOTBALL ELECTION to Francis Bowes Sayre 1909 Manag^e Team in 1908 ... ...et. The sophomores should have realized that it was useless to attack the "powerful, energetic and numberless class of 1910.'" ''To such a class as ou... ...; Stables Best of Service promised in every respect. Tblkphonh 45-3 LELAND POWERS Spring Strrht Williamstown Last Entertainment of Thompson Course Hel... ...ul fiancee, S and returns to France to marry the Princess Henriette. j Mr. Powers, in the three parts j into which the drama is divided, impersonated ... ...an iiiiiietiioH ity that is gieater than he can manage. As Lady .Mioy. Mr. Powers impersoniilccl a woniaii, who, th(! hearer is led to Indieve, was of... ...wly Furnished R. J. EAHAR,;Proprietor, "ij^ -^^y TheWoman ofRefinenjent iB nB rnreful to-ilny of her papers as Bhe 1» of her portiinies. Hniily ehoBC-... ...iil(ls,Prop.KAMt C. B. FOWLER, ..Trucking.. Harvard Medical School Boston, nB%%. With the completion of the new buildings, which trerc dedicated &t-pt...

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...} are subgroups of L. Hence L 5 (3) is a S-loop. THEOREM [72]: Every power associative loop is a S-loop. Proof: Obvious by the very definiti... ...associative loop is a S-loop. Proof: Obvious by the very definition of power associative loop as every element generates a group. Hence every pow... ...n, m – 1) = 1 with m < n is a S-loop. Proof: Since L n (m) ∈ L n are power associative loops as every element generates a group, we see each L ... ... obtain the relations between these concepts. THEOREM [72]: Let L be a power associative loop, L is then a S-commutative loop. Proof: Direct b... ...he claim. The following have been proved about zero divisor in case of power associative and diassociative loop algebras. THEOREM [56]: Let K... ... DEFINITION 3.5.1: Let L = [- ∝, ∝]. Define for a , b ∈ L, a ∗ b = ma + nb where m, n ∈ L. Clearly {L, ∗, (m, n)} is a groupoid of infinite order....

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...ood or bad, they are now tendered to the appropriation of your individual house, the Messrs. Ticknor & Fields, according to the amplest extent of any power to make such a transfer that I may be found to possess by law or custom in America....

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...ouse, the Messrs. TICKNOR & FIELDS, according to the amplest extent of any power to make such a transfer that I may be found to possess by law or cust... ...of human nature. But, on the contrary, the Peace Societies would, if their power kept pace with their guilty purposes, work degradation for man by dra... ...xactly as the motives and the interests rise on behalf of which such awful powers are invoked. Fighting for truth in its last recesses of sanctity, fo... ... that history has recorded. 10 Memorials, and Other Papers arrived at the power of reflecting and comparing, we look for something nobler. We English... ...ich features at that time lay—1st, in velocity unprecedented; 2dly, in the power and beauty of the horses: 3dly, in the official connection with the g...

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...I finally resolved to inquire whether there might be some real good having power to communicate it- self, which would affect the mind singly , to the ... ...ures of Sense: with these three the mind is so absorbed that it has little power to reflect on any different good. [4] (1) [4] (1) [4] (1) [4] (1) [4]... ...ified. IV IV IV IV IV . . . . . To compare this result with the nature and power of man. (4) (4) (4) (4) (4) We shall thus discern the high- est degre... ...6) (6) (6) (6) (6) We might thus vainly endeavor to prove that men have no power of working iron. [31] (1) [31] (1) [31] (1) [31] (1) [31] (1) But as ... ...era- tions, [l], and from these operations again fresh instruments, or the power of pushing its investiga- tions further, and thus gradually proceeds ...

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...e country privately by the old Queen, to enrich certain relatives in M-ckl-nb-rg Str- 7 Some Roundabout Papers l-tz? Not all the treasure went. Non o... ...r, and why in his travels this and that point struck him; wherein lies his power of pathos, humour, eloquence; — that Minister of State, and what move... ...different state of things! Credite posteri. To see these nymphs — gracious powers, how beautiful they were! That leering, painted, shrivelled, thin-ar... ...haplain! Now we haste away to merrier scenes: to Tattersall’s (ah gracious powers! what a funny fellow that actor was who performed Dicky Green in tha... ...e rain roars and pours! The darkness becomes awful, always deepened by the power of the music — and see — in the midst of a rush, and whirl, and screa...

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... and admired, they had both, constantly refused to submit to such despotic Power. After having so nobly disentangled themselves from the shackles of P... ...mire so promising a Disposition, and had endeavoured by every means on his power to pre- vent it encreasing with her Years. He had actually so far ex... ... receiving this Billet flew on the wings of Love to Macdonald-Hall, and so powerfully pleaded his At- tachment to her who inspired it, that after a fe... ...g beneath thy roof.” “Your Freindship for my Daughter has indeed been most powerfully exerted by throwing her into the arms of an unprincipled Fortune... ...ite, and the ten- derness she felt for Augustus so poignant that I had not power to start any other topic, justly fearing that it might in some unfors...

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...ere indeed been any act of one solitary insulated indi- vidual, armed with power so appalling over the hearts of men, as that exterminating murder, by... ...s a philosopher, and threw him into a profound reverie upon the tremendous power which is laid open in a moment to any man who can reconcile himself t... ... from No. 29 Ratcliffe High- way, and the poor vanquished imagination sank powerless before the fascinating rattlesnake eye of the murderer. There is ... ...lossal must have been his audacity! prob- ably, also, his skill and animal power! Moreover, the unknown enemy (whether single or double) would, doubtl... ...f the mist. T wo feet ahead of you, a man was entirely withdrawn from your power of identification; and, on overtaking him, you could not venture to c...

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...elow. The others, to whom I served as cushion, were not damaged beyond the power of yelling, and were quite restored in half-an-hour, but I was underm... ...o be varied, in the curious childish love of reiteration with its soothing power. I am afraid that any true picture of our parents, especially of my m... ...ried out consistently at whatever present cost to us both, that I owe such powers of locomotion as I possess, and the habits of exertion that have 6 ... ...ame his guardian friend, and protected him from all the roughnesses in his power. If there were a fault in that excel- lent Coles, it was that he made... ...urch, and it was recognised that some preparation was desirable beyond the power of repeating the Church Catechism. This was all that had been require...

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...ism that the faculties are suddenly excited to the highest degree of their power or driven to utter disorganization. Physiologists have long wondered ... ...ed by the rapidity of her impressions, the poor woman had the supernatural power of emitting more ideas and bring- ing to the surface more recollectio... ...grotesque effects which terrify the imagination at a moment when it has no power except to foresee misfortunes and exaggerate them? Madame Birotteau s... ...God can do no harm,” said Birotteau, gravely. “But the oil in nuts is also powerful, wife. I made this discovery just as I made that of the Double Pas... ... Sultans and the Carminative Balm are two operative compounds, of a motive power which acts without risk upon the internal energies and seconds them. ...

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...ve, and chancing to walk near the course through which it passed, has felt powerfully, in the silence and desertion of the streets and in the stagnati... ...ight to make the least resistance; for, according to himself, irresistible power creates the very highest species of right, so that it is rebellion of... ...nument of his genius; for such was the impression of awe, and the sense of power left behind, by the strength of conception manifested in this murder,... ...on him for three lives, and expressed his regret at not hav- ing it in his power to knight him. We, on our part—we amateurs, I mean—having heard that ... ...gain challenged with shouts the stormiest for the new glee. I made again a powerful effort to overrule the challenge. I might as well have talked to t...

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...f, and, above all, in that full and sympathetic masculine grasp of a man’s powerful mind, which is necessarily denied to me. But these fittest of all ... ...e has left few, if any, literary monuments to record what his intellectual powers were; and even in our common profession the ordinary course and prac... ... no more heinous fault than not having worked up to the full extent of his powers! He kept his promise of diligence, and never again incurred reproof,... ...was far more affected than he then had 20 Life of John Coleridge Patteson power to express. He says: ‘I heard Archdeacon Wilberforce in the morning, ... ...on to have his way, and the game, though closely contested, was saved. His powers of defence were indeed remarkable. I saw the famous professional cri...

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...querable chestnuts cluster ‘like herded elephants’ upon the spur of a mountain, is to rise to higher thoughts of the powers that are in Nature. Betwee... ...t night, a frequenter of hills and forests, an old honest countryman, has, in the end, a sense of communion with the powers of the universe, and amica... ...ice of logic; it is the poetry of the man’s experience, the philosophy of the history of his life. God, like a great power, like a great shining sun, ... ... sell. He had noticed the blue wool which hung out of my pack at either end; and from this he had decided, beyond my power to alter his decision, that... ...at I dealt in blue-wool collars, such as decorate the neck of the French draught-horse. I had hurried to the topmost powers of Modestine, for I dearly...

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...of the stars reaches to many actions and events which are not any way in the power of reason; as sickness, death, and what we com- monly call accident... ...armies abroad, which I might, and could very largely have done; but those in power have wisely discouraged men from meddling in publick concerns, and ... ... great men, and publick spirits, to the scandal of religion, and reproach of power; and if sover- eign princes and astrologers must make diversion for...

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...ls from my Reason and my more com posed Judgment to go home, yet I had no Power to do it. I know not what to call this, nor will I urge, that it is a... ...ore terrible in its Nature, or more immediately directing to the invisible Power which alone directs such Things, yet no sooner was the first Fright ov... ...me, humane and brutal, whence are we ? Sure we are all made by some secret Power, who form’d the Earth and Sea, the Air and Sky; and who is that? Then... ... He guides and governs them all, and all Things that concern them; for the Power that could make all Things, must certainly have Power to guide and di... ...ught to this miserable Circum stance by his Direction, he having the sole Power, not of me only, but of every Thing that happen’d in the World. Immed...