Description Details
...the research of Smarandache problems to readers, comprehensively and systematically, including the mean value of arithmetic functions, identities and inequalities, infinite series, the solutions of special equations, and put forward to some new interesting problems. We hope that the readers could be interested in these issues. At the same time, this book could open up th...

Description Details
...s; Arithmetic functions; Divisibility properties of numbers and functions; and Some irrationality results. Chapter 1 deals essentially with geometric inequalities for the remarkable elements of triangles or tetrahedrons. Other themes have an arithmetic character (as 9-12) on number theoretic problems in Geometry. Chapter 2 includes various diophantine equations, some of wh...

Full Text Search Details
...and Some irrationality results. Chapter 1 deals essentially with geometric inequalities for the remarkable elements of triangles or tetrahedrons. Othe... ...zed Bisector Theorem . . . . . . . . . . . . . . . . . . . . . . 11 3 Some inequalities for the elements of a triangle . . . . . . . . . . . . . . . .... ...os-Mordell inequality for tetrahedrons . . . . . . . . . . 25 8 On certain inequalities for the distances of a point to the vertices and the sides of ... .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 2 On certain inequalities involving the Smarandache function . . . . . . . . . 124 3 On ... ... involving the Smarandache function . . . . . . . . . 124 3 On certain new inequalities and limits for the Smarandache function . . . . 129 4 On two n... ...dache function . . . . . . . . . . . . . . . . . . . . . 156 12 On certain inequalities for Z(n) . . . . . . . . . . . . . . . . . . . . . . . . 159 1... ...al functions at factorials I . . . . . . . . . . . . . . 182 20 On certain inequalities for k . . . . . . . . . . . . . . . . . . . . . . . . . 189 ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 6 On Bonse-type inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 7 On... ...ies . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 7 On certain inequalities for primes . . . . . . . . . . . . . . . . . . . . . . . 241 8...

Table of Contents Details
...Smarandache n-expressions 54 T. Jayeo. la : Palindromic permutations and generalized Smarandache palindromic permutations 65 J. Sandor : On certain inequalities involving the Smarandache function 78 A. Muktibodh : Sequences of pentagonal numbers 81 J. Gao : On the additive analogues of the simple function 88 W. Zhu : On the primitive numbers of power p 92 A. Vyawahar...

Table of Contents Details
... of length 85 and related quantum error-correcting 9 J. Chen : Value distribution of the F.Smarandache LCM function 15 M. Bencze : About a chain of inequalities 19 C. Tian and N. Yuan : On the Smarandache LCM dual function 25 J. Caltenco, etc. : Riemannian 4-spaces of class two 29 Y. Xue : On a conjecture involving the function SL¤(n) 41 Y. Wang : Some identities inv...

Table of Contents Details
...ction 78 M. Bayat, H. Teimoori and M. Hassani : An extension of ABC-theorem 81 J. Ma : An equation involving the Smarandache function 89 C. Chen : Inequalities for the polygamma functions with application 91 W. Vasantha and M. Chetry : On the number of Smarandache zero-divisors and Smarandache weak zero-divisors in loop rings 96 M. Le : The function equation S(n) = Z(...

Description Details
...ralized to a set of preferences that can be transformed into a system of linear and/or non-linear homogeneous and/or non-homogeneous equations and/or inequalities. Many consistent, weak inconsistent, and strong inconsistent examples are given....

Description Details
...argument in favor of equality between the sexes. It offers both detailed argumentation and passionate eloquence in opposition to the social and legal inequalities commonly imposed upon women by a patriarchal culture. Just as in On Liberty, Mill defends the emancipation of women on utilitarian grounds, convinced that the moral and intellectual advancement of women would res...

Full Text Search Details
...society. And the elimination of discrimination would constitute a grave error. The inequalities among humans and the ensuing conflicts are the fuel... ...es of money create markets, labour, property, planning, wealth and capital. Mental inequalities lead to innovation and theory. Knowledge differenti... ...s in human society are all the results of incongruences, disparities, differences, inequalities and the negative and positive emotions attached to ...

Full Text Search Details
...use at home or their husbands extra marital sexual activities. These cultural inequalities thus increase the vulnerability of women to HIV infection... ...t of these women being uneducated and from rural areas suffer due to economic inequalities. When a husband dies the relatives are supposed to share ... ... thus: - 0 <= inf T + inf I + inf F <= sup T + sup I + sup F <= 3 + . The inequalities (2.1) and (2.4) of IFS are relaxed. This non-restricti...

Table of Contents Details
...al Sums, by Mihály Bencze and Florentin Smarandache, in Octogon Mathematical Magazine, Bra_ov, Romania, Vol. 15, No. 2, 810-812, 2007.......574 72. Inequalities for Integer and Fractional Parts, by Mihály Bencze, Florentin Smarandache, in “Octogon", Vol. 14, No. 1, 206-211, 2006..............577 73. Souvenirs from the Empire of Numbers, by Florentin Smarandache ..........

Table of Contents Details
...al Sums, by Mihály Bencze and Florentin Smarandache, in Octogon Mathematical Magazine, Bra_ov, Romania, Vol. 15, No. 2, 810-812, 2007.......574 72. Inequalities for Integer and Fractional Parts, by Mihály Bencze, Florentin Smarandache, in “Octogon", Vol. 14, No. 1, 206-211, 2006..............577 73. Souvenirs from the Empire of Numbers, by Florentin Smarandache ..........

Full Text Search Details
...> > > > > > : 0; if 0·k·4 [ p 4k+9¡5 2 ]; if k¸5and4 k+9is asquareofaninteger [ p 4k+9¡5 2 ]+1; if k¸5and4 k+9is not a square of an integer (1) and th... ...)´1, because [ n k ] § s=1 f(ks)¡ [ n¡1 k ] § s=1 f(ks) f(n) = 8 < : 1; if k is a divisor of n 0; otherwise 55 Finally, we use (12) to obtain some ne... ...k)) Hn ¸ n Y k=1 ( k p k) H [ n k ] ; where H m denotes the m-th partial sum of the harmonic series, i.e., H m = 1 1 + 1 2 +:::+ 1 m : All of the abov... ...r solv- ability of (22) is: p´5(mod24) or p´23(mod24) (23); (2) If (23) is valid, then there exists exactly one solution<x;y>2N 2 of (22) such that th... ...l-known fact that ³(0) = ¡ 1 2 and ³(¡2m)=0form2N(see [12]). Sincenumbers'(6k¡1);'(6k+1);Ã(6k¡1);Ã(6k+1);2¾(6k¡ 1);2¾(6k+1) are even, and the followin... ...(6k+1) are even, and the following inequalities '(6k¡1)+'(6k+1)·12k¡2 Ã(6k¡1)+Ã(6k+1)¸12k+2 ¾(6k¡1)+¾(6k+1)¸12k+2 are valid, and the fact that the las... .... First, we use that '(36k 2 ¡1)='(6k¡1):'(6k+1) and Ã(36k 2 ¡1) =Ã(6k¡1):Ã(6k+1); since, the functions ' and à are multiplicative. Second, we use tha... ...l k2N ¼( a g ¼( a g 1 )+k )=0: (23) But g ¼( a g 1 )+k 6 2 W.Thenwehavethatg ¼( a g 1 )+k > a g 1 . Hence, for all k2N a g ¼( a g 1 )+k <g 1 : The las... ...ctions, M. Abramowitz and I. Stegun (Eds.), National Bureau of Standards, Applied Mathe- matics Series, Vol. 55, 1964. [16] Mitrinovi¶ c,D.,M.Popadi¶ ...

Full Text Search Details
...ate Lord pervading everywhere.) This was the time when caste and religious inequalities were barbaric and humiliation of the downtrodden through sub...

Full Text Search Details
...at it is true for all n . Therefore if we prove that the (1) and (2) inequalities are true for all n which seems to be very probable; we will...

Full Text Search Details
...cients A, b and c are fuzzy numbers, the constraints may be considered as fuzzy inequalities with variables x and z. We use fuzzy linear programming... ...cients A, b and c are fuzzy numbers, the constraints may be considered as fuzzy inequalities. The decision space is defined by the constraints with ... ... [0, 1]}, where S α = {γ / (µ (γ) ≥ α)}. The parameter α is used to turn fuzzy inequalities into crisp inequalities. So we can rewrite equation (1)... ...icients A, b, c, ∈ FN ∪ [0 1] the constraints may be considered as neutrosophic inequalities with variables x and z. Construct a neutrosophic linear... ...ts A, b, c, ∈ FN ∪ [ 0 1 ] t h e constraints may be considered as neutrosophic inequalities with variables x and z. 25. Use the neutrosophic lin...

Full Text Search Details
...cients A, b and c are fuzzy numbers, the constraints may be considered as fuzzy inequalities with variables x and z. We use fuzzy linear programming... ...cients A, b and c are fuzzy numbers, the constraints may be considered as fuzzy inequalities. The decision space is defined by the constraints with ... ... [0, 1]}, where S α = {γ / (µ (γ) ≥ α)}. The parameter α is used to turn fuzzy inequalities into crisp inequalities. So we can rewrite equation (1)... ...icients A, b, c, ∈ FN ∪ [0 1] the constraints may be considered as neutrosophic inequalities with variables x and z. Construct a neutrosophic linear... ...ts A, b, c, ∈ FN ∪ [ 0 1 ] t h e constraints may be considered as neutrosophic inequalities with variables x and z. 25. Use the neutrosophic lin...

Full Text Search Details
...ems, we need the following lemma: Lemma. For any real number , we have the inequalities where is a positive integer such that . Proof. Let , then . A...

Full Text Search Details
... k)! n 2 Since the largest is known to converge, we can put the inequalities together and conclude that also converges. 115 To determine t...

Full Text Search Details
... . . f DP (4) – f DP (3) ≥ f DP (3) + 1. Summing up all the above inequalities, we have n-1 n-1 f... ...the inequality p i ki ≤ n 1/3 < p i ki +1 . Multiplying all the inequalities we get p 1 k1 *p 2 k2 p 3 k3 . . . p s ks ≤ {n 1/3...

Full Text Search Details
...plications of his policies, in poverty alleviation and in growing economic inequalities and social tensions. Napoleon III was a man for all seasons...