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...e Numbers also clearly accumulate on defined spiral-graphs. A mathematical analysis shows, that all these spiral graphs represent special quadratic polynomials. Even the famous Fibonacci Numbers play a part in the structure of the Square Root Spiral. Fibonacci Numbers divide the Square Root Spiral into areas and angles with constant proportions, which are linked to t...

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...This book has eight chapters. The first chapter is introductory in nature. Polynomials with matrix coefficients are introduced in chapter two. Algebraic structures on these polynomials with matrix coefficients is defined and described in chapter three. Chapter four introduces natural product on matr...

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...umbers. Chapter two introduces the notion of finite complex numbers; algebraic structures like groups, rings etc are defined using them. Matrices and polynomials are constructed using these finite complex numbers. Chapter three introduces the notion of neutrosophic complex modulo integers. Algebraic structures using neutrosophic complex modulo integers are built and around...

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...Abstract In this paper, we use 4-cyclotomic cosets of modulo n and generator polynomials to describe quaternary simple-root cyclic codes of length n = 85. We discuss the conditions under which a quaternary cyclic codes contain its dual, and obtain some quantum error-correcting codes of length n = 85, ...

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...ee BY MUDDEBIHAL M. H. AND SYED BABAJAN. . . . . . . . . . . . . . . . . . 64 Varieties of Groupoids and Quasigroups Generated by Linear-Bivariate Polynomials Over Ring Zn BY E.ILOJIDE, T.G.JAIYEOLA AND O.O.OWOJORI . . . . . 79 New Characterizations for Bertrand Curves in Minkowski 3-Space BY BAHADDIN BUKCU, MURAT KEMAL KARACAN AND NURAL YUKSEL . . 98 Respectable Gra...

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...uadratic forms; 2. W. Liang, An identity for Stirling numbers of the second kind; 3. Y. Wang and Z. Sheng, Two formulas for x^n in terms of Chebyshev polynomials. Other papers are concerned with the number-theoretic Smarandache problems and will enrich the already rich stock of results on them. Readers can learn various techniques used in number theory and will get familia...

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... a Natural Number 55 P. Zhang : Some identities on k-power complement 60 Y. Wang and Z. Sheng : Two Formulas for xn being Represented by Chebyshev Polynomials 64 X. Chen : Two Problems About 2-Power Free Numbers 70 X. Ma : The Asymptotic Formula of P n·x log Pad(n) 72 Q. Tian : On the K-power free number sequence 77 C. Lv : On a generalized equation of Smarandach...

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.... The law of distributivity is not true. Further the MOD function in general do not obey all the laws of integration or differentiation. Likewise MOD polynomials in general do not satisfy the basic properties of polynomials like its roots etc. Thus over all this study is not only innovative and interesting but challenging. So this book which is only full of problems based ...

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...d product and the other MOD natural neutrosophic zero dominated product. Both the product are different. Finally using these MOD subsets matrices and polynomials are defined and developed. The new notion of MOD subset matrices are defined and these collections also under ‘+’ (or ‘x’) are only semigroups. On similar lines MOD subset polynomials are defined and they also for...

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... . . . 46 The Forcing Domination Number of Hamiltonian Cubic Graphs BY H.ABDOLLAHZADEH AHANGAR AND PUSHPALATHA L. . . . . . . . . . 53 Permutation Polynomials modulo n, n 6= 2w and Latin Squares BY VADIRAJA BHATTA G. R. and SHANKAR B. R. . . . . . . . . . . . . . . . . 58 Graphoidal Tree d - Cover BY S.SOMASUNDARAM, A.NAGARAJAN AND G.MAHADEVAN . . . . . . . . 66 Com...

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...e Buchsteiner Loops BY J.O.AD´EN´IRN and Y.T.OYEBO. . . . . . . . . . . . . . . . . . . . . . . . .01 Generalizations of Poly-Bernoulli Numbers and Polynomials BY HASSAN JOLANY, M.R.DARAFSHEH AND R.EIZADI ALIKELAYE . . . . 07 Open Alliance in Graphs BY N.JAFARI RAD AND H.REZAZADEH . . . . . . . . . . . . . . . . . . . . . 15 The Forcing Weak Edge Detour Number of a G...

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... the Smarandache prime part 74 W. He : The existence of solutions to the Direchlet problem 78 C. Aguilar-Chavez, etc. : On the fifth-kind Chebyshev polynomials 82 S. Luo : Deficient functions of random Dirichlet series of finite order in the half plane 85 J. Li : On the Pseudo-Smarandache-Squarefree function 93 L. Cheng : On the mean value of the Pseudo-Smarandache fu...

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... Theorems in Fuzzy n-Normed Spaces BY SAYED KHALIL ELAGAN and MOHAMAD RAFI SEGI RAHMAT . . . . . . . . . . 45 A Result of Ramanujan and Brahmagupta Polynomials Described by a Matrix Identity BY R. RANGARAJAN . . . . . . . . . . . . . . . . 57 Biharmonic Slant Helices According to Bishop Frame in E3 BY ESSIN TURHAN and TALAT K¨ORPINAR . . . . . . . . . . . . . . . . . ...

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...ery simple functions like y = nx has several zeros in MOD planes where as they are nice single line graphs with only (0, 0) as the only zero. Further polynomials in MOD planes do not in general follows the usual or classical laws of differentiation or integration. These drawbacks becomes a challenging issue. So study in this direction is open. In fact several open conjectu...

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...r only on these new structures, study, develop, describe and define the notion of SSS-linear functionals, SSS-eigen values, SSS-eigen vectors and SSS-polynomials....

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... n roots”, n is untrue in case of polynomial in polynomial quaternion rings in general. Further the very concept of derivative and integrals of these polynomials are untrue. Finally interval pseudo quaternion rings also behave in an erratic way. Not only finite real quaternion rings are studied, but also finite complex modulo integer quaternion rings, neutrosophic finite q...

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... dimensional vector space V. Let p and f be the minimal and characteristic polynomials for T, respectively i. p divides f. ii. p and f have the s... ...trong basis of V. Proof: By an example. Let V = Q [x] be the set of all polynomials of degree less than or equal to 10. V is a vector space over ... ...ere 10 serves as the unit element of P. 8 15 Z [x] is a additive group of polynomials of degree less than or equal to 8 and M 3 ×3 is the additive... ... a ij ∈ Q}, collection of all 2 × 2 matrices with entries from Q. V 2 = {Polynomials of degree less than or equal to 4}. Clearly V is a finite d... ...2 × 2 matrices with entries from Q where as V 2 is the collection of all polynomials of degree less than or equal to 4. So no relation among eleme... ...sion 2, say V 1 = F × F and V 2 is a vector space of dimension say 6 all polynomials of degree less than or equal to 5 with coefficients from F. W... ...f dimension 8 where W 1 is a vector space of dimension 3 i.e. W 1 = {all polynomials of degree less than or equal to 2} with coefficients from F a... ... 1 = {the set of all 2 × 5 matrices with entries from Q} and V 2 = {all polynomials of degree less than or equal to 5 with entries from Q}. Clear... ...ere V 1 = Q × Q × Q × Q be the group under ‘+’ and V 2 = The set of all polynomials over the field Q ( ) 2 . Now V 1 is a linear algebra over Q ...

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... [] > + < 1 x x Z 2 . 10. Find all principal ideals in [] x Z 5 3 = {all polynomials of degree ≤ less than or equal to 5}. (Hint: We say any idea... ... Let F n [x] be a vector space over the field F; where F n [x] contains all polynomials of degree less or equal to n. Now β = {1, x, x 2 , … , x n } ... ...hen p(x) = q(x) if and only if for every integer i ≥ 0; a i = b i . Two polynomials are declared equal if and only if their corresponding coeffic... ...der +. It is left for the reader to verify S[x] under multiplication of two polynomials is a commutative semigroup. So (S[x], +, •) is a commutative... ...ook any special properties enjoyed by the polynomial semirings or about the polynomials in them. For more about polynomial semirings one can refer L... ... p(a) = 0 for some a ∈ Z 0 \{0}? 7. Define concepts like reducibility of polynomials in Q o [x] or in general in a polynomial semiring. 2.5... ... Example 3.3.3: Let Z o be a semifield. o n Z [x] denote the set of all polynomials of degree less than or equal to n. o n Z [x] is a semivecto... ...arly dimension of V is 5. Example 3.3.5: Let V = 0 7 Z [x] = {set of all polynomials of degree less than or equal to 7 with coefficient from Z o ... ...pace of R o . Example 3.3.7: Z o [x] is a semivector space over Z o . All polynomials of even degree in Z o [x] denoted by S is a subsemivector sp...

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.../ a ij ∈ Q}, collection of all 2 × 2 matrices with entries from Q. V 2 = {Polynomials of degree less than or equal to 4}. Clearly V is a finite ... ... 2 × 2 matrices with entries from Q where as V 2 is the collection of all polynomials of degree less than or equal to 4. So no relation among ele... ...nsion 2, say V 1 = F × F and V 2 is a vector space of dimension 6 say all polynomials of degree less than or equal to 5 with coefficients from F. ... ...imension 8 where W 1 is a vector space of dimension 3 i.e. W 1 = {all polynomials of degree less than or equal to 2} with coefficients from F ... ...al is a pair given by (det (xI 1 – A 1 ), det (xI 2 – A 2 )). If both the polynomials in the bipolynomial has roots in the field over which the bi... ... characteristic bivalues in (For is the algebraically closed field and all polynomials are linearly reducible over , the field of complex numbers.)... ...44 11 22 det xI A det xI A ×× −∪ − = ((x 2 – 4) ∪ x 2 (x –1) 2 ). Both the polynomials has real roots so T = T 1 ∪ T 2 (or A = A 1 ∪ A 2 ) has c... ...... f C β++ β ∪ f 2 ( ) ( ) 22 22 11 2k 'k ' C ... f C β ++ β Choose polynomials 11 1 ... k f f and 22 1' ... k f f , such that 111 iji j ... ... bioperator T = T 1 ∪ T 2 is the unique monic generator of the biideal of polynomials over F which annihilate T = T 1 ∪ T 2 . The reader is reque...

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...∪ Z 3 ; R[x] = Z' 2 [x] ∪ Z' 3 [x]. Here Z' 2 [x] denotes the set of all polynomials in x of degree less than or equal to 7 and Z' 3 [x] = set of al... ...ynomials in x of degree less than or equal to 7 and Z' 3 [x] = set of all polynomials of degree less than or equal to 5. Let x 2 + 1 ∈ R[x]; the ... ...g can be defined and the notions of reducibility or irreducibility of the polynomials can be defined. Here it has become important for us to define ... ..., 1)} ∪ {1, x, x 2 , … , x 8 }. ] x [ Z o 8 denotes the collection of all polynomials of degree less than or equal to 8 with coefficient from Z o . ... ...e, Let V = V 1 ∪ V 2 , V 1 = Z o × Z o and V 2 = ] x [ Z o 3 = {all polynomials of degree less than or equal to 3). V is a bisemivector space o... ...s a strict bisemigroup. Example 7.4.9: Let V = Z[x] ∪ Z 2 ×3 i.e. the polynomials over Z under '+' and the set of all 2 × 3 matrices with entrie... ...pseudo semivector space. Example 7.4.11: Let Q 7 [x] be the set of all polynomials of degree less than or equal to 7. S = ] x [ Z o 7 ∪ Q o is ... ...(N, +, ) where N = Z ∪ R[x] where Z is the near-ring and R[x] the ring of polynomials over the reals R, a free binear-ring? 5. What is the smallest ... ...a ij ∈ Q}, collection of all 2 × 2 matrices with entries from Q. V 2 = {Polynomials of degree less than or equal to 4}. Clearly V is a finite ...

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...+k)::: (n + 1)=x n+k+1 = = e x G(x) H(x) x 2n+1 ; where G(x) and H(x) are polynomials with integer coe cients. By taking the derivative of the right ... ...f many numbers may be easily established by using a well known property of polynomials with integer coe cients. Proposition 1. If the polynomial P (x)... ... x m +P m (x) 1 +Q m (x) ; where x = tg ; andP m (x);Q m (x) are certain polynomials of integer coe cients, of order at most m 1. This assertion fo... ...nd only if n62f1; 2; 3; 4; 6g. Remark. Many other properties of cyclotomic polynomials appear in [1]. See also article 3.8. For applications of irrati...

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...q ... x q q ) x ( q + + + = Addition is performed as in the case of usual polynomials. Define ☼ on R [x [0 1] ] by p(x) ☼ q(x) = p(x) for p(x), ... ...x s y t + y t s s where s + t = 1. s, t, r ∈ [0, 1]. We have like other polynomials we can extend the fuzzy polynomials to any number of variables ... ... s j , t p ∈ [0, 1], 1 ≤ i ≤ n, 1 ≤ j ≤ m and 1 ≤ p ≤ k. i.e. we add two polynomials as in case of usual polynomials by adding the like powers of x... ... be denoted by 0. Now we have to define multiplication ‘o’ of two fuzzy polynomials in R[x [0, 1] ]. For R x and x 1 1 t s ∈ [x [0, 1] ] 1 t s... ...eg (p(x) o q(x)) in general are not equal to deg p(x) + deg q(x) in fuzzy polynomials. 196 We have nice results analogous to usual polynomial ri... ...t a field or an integral domain we have to show R[x [0, 1] ] contains two polynomials 0 ≠ p(x), 0 ≠ q(x) ∈ R[x [0, 1] ] with p(x) o q(x) = 0. Take p(... ... L where for each c i = p i + q i provided γ i = s i i.e. we add two polynomials by adding terms of x which have equal powers. Define multi... ... p 1 s if p s j i j i j i γ γ γ Example 1.12.9: Let p(x), q(x) be fuzzy polynomials in a fuzzy polynomial semiring Z[x [0, 1] ]. p(x) = x 0.11 + ... ...ve S-semigroup of finite order. Example 2.1.4: Let Q[x] be the ring of polynomials; Q[x] is a semigroup under multiplication. P = Q \ {0} ⊂ Q[x] ...

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... that R[x] is a commutative ring with unit 1. Elements of R[x] are called polynomials and R[x] is a polynomial ring in the indeterminate x with coef... ...n) …; (n) denotes ideals of Z where n ∈ Z. Example 2.7.2: The ring of polynomials in finitely many variables over a Noetherian ring is Noetheria... ...3.6.5: Let Z[x] be the S-ring II, M = Z[x], the polynomial ring with only polynomials of even degree. Then M is a S-module II over the S-subring Z bu...

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...Q, V 2 = Q[x] is a vector space over Q and V 3 = {set of all odd degree polynomials with coefficients from Q}. V is a S-Q-vectorial space and 1 V... ...trong basis of V. Proof: By an example. Let V = Q [x] be the set of all polynomials of degree less than or equal to 10. V is a vector space over Q...

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... 2 that admits straightforward solutions in terms of Gaussians and Hermite polynomials similar to the ordinary point-particle oscillator. There are no...