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...y the properties of finite real quaternion ring which was introduced in [2000]. Here a complete study of these finite quaternion rings are made. Also polynomial quaternion rings are defined, they happen to behave in a very different way. In the first place the fundamental theorem of algebra, “a nth degree polynomial has n and only n roots”, n is untrue in case of polynomia...

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...a circulant matrix over GF(2)). We denote the ‘norm’ of v by r(v). We just prove the following theorem. THEOREM 1.2: Suppose ____GF(2N) has the polynomial representation g(x) over GF(2) such that the gcd(g(x), xN +1) has degree N – k, where 0 _ k _ N. Then the ‘norm’ of the word generated by _ is ‘k’. ...

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...g the new class of loops. All these non-associative linear algebras are defined over the prime characteristic field Zp, p a prime. However if we take polynomial non associative, linear algebras over Zp, p a prime; they are of infinite dimension over Zp. We in chapter four introduce the notion of groupoid vector spaces of finite and infinite order and their generalizati...

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...roduces for the first time the new notion of neutrosophic rings and some special neutrosophic rings like neutrosophic ring of matrix and neutrosophic polynomial rings. Chapter three gives some new classes of neutrosophic rings like group neutrosophic rings, neutrosophic group neutrosophic rings, semigroup neutrosophic rings, S-semigroup neutrosophic rings which can be real...

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...R ALI M.M. . . . . . 71 Some Results on Super Mean Graphs BY R.VASUKI AND A.NAGARAJAN. . . . . . . . . . . . . . . . . . . . . . . . .82 Chromatic Polynomial of Smarandache νE-Product of Graphs BY KHALIL PARYAB AND EBRAHIM ZARE. . . . . . . . . . . . . . . . . . . .97 Open Distance-Pattern Uniform Graphs BY BIBIN K. JOSE . . . . . . . . . . . . . . . . . . . . . . . ...

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...ly these finite structures can find applications in several important problems related to automaton theory. The authors also mention about matrix and polynomial MOD rectangular natural neutrosophic semigroups. Several innovative results are obtained....

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...ina(60472068). 14 RESEARCHONSMARANDACHEPROBLEMSINNUMBERTHEORYII f(y) is a polynomial ofy with degreem = [ k+1 2 ], andε is any fixed positive number. ... ...(m) = mx 1 m ζ m+1 (2) R 1 m f(logx)+O x 1 2m + , where f(y) is a polynomial of y with degree k, and ε is any fixed positive number. 16 RESEA...

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...ults . . . . . . . . . 488 18.6 Discussion and conclusion . . . . . . . 491 18.7 References . . . . . . . . . . . 492 18.8 Appendix: PCR rule with polynomial complexity . . . 495 18.8.1 Algorithm for the conjunctive rule . . . . . 496 18.8.2 Proposed PCR rule of combination . . . . . 499 18.8.3 Conclusion . . . . . . . . . 502 Chapter 19 Fusion of ESM allegiance rep...

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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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...gs and its properties 33 2.3 Semirings using distributive lattices 38 2.4 Polynomial semirings 44 2.5 Group semirings and semigroup semirings 46 ... ...subring of Z. Example 1.3.13: Let R be a commutative ring and R[x] be the polynomial ring. R ⊂ R[x] is a subring of R[x]. In fact R[x] is an integ... ... {0, 2, 4, 6, 8, 10} is an ideal of Z 12 . Example 1.3.16: Let R[x] be a polynomial ring. p(x) = p 0 + p 1 x + … + p n x n be a polynomial of d... ...bring of M 2 ×2 , which is not an ideal of M 2 ×2 . 6. Let Z 7 [x] be the polynomial ring. Suppose p(x) = x 2 + 3. Find the ideal I generated by ... ... the quotient ring Z 12 / I. Is Z 12 / I a field? 8. Let Z 7 [x] be the polynomial ring over Z 7 . I = 〈x 3 + 1 〉 be the ideal generated by the... ... [] > + < 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... ...y 'V is a vector space over the field F'. Example 1.4.1: Let R[x] be the polynomial ring where R is the field of reals. R[x] is a vector space ov... ... 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 } ... ... n [x]. The dimension of F n [x] is n + 1. Example 1.4.8: Let F[x] be the polynomial ring which is a vector space over F. Now the set {1, x, x 2 ,...

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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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...perator on a finite dimensional vector space V. If f is the characteristic polynomial for T, then f(T) = 0, in other words the minimal polynomial di... ...(T) = 0, in other words the minimal polynomial divides the characteristic polynomial for T. THEOREM: (CYCLIC DECOMPOSITION THEOREM): Let T be a ... ... 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... ...he ring of integers modulo n, n a composite number Z n [x] will denote the polynomial ring in the variable x with coefficients from Z n . DEFINIT... ...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 ... ...ing examples. Example 1.2.4: Let M = R[x] × R[x] be a direct product of polynomial rings, over the ring R × R. Clearly M = R[x] × R[x] is a S-vec... ... = {0, 1, 2, … , 10, 11} be the S-ring of characteristic two. Consider the polynomial ring Z 12 [x] . Clearly Z 12 [x] is a S-vector space over the ... ...lti vector space). For consider the vector space Z 6 [x]. Z 6 [x] is the polynomial ring in the indeterminate x with coefficients 25 from Z 6 . ... ...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...

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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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... . . . 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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.../ 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 ... ...le: Example 2.4.1: Let T be a linear bioperator on R × R ∪ {Space of all polynomial functions from R into R which have degree less than or equal ... ...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... ...ld F is algebraically closed for example the field of complex numbers every polynomial over F can be factored, however if F is not algebraically clo...

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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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...braic structures. One such is the reducibility or the irreducibility of a polynomial, or we can say in some cases a polynomial is such that it canno... ...t of notation we just denote it as S-ring. Example 1.4.1: Let F[x] be a polynomial ring over a field F. F[x] is a S-ring. 22 Example 1.4.2: L... ...field given in example 1.5.5, L has no characteristic associated with it. Polynomial semirings are defined analogous to polynomial rings. If S[x] is... ...ynomial semirings are defined analogous to polynomial rings. If S[x] is a polynomial semiring then S[x] is a commutative semiring with 1. To create ... ...if and only if for all s ∈ S; s is quasi regular. The concept of near polynomial rings is defined analogous to polynomial rings similarly matrix... ...gely as a matter of convenience. Example 1.7.1: F be a field, F[x] the polynomial ring; F[x] is a vector space over F. Example 1.7.2: Let V = ... ...rime bifield as F has no proper sub-bifields. Now we proceed on to define polynomial birings. DEFINITION 6.1.11: Let R = R 1 ∪ R 2 where R is ei... ...ominal biring R [x] = R 1 [x] ∪ R 2 [x]. Here R 1 [x] and R 2 [x] are polynomial rings over the rings or fields in the same indeterminate x. If ... ...e rings or fields in the same indeterminate x. If is easily verified that polynomial biring is a biring. DEFINITION 6.1.12: Let D = D 1 ∪ D 2 wh...

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...gs which are specially formed and illustrate them with examples. They are polynomial rings, matrix rings, direct product of rings, ring of Gaussian ... ... 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... ...cients from R. Example 2.5.1: Let Z be the ring of integers. Z[x] is a polynomial ring in the variable x. Z[x] is an integral domain. Example 2.... ...n. Example 2.5.2: Let Z n be the ring of integers modulo n. Z n [x] is a polynomial ring with coefficients from Z n . Z n [x] is a commutative ring... ...divisors if n is a non-prime. 31 It is interesting to note in the polynomial ring Z[x] every ideal is principal. Polynomial rings in seve... ... several variables can also be defined in a similar way. For if R[x] is a polynomial ring. Suppose y is another indeterminate then (R[x]) [y] is a p... ... = yx) denoted by R[x, y]. Suppose x 1 , …, x n are n variables then the polynomial ring in n variables is R[x 1 , x 2 , …, x n ] where we assume x... ...n 1940 by G Higman [33]. PROBLEMS: 1. Find ideals in Z 7 [x], the polynomial ring in the variable x. 2. Can Z 8 [x] have zero divisors? Fin... ...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...

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... A i = ) over ( L K T L i δ ⊗ ∈ . Now we proceed on to define fuzzy polynomial subring and fuzzy power series subring. DEFINITION [40]: Let... ... Let S be a commutative ring with identity. Let R = S[x 1 , …, x n ] be a polynomial ring in the indeterminates x 1 , x 2 , …, x n over S. Let A b... ...i n 2 1 K K K = = , min {t j j = 1, 2, …, n}}, then A is called a fuzzy polynomial subring of R where t j = A(x j ) if x j appears non-trivially... ...e following theorems. THEOREM [40]: Let R = S[x 1 , x 2 , …, x n ] be a polynomial ring over S and M = 〈 x 1 , …, x n 〉. Let A be a fuzzy subring ... ...nd M = 〈 x 1 , …, x n 〉. Let A be a fuzzy subring of R. Then A is a fuzzy polynomial subring of R if and only if A = A| S + A| M and for all r ∈ R... ...es subring of R. Now we proceed on to recall some more notions on fuzzy polynomial subrings. THEOREM [40]: Suppose that R = S[x](S[[x]]) is a po... ...(S[[x]]) is a polynomial (power series) ring over S and that A is a fuzzy polynomial (power series) subring of R. If A/ M is a fuzzy ideal of R, th... ...S 2 , +, ) be a bisemiring which is commutative and has unit element. The polynomial bisemiring is denoted by S[x] = S 1 [x] ∪ S 2 [x] where S 1 [x]... ...ring is denoted by S[x] = S 1 [x] ∪ S 2 [x] where S 1 [x] and S 2 [x] are polynomial semirings. 1 a b 0 Figure 1.7.1 90 DEFINITION 1.7.9:...

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... n + 1 = prime ) ’(n + 1) =n: Proposition 9. Let n be the nth cyclotomic polynomial. Then for each a 2 (integer) one has F ’ (n) n (a) for all ... ...2 (integer) one has F ’ (n) n (a) for all n: (31) Proof. The cyclotomic polynomial is the irreducible polynomial of grade ’(n) with integer coe cie... ...+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... ...os 2 s (s2Q) Let n (x) = Y (r;n)=1 (x e 2 ir=n ) be the n-th cyclotomic polynomial (where e 2 ir=n for (r;n) = 1 are then-th primitive roots of uni... ...1 are then-th primitive roots of unity). It is well-known that n (x) is a polynomial with integer coe cients, and that this is an irreducible polynom... ... = 0 (2) where n 7, and b 1 ;:::;b ’(n)=2 are some integers. This gives a polynomial in 2 cos 2 r n , which cannot be reducible overZ[x], since then ... ...uld be reducible, too (impossible, by the irreducibility of the cyclotomic polynomial K n ). Since the leader coe cient in (2) is 1, the number 2 cos ...

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...nd rooted maps of a graph on genus. A general equation for the total genus polynomial and rooted total map polynomial is found in this section. Asacom... ...d rooted maps of a graph on genus20 4.1 The rooted total map and embedding polynomial of a graph................20 4.2 The number of rooted maps under... ...and rooted maps of a graph on genus 4.1 The rooted total map and embedding polynomial of a graph Fora given graph Γ with maximum valency≥3, assume tha... ...ntable genus of the graph Γ, respectively. Define its rooted orientable map polynomial r[Γ](x) , rooted non-orientable map polynomial r[Γ](x) and root... ...Γ](x) , rooted non-orientable map polynomial r[Γ](x) and rooted total map polynomialR[Γ](x) on genus by: r[Γ](x) = i≥0 r i (Γ)x i , r[Γ](x) = i≥... ...i−1and onthenon-orientablesurfacewith genus γ(Γ)+i−1. The orientable genus polynomialg[Γ](x) , non-orientable genus polynomial g[Γ](x) and total genus... ...olynomialg[Γ](x) , non-orientable genus polynomial g[Γ](x) and total genus polynomial G[Γ](x) of Γ are defined by g[Γ](x)= i≥0 g i (Γ)x i , g[Γ](x) ... ...and G[Γ](x) = i≥0 g i (Γ)x i + i≥1 g i (Γ)x −i . The orientable genus polynomial g[Γ](x) is introduced by Gross and Furst in [23],andin[19],[23]−... ...ntroducedbyChernetal. in[13],andin[31],recurrencerelationsforthetotalgenus polynomial ofabouquet andadipolearefound. The rootedorientablemap polyno- m...

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...with respect to the same induced operation and the ‘ ×’ in V 2 being the polynomial multiplication. Here K = Q. Example 1.2.2: V = V 1 × V 2 × ... ...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... ...ld theory and stochastic quantization. arXiv: hep-th/0311048. 88. Tapia V. Polynomial identities for hypermatrices. arXiv: math-ph/0208010. 89. Defari...

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...oint then f (G ∪ H, λ) = f(G; λ); f (H; λ) f (G; λ) is called the chromatic polynomial of the graph G. THEOREM 2.7.5: For a simple graph G of ord... ...EOREM 2.7.5: For a simple graph G of order n and size m. f(G; λ) is a monic polynomial of degree n in λ with integer coefficients and constant term ... ...atic index, 53-4 Chromatic number, 52 Chromatic partition, 52-3 Chromatic polynomial, 55 Circumference of a graph, 27 Closed walk, 26 Combined F...

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...that j • Γ ( = 0 and b j ≠ 0. Algorithm 1 Step 1: Write down the pseudo-polynomial form of min-SM: P = C ( 0 ≠ ∈ ∈ ∑ Γ j n m b N j N i ii... ... multiplication, division) are symbolic. Step 2: Calculate P according to the polynomial multiplication in symbolic form. Step 3: Simplify P by... ...rcise for the reader to prove. Algorithm 2 Step 1: Write done the pseudo-polynomial form of Γ: P = C ) 0 ≠ ∈ = Γ ∈ ∑ j n j ij m b N j r N... ...j ij m b N j r N j j ( 1 7 ) 48 Step 2: Calculate P according to the polynomial multiplication in symbolic form. Step 3: Simplify P by... ...on-convexity nature of its feasible domain, we tend to believe that there is no polynomial-time algorithm for this problem. The best we can do here ...

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...that j • Γ ( = 0 and b j ≠ 0. Algorithm 1 Step 1: Write down the pseudo-polynomial form of min-SM: P = C ( 0 ≠ ∈ ∈ ∑ Γ j n m b N j N i ii... ... multiplication, division) are symbolic. Step 2: Calculate P according to the polynomial multiplication in symbolic form. Step 3: Simplify P by... ...rcise for the reader to prove. Algorithm 2 Step 1: Write done the pseudo-polynomial form of Γ: P = C ) 0 ≠ ∈ = Γ ∈ ∑ j n j ij m b N j r N... ...j ij m b N j r N j j ( 1 7 ) 48 Step 2: Calculate P according to the polynomial multiplication in symbolic form. Step 3: Simplify P by... ...on-convexity nature of its feasible domain, we tend to believe that there is no polynomial-time algorithm for this problem. The best we can do here ...

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...elated to graph theory and mathematically tend to be NP- complete, insoluble in polynomial time e.g.[Crescenzi & Kann]. 6. Beyond Aristotle For ...

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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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.... 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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...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....