#### Volume-3 ~ Issue-4

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Paper Type |
: | Research Paper |

Title |
: | Precision Angular Measurements Using Scale of Chords |

Country |
: | India |

Authors |
: | Dr A. M. Chandra |

: | 10.9790/5728-0340103 |

**Abstract :**Presently angular measurements are made using protractors having a normal accuracy of 1 or at the most ½ . The scale of chords, as linear scale, can be constructed for measurement and construction of angles having accuracy equal to that of a normal protractor. This paper presents a new concept of construction of a diagonal scale of chords that can have an accuracy of 10 or less, and thus, using diagonal scale of chords, angles can be constructed or measured to a higher accuracy which is not possible using normal size protractors.

**Keywords -**Angle measurements, Protractor, Scale of chords, Diagonal scale of chords

[1] A. M. Chandra and Satish Chandra, Engineering Graphics (Narosa Publishing House, New Delhi, 2003)

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Paper Type |
: | Research Paper |

Title |
: | Dominating Sets and Domination Polynomials of Square Of Cycles |

Country |
: | India |

Authors |
: | A. Vijayan, K. Lal Gipson |

: | 10.9790/5728-0340414 |

**Abstract :**Let G = (V, E) be a simple graph. A set S V is a dominating set of G, if every vertex in V-S is adjacent to atleast one vertex in S. Let 2 n C be the square of the Cycle n C and let 2 ( , ) n D C i denote the family of all dominating sets of 2 n C with cardinality i. Let 2 ( , ) n d C i = | 2 ( , ) n D C i |. In this paper, we obtain a recursive formula for 2 ( , ) n d C i . Using this recursive formula, we construct the polynomial, 2 ( , ) n D C x = 2 5 ( , ) n i n i n d C i x , which we call domination polynomial of 2 n C and obtain some properties of this polynomial.

.

**Keywords:**domination set, domination number, domination polynomials.

[1]. S.Alikhani and Y.H.Peng, Introduction to domination polynomial of a graph. arXiv:0905.2251v1[math.CO] 14 May 2009.

[2]. S.Alikhani and Y.H.Peng, 2009, Domination sets and Domination Polynomials of paths, International journal of Mathematics and

Mathematical Sciences. Article ID 542040.

[3]. G.Chartand and P.Zhang, Introduction to Graph Theory, McGraw-Hill, Boston, Mass, USA, 2005.

[4]. T.W.Haynes ,S.T.hedetniemi,and P.J.Slater, Fundamental of Domination in graphs,vol.208 of Monographs and Textbooks in Pure

and Applied Mathematics, Marcel Dekker, New York,NY,USA,1998.

[5]. S.Alikhani and Y.H.Peng, Domination sets and Domination Polynomials of cycles, arXiv: 0905.3268v [math.CO] 20 May 2009.

[6]. A.Vijayan and K.Lal Gipson, Domination sets and Domination Polynomials of Square paths, accepted in "Open Access journal of

Discrete Mathematics." – USA.

[2]. S.Alikhani and Y.H.Peng, 2009, Domination sets and Domination Polynomials of paths, International journal of Mathematics and

Mathematical Sciences. Article ID 542040.

[3]. G.Chartand and P.Zhang, Introduction to Graph Theory, McGraw-Hill, Boston, Mass, USA, 2005.

[4]. T.W.Haynes ,S.T.hedetniemi,and P.J.Slater, Fundamental of Domination in graphs,vol.208 of Monographs and Textbooks in Pure

and Applied Mathematics, Marcel Dekker, New York,NY,USA,1998.

[5]. S.Alikhani and Y.H.Peng, Domination sets and Domination Polynomials of cycles, arXiv: 0905.3268v [math.CO] 20 May 2009.

[6]. A.Vijayan and K.Lal Gipson, Domination sets and Domination Polynomials of Square paths, accepted in "Open Access journal of

Discrete Mathematics." – USA.

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**Abstract:**This paper is concerned with the determination of the distribution of temperature and displacement in a thin semi-infinite elastic rod when its free end is subjected to periodic heating. It has been pointed out by P.Chadwick(1960) that the rigorous approach,i.c. the approach by way of the coupled equations, to the thermal boundary value problem. In this paper the one dimensional problem of the periodic heating of the free surface of a semi-infinite rod has been solved by a perturbation procedure, approximations upto the first order being retained.

[1] ATKINSON,K.E.(1976); A survey of Numerical Methods for the Solution of Fredholm- Integral Equations of the Second – Kind.

Society of Industial and applied Mathematics, Philadelphia,Pa.

[2] CARSLAW,H.S. AND JAEGER, J.C. (1959); Conduction of heat in Solids, 2nd Edn. O.U.P.

[3] SNEDDON, I.N.(1972); The Use of Integral Transform, McGraw Hill, New-York.

[4] SOKOLNIKOFF,I.S.(1956); Mathematical theory of Elasticity, McGraw Hill Book Co.

[5] NOWACKI, W.(1986); Thermoelasticity, 2nd End. Pergamon Press.

[6] WASTON,G.N. (1978) ATreatise on the Theory of Bessel Functions, 2nd Edn. C.U.P.

[7] LOVE,A.E.H.(1927) A Treatise on the Mathematical Theory of Elasticity, 4th Edn. Dover Publication.

[8] PARIA.G. (1968): Instantaneous heat sources in an infinite solid, India, J.Mech.Math.(spl. Issue), partI,41.

[9] LESSEN. M.(1968) ; j.Mech.Phys.solids.5,p.

[10] SNEDDON,I.N.(1958); Prog.Roy.Soc.Edin.1959,pp 121-142.

Society of Industial and applied Mathematics, Philadelphia,Pa.

[2] CARSLAW,H.S. AND JAEGER, J.C. (1959); Conduction of heat in Solids, 2nd Edn. O.U.P.

[3] SNEDDON, I.N.(1972); The Use of Integral Transform, McGraw Hill, New-York.

[4] SOKOLNIKOFF,I.S.(1956); Mathematical theory of Elasticity, McGraw Hill Book Co.

[5] NOWACKI, W.(1986); Thermoelasticity, 2nd End. Pergamon Press.

[6] WASTON,G.N. (1978) ATreatise on the Theory of Bessel Functions, 2nd Edn. C.U.P.

[7] LOVE,A.E.H.(1927) A Treatise on the Mathematical Theory of Elasticity, 4th Edn. Dover Publication.

[8] PARIA.G. (1968): Instantaneous heat sources in an infinite solid, India, J.Mech.Math.(spl. Issue), partI,41.

[9] LESSEN. M.(1968) ; j.Mech.Phys.solids.5,p.

[10] SNEDDON,I.N.(1958); Prog.Roy.Soc.Edin.1959,pp 121-142.

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**Abstract :**In this paper we prove two fixed point theorems in topological vector space valued cone metric spaces (briefly TVS-CMS). To that end we introduce the concept of complete topological algebra cone (briefly CTA cone). Our theorems are generalizations of corresponding theorems in [8] and [13]. The paper also gives answers to the open problems posed in [15]. Finally we give an application of our first theorem. AMS Mathematics Subject Classification (2010): 47H10, 54H25, 46A40

**Keywords -**topological vector space, ordered topological vector space, algebra over a field, topological algebra over a field, topological vector space valued cone metric space, scalarization function, CTA cone

[1] C.D. Aliprantis and R. Tourky , Cones and Duality (American Mathematical Society, 2007).

[2] I. D.Arandelovic and D. J. Keckic, TVS-Cone Metric Spaces as a Special case of Metric Spaces, arXiv: 1202.5930v1 [math.FA],

2012.

[3] I. Beg, A. Azam and M. Arshad, Common fixed points for maps on topological vector space valued cone metric spaces, Internat. J.

Math. Math. Sciences, 2009 (2009).

[4] M. M. Deza and E. Deza, Encyclopedia of Distances (Springer-Verlag, 2009), i – x.

[5] W. S.Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Analysis, 72 (5) (2010), 2259-2261.

[6] M. Fréchet, Sur quelques points du calcul fonctionnel. Rendi. Circ. Mat. Palermo, 22(1906), 1- 74.

[7] F. Hausdorff, Grundzüge der Mengenlehre, Verlag Von Veit & Company, Leipzig (1914). Reprinted by Chelsea Publishing

Company, New York (1949).

[8] L.G. Huang and X.Zhang, Cone Metric Spaces and Fixed Point Theorems of Contractive mappings, J. Math. Anal. Appl. , 332

(2007), 1467 - 1475.

[9] M.C. Joshi and R.K. Bose, Some Topics in Nonlinear Functional Analysis (Wiley Eastern Ltd., New Delhi, 1985).

[10] D.R. Kurepa, Tableaux ramifies d'ensembles. Espaces pseudo-distancies, C. R. Acad. Sci. Paris, 198 (1934), 1563–1565.

[2] I. D.Arandelovic and D. J. Keckic, TVS-Cone Metric Spaces as a Special case of Metric Spaces, arXiv: 1202.5930v1 [math.FA],

2012.

[3] I. Beg, A. Azam and M. Arshad, Common fixed points for maps on topological vector space valued cone metric spaces, Internat. J.

Math. Math. Sciences, 2009 (2009).

[4] M. M. Deza and E. Deza, Encyclopedia of Distances (Springer-Verlag, 2009), i – x.

[5] W. S.Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Analysis, 72 (5) (2010), 2259-2261.

[6] M. Fréchet, Sur quelques points du calcul fonctionnel. Rendi. Circ. Mat. Palermo, 22(1906), 1- 74.

[7] F. Hausdorff, Grundzüge der Mengenlehre, Verlag Von Veit & Company, Leipzig (1914). Reprinted by Chelsea Publishing

Company, New York (1949).

[8] L.G. Huang and X.Zhang, Cone Metric Spaces and Fixed Point Theorems of Contractive mappings, J. Math. Anal. Appl. , 332

(2007), 1467 - 1475.

[9] M.C. Joshi and R.K. Bose, Some Topics in Nonlinear Functional Analysis (Wiley Eastern Ltd., New Delhi, 1985).

[10] D.R. Kurepa, Tableaux ramifies d'ensembles. Espaces pseudo-distancies, C. R. Acad. Sci. Paris, 198 (1934), 1563–1565.

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Paper Type |
: | Research Paper |

Title |
: | On the generalized bilinear differential equations |

Country |
: | Nigeria. |

Authors |
: | M. Y. Adamu, E. Suleiman |

: | 10.9790/5728-0342430 |

**Abstract :**By using the generalized Hirrota bilinear operators a kind of bilinear differential equations is established and examined when the linear super position principle can apply to the resulting generalized bilinear differential equations. Examples of generalized bilinear differential equations together with an algorithm using weights are computed using a 1+1 and 2+1 dimensional equations in order to shed more lights on the presented general scheme for the construction of the bilinear differential equations which posses linear subspaces of solutions.

[1] Ma, W. X., Huang,T. W. Zhang, Y., A multiple Exp-function Method for nonlinear differential equations and its application

Physica Scipta 82 2010 065003.

[2] Ma, W. X. and Fan, E., Linear superposition principle applying to Hirota bilinear equations. Computers And Mathematics With

Appllications, 61,(2011 950-959

[3] W. X., Ma, Y .Zhang,, Y .Tang ,. and J . Tu,. Hirota bilinear equations with linear subspaces of solutions. Appllied Mathematics

And Computations 218, 2012, 7174-7183

[4] W. X, Ma, , and Y . You,Solving Korteweg de-Vries equation by its bilinear form: Wronskian solution. Trans. Amer. Math. Soc.,

357,2005:1753-1778

[5] R. Hirota.. The Direct Method in Soliton Theory. (Cambridge University Press) (2004)

[6] Asaad, M. G and Ma, W. X., , Pfaffian solution to a (3+1)-dimensional generalized B.typeKadomtsev-Petviashvilli equation and

its modified counterpart, applied Math. And Comp.218, 2012, 5524-5542

[7] R. Hirota,., a new form of Baclund Transformation and its relation to the inverse scattering problem. Progr. Of Theoret. Phys., 52,

1974, 329-338

[8] N. C, Freeman, and J.J.C Nimmo, , the use of Backlund transformation in obtaining N-soliton solutions in Wronskian form. Phys.

lett. A 95, 1983, 1-3

[9] S . Zangh, and T. C, Xia,., A generalized new auxiliary equation and its application to nonlinear partial differential equations.

Phys. Lett. A, 363, 2007, 356-360

[10] M. Jimbo. and T. Miwa, solitons and infinite dimensional Lie algebra, publications in research Institute for Mathematical

Sciences. 19, 1983, 943-1001

Physica Scipta 82 2010 065003.

[2] Ma, W. X. and Fan, E., Linear superposition principle applying to Hirota bilinear equations. Computers And Mathematics With

Appllications, 61,(2011 950-959

[3] W. X., Ma, Y .Zhang,, Y .Tang ,. and J . Tu,. Hirota bilinear equations with linear subspaces of solutions. Appllied Mathematics

And Computations 218, 2012, 7174-7183

[4] W. X, Ma, , and Y . You,Solving Korteweg de-Vries equation by its bilinear form: Wronskian solution. Trans. Amer. Math. Soc.,

357,2005:1753-1778

[5] R. Hirota.. The Direct Method in Soliton Theory. (Cambridge University Press) (2004)

[6] Asaad, M. G and Ma, W. X., , Pfaffian solution to a (3+1)-dimensional generalized B.typeKadomtsev-Petviashvilli equation and

its modified counterpart, applied Math. And Comp.218, 2012, 5524-5542

[7] R. Hirota,., a new form of Baclund Transformation and its relation to the inverse scattering problem. Progr. Of Theoret. Phys., 52,

1974, 329-338

[8] N. C, Freeman, and J.J.C Nimmo, , the use of Backlund transformation in obtaining N-soliton solutions in Wronskian form. Phys.

lett. A 95, 1983, 1-3

[9] S . Zangh, and T. C, Xia,., A generalized new auxiliary equation and its application to nonlinear partial differential equations.

Phys. Lett. A, 363, 2007, 356-360

[10] M. Jimbo. and T. Miwa, solitons and infinite dimensional Lie algebra, publications in research Institute for Mathematical

Sciences. 19, 1983, 943-1001

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Paper Type |
: | Research Paper |

Title |
: | Neutrosophic Set and Neutrosophic Topological Spaces |

Country |
: | Saudi Arabia |

Authors |
: | A.A.Salama, S.A.Alblowi |

: | 10.9790/5728-0343135 |

**Abstract :**Neutrosophy has been introduced by Smarandache [7, 8] as a new branch of philosophy. The purpose of this paper is to construct a new set theory called the neutrosophic set. After given the fundamental definitions of neutrosophic set operations, we obtain several properties, and discussed the relationship between neutrosophic sets and others. Finally, we extend the concepts of fuzzy topological space [4], and intuitionistic fuzzy topological space [5, 6] to the case of neutrosophic sets. Possible application to superstrings and space–time are touched upon.

**Keywords -**Fuzzy topology; fuzzy set; neutrosophic set; neutrosophic topology

[1] K. Atanassov, intuitionistic fuzzy sets, in V.Sgurev, ed.,Vii ITKRS Session, Sofia(June 1983 central Sci. and Techn. Library, Bulg.

Academy of Sciences( 1984)).

[2] K. Atanassov, intuitionistic fuzzy sets, Fuzzy Sets and Systems 20(1986)87-96.

[3] K. Atanassov, Review and new result on intuitionistic fuzzy sets , preprint IM-MFAIS-1-88, Sofia, 1988.

[4] C.L. Chang, Fuzzy Topological Spaces, J. Math. Anal. Appl. 24 (1968)182-1 90.

[5] Dogan Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems. 88(1997)81-89.

[6] Reza Saadati, Jin HanPark, On the intuitionistic fuzzy topological space, Chaos, Solitons and Fractals 27(2006)331-344 .

[7] Florentin Smarandache , Neutrosophy and Neutrosophic Logic , First International Conference on Neutrosophy , Neutrosophic Logic , Set, Probability, and Statistics University of New Mexico, Gallup, NM 87301, USA(2002) , smarand@unm.edu

[8] F. Smarandache. A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability.

American Research Press, Rehoboth, NM, 1999.

[9] L.A. Zadeh, Fuzzy Sets, Inform and Control 8(1965)338-353

Academy of Sciences( 1984)).

[2] K. Atanassov, intuitionistic fuzzy sets, Fuzzy Sets and Systems 20(1986)87-96.

[3] K. Atanassov, Review and new result on intuitionistic fuzzy sets , preprint IM-MFAIS-1-88, Sofia, 1988.

[4] C.L. Chang, Fuzzy Topological Spaces, J. Math. Anal. Appl. 24 (1968)182-1 90.

[5] Dogan Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems. 88(1997)81-89.

[6] Reza Saadati, Jin HanPark, On the intuitionistic fuzzy topological space, Chaos, Solitons and Fractals 27(2006)331-344 .

[7] Florentin Smarandache , Neutrosophy and Neutrosophic Logic , First International Conference on Neutrosophy , Neutrosophic Logic , Set, Probability, and Statistics University of New Mexico, Gallup, NM 87301, USA(2002) , smarand@unm.edu

[8] F. Smarandache. A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability.

American Research Press, Rehoboth, NM, 1999.

[9] L.A. Zadeh, Fuzzy Sets, Inform and Control 8(1965)338-353

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**Abstract :**Operator regularization is one of the excellent prescriptions for studying gauge theories. Among the many regularization prescriptions Dimensional regularization and Pre-regularization are the best methods for evaluating loop diagrams perturbatively. On the other hand Operator regularization can also be said one of the best methods for studying gauge theories because of its two-fold use. With this prescription one can adopt pathintegral method with the combination of background field quantization and Schwinger expansion to find the result of the required problem without considering any Feynman diagrams. Also from this prescription one can consider Feynman diagrams and evaluating these diagrams using the Operator regularization prescription. In this paper we have shown how one can use both the options of Operator regularization method to evaluate Feynman diagrams in QED in (3+1) dimensional space-time.

**Keywords-**Operator regularization, Dimensional regularization, Feynman diagrams in QED, Path-integral method, Background field quantization and Generating functional..

[1] D. G. C. McKeon and T. N. Sherry. 1987. Operator Regularization of Green's Functions, Phys. Rev. Lett. 59, p.532.

[2] G.'t Hooft and M. Veltman,. 1972. Regularization and Renormalization of Gauge Fields, Nucl. Phys. B44, 189

[3] D. G. C. McKeon and T. N. Sherry.1987.Operator Regularization and One-Loop Green's Functions, Phys. Rev. D 35, p.3854.

[4] A. Y. Shiekh. 2010. Operator Regularization of Feynman Diagrams in One Loop QED, arXiv: 1006. 1806v3 [phys.gen-ph].

[5] J. Schwinger. 1951. On Gauge Invariance and Vacuum Polarization, Phys. Rev.82, 664.

[6] A. Salam and J. Strathdee. (1975). Transition Electromagnetic Fields in Particle Physics, Nucl. Phys.B90, 203.

[7] J. Dowker and R. Critchley. 1976. Effective Langrangian and Energy-momentum Tensor in De sitter Space, Phys. Rev. D 13, 3224.

[8] S. Hawking. 1977. Zeta Function Regularization of Path Integral in Curved Space, Commun. Math. Phys. 55, 133.

[9] M. Reuter. 1985. Chiral Anomalies and Zeta-function Regularization, Phys. Rev. D 31, 1374 .

[10] B. De Witt. 1967. Quantum Theory of Gravity. II. The Manifest Covariant Theory, Phys. Rev. 162, 1195.

[2] G.'t Hooft and M. Veltman,. 1972. Regularization and Renormalization of Gauge Fields, Nucl. Phys. B44, 189

[3] D. G. C. McKeon and T. N. Sherry.1987.Operator Regularization and One-Loop Green's Functions, Phys. Rev. D 35, p.3854.

[4] A. Y. Shiekh. 2010. Operator Regularization of Feynman Diagrams in One Loop QED, arXiv: 1006. 1806v3 [phys.gen-ph].

[5] J. Schwinger. 1951. On Gauge Invariance and Vacuum Polarization, Phys. Rev.82, 664.

[6] A. Salam and J. Strathdee. (1975). Transition Electromagnetic Fields in Particle Physics, Nucl. Phys.B90, 203.

[7] J. Dowker and R. Critchley. 1976. Effective Langrangian and Energy-momentum Tensor in De sitter Space, Phys. Rev. D 13, 3224.

[8] S. Hawking. 1977. Zeta Function Regularization of Path Integral in Curved Space, Commun. Math. Phys. 55, 133.

[9] M. Reuter. 1985. Chiral Anomalies and Zeta-function Regularization, Phys. Rev. D 31, 1374 .

[10] B. De Witt. 1967. Quantum Theory of Gravity. II. The Manifest Covariant Theory, Phys. Rev. 162, 1195.