ISSN (Print): 2333-4878

ISSN (Online): 2333-4886

Editor-in-Chief: Vishwa Nath Maurya

Website: http://www.sciepub.com/journal/AMP

   

Article

Atom Bond Connectivity Index of Carbon Nanocones and An Algorithm

1Department of Mathematics, Faculty of Science and Arts, Celal Bayar University, Manisa, Turkey

2Turgutlu Vocational Training School, Celal Bayar University, Manisa, Turkey


Applied Mathematics and Physics. 2015, 3(1), 6-9
doi: 10.12691/amp-3-1-2
Copyright © 2015 Science and Education Publishing

Cite this paper:
Ömür Kıvanç Kürkçü, Ersin Aslan. Atom Bond Connectivity Index of Carbon Nanocones and An Algorithm. Applied Mathematics and Physics. 2015; 3(1):6-9. doi: 10.12691/amp-3-1-2.

Correspondence to: Ömür  Kıvanç Kürkçü, Department of Mathematics, Faculty of Science and Arts, Celal Bayar University, Manisa, Turkey. Email: omurkivanc@outlook.com

Abstract

Let G be a chemical graph, where V(G) and E(G) are represented set of vertices and edges respectively. Atom bond connectivity index ABC(G) is related to degree of vertices of graph G. In this paper, we calculate the index for generalized carbon nanocones. Subsequently, an useful algorithm (pseudocode) are given. The goal of this paper is to further the study of ABC(G) index for generalized carbon nanocones.

Keywords

References

[1]  Khaksar, A., Ghorbani, M. and Maimani, H.R., “On atom bond connectivity and GA indices of nanocones,” Optoelectron. Adv. Mater. – Rapid Commun., 4 (11). 1868-1870. 2010.
 
[2]  Furtula, B., Graovac, A. and Vukicevic, D., “Atom–bond connectivity index of trees,” Disc. Appl. Math., 157. 2828-2835. 2009.
 
[3]  Trinajstic, N. and Gutman, I., “Mathematical Chemistry,” Croat. Chem. Acta, 75. 329-356. 2002.
 
[4]  West, D.B., Introduction to Graph theory, Prentice Hall, Upper Saddle River, 1996.
 
[5]  Harary, F., Graph Theory, Addison-Wesley, Reading MA, 1969.
 
Show More References
[6]  Iijima, S., “Helical microtubules of graphitic carbon,” Nature, 354. 56-58. 1991.
 
[7]  Ge, M. and Sattler, K., “Observation of fullerene cones,” Chem. Phys. Lett., 220. 192. 1994.
 
[8]  Todeschini, R. and Consonni, V., Handbook of Molecular Descriptors, Wiley-VCH: Weinheim, 2000.
 
[9]  Diudea, M.V. QSPR/QSAR Studies by Molecular Descriptors, Nova Science Publishers: Huntington, NY, 2000.
 
[10]  Wiener, H., “Structural determination of paraffin boiling points,” J. Am. Chem. Soc., 69. 17-20. 1947.
 
[11]  Alipour, M.A. and Ashrafi, A.R., “A numerical method for computing the Wiener index of one heptagonal carbon nanocone,” Journal of Computational and Theoretical Nanoscience, 6. 1-4. 2009.
 
[12]  Ashrafi, A.R. and Gholami-Nezhaad, F., “The PI and Edge Szeged Indices of One-Heptagonal Carbon Nanocones,” Current Nanoscience, 5 (1). 51-53. 2009.
 
[13]  Gan, L., Hou, H. and Liu, B., “Some results on atom-bond connectivity index of graphs,” MATCH Commun. Math. Comput. Chem., 69. 669-680. 2011.
 
[14]  Ediz, S., “The Ediz eccentric connectivity index of one pentagonal carbon nanocones,” Fullerenes, Nanotubes, and Carbon Nanostructures, 21. 113-116. 2013.
 
[15]  Nejati, A. and Alaeiyan, M., “The edge version of MEC index of one-pentagonal carbon nanocones,” Bulgarian Chemical Communications, 46 (3). 462-464. 2014.
 
[16]  Yang, L. and Hua, H., “The harmonic index of general graphs, nanocones and triangular benzenoid graphs,” Optoelectron. Adv. Mater. – Rapid Commun., 6 (5-6). 660-663. 2012.
 
[17]  Alaeiyan, M., Asadpour, J. and Mojarad, R., “A Study of an Infinite Family of Carbon Nanocones by the Eccentricity Connectivity Polynomial,” Fullerenes, Nanotubes, and Carbon Nanostructures, 21. 849-860. 2013.
 
[18]  Das, K.Ch., Gutman, I. and Furtula, B., “On atom–bond connectivity index,” Filomat, 26 (4). 733-738. 2012.
 
[19]  Farahani, M.R., “The Edge Version of Geometric–Arithmetic Index of Benzenoid Graph,” Proc. Rom. Acad., Series B, 15 (2). 83-86. 2013.
 
[20]  Ashrafi, A.R. and Mohammad-Abadi, Z., “On Wiener Index of One-Heptagonal Nanocone,” Current Nanoscience, 8 (1). 180-185. 2012.
 
Show Less References

Article

On Instability of Dynamic Equilibrium States of Vlasov-Poisson Plasma

1Laboratory for Fluid and Gas Vortical Motions, Lavrentyev Institute for Hydrodynamics, Novosibirsk, Russian Federation

2Department for Differential Equations, Novosibirsk State University, Novosibirsk, Russian Federation


Applied Mathematics and Physics. 2016, 4(1), 1-8
doi: 10.12691/amp-4-1-1
Copyright © 2016 Science and Education Publishing

Cite this paper:
Yuriy G. Gubarev. On Instability of Dynamic Equilibrium States of Vlasov-Poisson Plasma. Applied Mathematics and Physics. 2016; 4(1):1-8. doi: 10.12691/amp-4-1-1.

Correspondence to: Yuriy  G. Gubarev, Laboratory for Fluid and Gas Vortical Motions, Lavrentyev Institute for Hydrodynamics, Novosibirsk, Russian Federation. Email: gubarev@hydro.nsc.ru

Abstract

The problem on linear stability of one–dimensional (1D) states of dynamic equilibrium boundless electrically neutral collisionless plasma in electrostatic approximation (the Vlasov–Poisson plasma) is studied. It is proved by the direct Lyapunov method that these equilibrium states are absolutely unstable with respect to small 1D perturbations in the case when the Vlasov–Poisson plasma contains electrons with stationary distribution function, which is constant over the physical space and variable in velocities, and one variety of ions whose distribution function is constant over the phase space as a whole. In addition, sufficient conditions for linear practical instability are obtained, the a priori exponential lower estimate is constructed, and initial data for perturbations, growing in time, are described. Finally, the illustrative analytical example of considered 1D states of dynamic equilibrium and superimposed small 1D perturbations, which grow on time in accordance with the obtained estimate, is constructed.

Keywords

References

[1]  Krall, N.A. and Trivelpiece, A.W., Principles of Plasma Physics, McGraw–Hill Book Company, Inc., New York, 1973.
 
[2]  Holm, D.D., Marsden, J.E., Ratiu, T., Weinstein, A., “Nonlinear stability of fluid and plasma equilibria,” Phys. Reports, 123 (1 & 2), 1-116, 1985.
 
[3]  Zakharov, V.E. and Kuznetsov, E.A., “Hamiltonian formalism for nonlinear waves,” Phys. Usp., 40, 1087-1116, 1997.
 
[4]  Pankavich, S. and Allen, R., “Instability conditions for some periodic BGK waves in the Vlasov–Poisson system,” Eur. Phys. J. D, 68 (12), 363-369, 2014.
 
[5]  Han–Kwan, D. and Hauray, M. “Stability issues in the quasineutral limit of the one–dimensional Vlasov–Poisson equation,” Comm. Math. Phys., 334 (2), 1101-1152, 2015.
 
Show More References
[6]  Esenturk, E. and Hwang, H.–J., “Linear stability of the Vlasov–Poisson system with boundary conditions”, Nonlinear Anal.–Theor., 139, 75-105, 2016.
 
[7]  Bernstein, I.B., “Waves in a plasma in a magnetic field,” Phys. Rev., 109, 10-21, 1958.
 
[8]  Gardner, C.S., “Bound on the energy available from a plasma,” Phys. Fluids, 6, 839-840, 1963.
 
[9]  Rosenbluth, M.N., “Topics in microinstabilities,” in: Advanced Plasma Theory, Academic Press, New York, 137-158.
 
[10]  Gubarev, Yu.G. and Gubkin, A.A., “On the stability for a class of 1D states of dynamical equilibrium of the Vlasov–Poisson plasma,” in: Differential Equations. Functional Spaces. Theory of Approximations, Institute for Mathematics SB RAS, Novosibirsk, 122-122 (in Russian).
 
[11]  Gubarev, Yu.G. and Gubkin, A.A., “On the stability for a class of 1–D states of dynamical equilibrium of the Vlasov–Poisson plasma,” in: Advanced Mathematics, Computations, and Applications, Academizdat, Novosibirsk, 79-80.
 
[12]  Zakharov, V.E. and Kuznetsov, E.A., Hamiltonian Formalism for Systems of Hydrodynamic Type, Preprint No. 186, Institute for Automation and Electrometry SB AS USSR, Novosibirsk, 1982 (in Russian).
 
[13]  Zakharov, V.E., “Benney's equations and quasiclassical approximation in the inverse problem method,” Funktsional. Anal. i Prilozgen., 14, 1980, 15-24 (in Russian).
 
[14]  Gubarev, Yu.G., “On an analogy between the Benney's equations and the Vlasov–Poisson's equations,” Dinamika Sploshn. Sredy, 110, 1995, 78-90 (in Russian).
 
[15]  Yakubovich, V.A. and Starzginskiyi, V.M., Linear Differential Equations with Periodical Coefficients and Its Applications, Nauka, Moscow, 1972 (in Russian).
 
[16]  Lyapunov, A.M., The general problem of the stability of motion, Taylor & Francis, London, 1992.
 
[17]  Chetaev, N.G., Stability of motion, Nauka, Moscow, 1990 (in Russian).
 
[18]  Gubarev, Yu.G., The direct Lyapunov method. The stability of quiescent states and steadystate flows of fluids and gases, Palmarium Academic Publishing, Saarbrücken, 2012 (in Russian).
 
[19]  Karacharov, K.A. and Pilyutik, A.G., Introduction in technical theory of motion stability, Fizmatgiz, Moscow, 1962 (in Russian).
 
[20]  La Salle, J. and Lefschetz, S., Stability by Liapunov's direct method with applications, Academic Press, New York, 1961.
 
[21]  Chandrasekhar, S., Ellipsoidal figures of equilibrium, Yale University Press, New Haven, 1969.
 
[22]  Gubarev, Yu.G., “Linear stability criterion for steady screw magnetohydrodynamic flows of ideal fluid,” T & A, 16 (3), 407-418, 2009.
 
[23]  Gavrilieva, A.A. and Gubarev, Yu.G., “Stability of steady–state plane–parallel shear flows of an ideal stratified fluid in the gravity field,” Vestnik of the NEFU named after M.K. Ammosov, 9 (3), 15-21, 2012 (in Russian).
 
[24]  Gubarev, Yu.G., “The problem of adequate mathematical modeling for liquids fluidity,” AJFD, 3 (3), 67-74, 2013.
 
[25]  Gavrilieva, A.A., Gubarev, Yu.G., Lebedev, M.P., “Rapid approach to resolving the adequacy problem of mathematical modeling of physical phenomena by the example of solving one problem of hydrodynamic instability,” IJTMP, 3 (4), 123-129, 2013.
 
[26]  Chandrasekhar, S., Hydrodynamic and hydromagnetic stability, Clarendon Press, Oxford, 1961.
 
[27]  Godunov, S.K., Equations of mathematical physics, Nauka, Moscow, 1979 (in Russian).
 
Show Less References

Article

Proca-Maxwell Equations for Dyons with Quaternion

1Department of Physics, G.B. Pant University of Agriculture & Technology, Pantnagar-263145 (U.K.) India

2Department of Mathematics, Kumaun University, D.S.B. Campus Nainital-263001 (U.K.) India

3Department of Physics, H.N.B. Garhwal University, Pauri Campus Pauri-246001 (U.K.) India


Applied Mathematics and Physics. 2016, 4(1), 9-15
doi: 10.12691/amp-4-1-2
Copyright © 2016 Science and Education Publishing

Cite this paper:
B. C. Chanyal, S. K. Chanyal, Virendra Singh, A. S. Rawat. Proca-Maxwell Equations for Dyons with Quaternion. Applied Mathematics and Physics. 2016; 4(1):9-15. doi: 10.12691/amp-4-1-2.

Correspondence to: B.  C. Chanyal, Department of Physics, G.B. Pant University of Agriculture & Technology, Pantnagar-263145 (U.K.) India. Email: bcchanyal@gmail.com, bcchanyal@gbpuat.ac.in

Abstract

The quaternions are first hyper-complex numbers, having four-dimensional structure, which may be useful to express the 4-dimensional theory of dyons carrying both electric and magnetic charges. Keeping in mind t’Hooft’s monopole solutions and the fact that despite the potential importance of massive monopole, we discuss a connection between quaternionic complex field, to the generalized electromagnetic field equations of massive dyons. Starting with the Euclidean space-time structure and two four-components theory of dyons, we represent the generalized charge, potential, field and current source in quaternion form with real and imaginary part of electric and magnetic constituents of dyons. We have established the quaternionic formulation of generalized complex-electromagnetic fields equations, generalized Proca-Maxwell’s (GPM) equations and potential wave equations for massive dyons. Thus, the quaternion formulation be adopted in a better way to understand the explanation of complex-field equations as the candidate for the existence of massive monopoles and dyons where the complex parameters be described as the constituents of quaternion.

Keywords

References

[1]  P. A. M. Dirac, “Quantized singularities in the electromagnetic field”, Proc. Roy. Soc. London, A133 (1931), 60.
 
[2]  J. Schwinger, “Dyons Versus Quarks”, Science, 166 (1969), 690.
 
[3]  G. pt Hooft, “Magnetic monopoles in unified gauge theories”, Nucl. Phys., B79 (1974), 276.
 
[4]  P. S. Bisht, O. P. S. Negi and B. S. Rajput, “Null-tetrad formulation of dyons”, IL Nuovo Cimento 104A (1991), 337.
 
[5]  B. S. Rajput, S. R. Kumar and O. P. S. Negi, “Quaternionic formulation for dyons”, Lett. Nuovo Cimento, 34 (1982), 180.
 
Show More References
[6]  J. S. Dowker and J. A. Roche, “The Gravitational Analogues of Magnetic Monopoles”, Proc. R. Soc., 92 (1967), 1.
 
[7]  L. E. Dickson, “On Quaternions and Their Generalization and the History of the Eight Square Theorem”, Ann. Math., 20 (1919), 155.
 
[8]  W. R. Hamilton, “Elements of quaternions”, Chelsea Publications Co., NY, (1969).
 
[9]  P. G. Tait, “An elementary Treatise on Quaternions”, Oxford Univ. Press, NY, (1875).
 
[10]  D. Finklestein, J. M. Jauch, S. Schiminovich and D. Speiser, “Principle of general quaternion covariance”, J. Math. Phys., 4 (1963), 788.
 
[11]  S. L. Adler, "Quaternion Quantum Mechanics and Quantum Fields", Oxford Univ. Press, NY, (1995).
 
[12]  P. S. Bisht, O. P. S. Negi and B. S. Rajput, “Quaternion Gauge Theory of Dyonic Fields” Prog. Theor. Phys., 85 (1991), 157.
 
[13]  B. C. Chanyal, P. S. Bisht and O. P. S. Negi , “Generalized Octonion Electrodynamics”, Int. J. Theor. Phys., 49 (2010), 1333.
 
[14]  B. C. Chanyal, P. S. Bisht and O. P. S. Negi, “Octonion electrodynamics in isotropic and chiral medium”, Int. J. Mod. Phys. A 29 (2014), 1450008.
 
[15]  B. C. Chanyal, S. K. Chanyal, ö. Bektas, S. Yüce, “A new approach on electromagnetism with dual number coefficient octonion algebra”, Int. J. Geom. Meth. Mod. Phys. 13 (2016), 1630013.
 
[16]  N. Cabibbo and E. Ferrari, “Quantum electrodynamics with Dirac monopoles”, Nuovo Cim., 23 (1962), 1147.
 
[17]  B. S. Rajput, S. Kumar and O. P. S. Negi, “Quaternionic formulation for generalized field equations in the presence of dyons”, Lett. Nuovo Cimento, 34 (1982), 180.
 
[18]  A. Proca, “Sur 1’equation de Dirac” Compt. Rend. 190 (1930), 1377.
 
[19]  A.Yu. Ignatiev and G. C. Joshi, “Massive electrodynamics and the magnetic monopoles”, Phys. Rev., D 53 (1996), 984.
 
[20]  G. pt Hooft, “A property of electric and magnetic flux in non-Abelian gauge theories”, Nucl. Phys., B153 (1979), 141.
 
[21]  A. M. Polyakov, “Particle spectrum in the quantum field theory”, JETP. Lett., 20 (1974), 194.
 
[22]  E. B. Bogomolny, “Stability of Classical Solutions”, Sov. J. Nucl. Phys., 24 (1976), 449.
 
[23]  M. K. Prasad and C. M. Sommerfield, “Exact Classical Solution for the ’t Hooft Monopole and the Julia-Zee Dyon”, Phys. Rev. Lett., 35 (1975), 760.
 
Show Less References