Math. Interdisc. Res. University of Kashan Mathematics Interdisciplinary Research 2538-3639 University of Kashan 106 10.22052/mir.2016.34108 Mathematical Physics Motion of Particles under Pseudo-Deformation Motion of Particles under Pseudo-Deformation Yadav Akhilesh Chandra M G Kashi Vidyapith Varanasi 01 07 2016 1 2 273 277 01 04 2016 28 04 2016 Copyright © 2016, University of Kashan. 2016 http://mir.kashanu.ac.ir/article_34108.html

In this short article, we observe that the path of particle of mass \$m\$ moving along \$mathbf{r}= mathbf{r}(t)\$ under pseudo-force \$mathbf{A}(t)\$, \$t\$ denotes the time, is given by \$mathbf{r}_d= int(frac{dmathbf{r}}{dt} mathbf{A}(t)) dt +mathbf{c}\$. We also observe that the effective force \$mathbf{F}_e\$ on that particle due to pseudo-force \$mathbf{A}(t)\$, is given by \$ mathbf{F}_e= mathbf{F} mathbf{A}(t)+ mathbf{L} dmathbf{A}(t)/dt\$, where \$mathbf{F}= m d^2mathbf{r}/dt^2 \$ and \$mathbf{L}= m dmathbf{r}/dt\$. We have discussed stream lines under pseudo-force.

Right loops right transversals gyrotransversals
1. H‎. ‎Kiechle‎, ‎Theory of \$K\$-loops‎,‎ Lecture Notes in Mathematics‎, ‎1778‎, ‎Springer-Verlag‎, ‎Berlin‎, ‎2002‎. 2. R. Lal, Transversals in groups, J. Algebra 181 (1996) 70–81. 3. R. Lal, A. C. Yadav, Topological right gyrogroups and gyrotransversals, Comm. Algebra 41 (2013) 3559–3575. 4. A. C. Yadav, R. Lal, Smooth right quasigroup structures on 1-manifolds, J. Math. Sci. Univ. Tokyo 17 (2010) 313–321.
Math. Interdisc. Res. University of Kashan Mathematics Interdisciplinary Research 2538-3639 University of Kashan 106 10.22052/mir.2016.34106 Applied Algebra C-Class Functions and Remarks on Fixed Points of Weakly Compatible Mappings in G-Metric Spaces Satisfying Common Limit Range Property C-Class Functions and Weakly Compatible Mappings in G-Metric Spaces Hojat Ansari Arslan Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran, Dolicanin-Dekic Diana Faculty of Technical Science, 38000 Kosovska Mitrovica, Gu Feng Institute of Applied Mathematics and Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, Popovic Branislav 4Faculty of Science, University of Kragujevac, Radoja Domanovica 12, 34000 Kragujevac, Serbia, Radenovic Stojan N Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Beograd, Serbia 01 07 2016 1 2 279 290 09 04 2016 12 06 2016 Copyright © 2016, University of Kashan. 2016 http://mir.kashanu.ac.ir/article_34106.html

In this paper, using the contexts of C-class functions and common limitrange property, common fixed point result for some operator are obtained.Our results generalize several results in the existing literature. Some examplesare given to illustrate the usability of our approach.

Generalized metric space common fixed point generalized weakly G-contraction weakly compatible mappings common (CLRST ) property C-class functions
1. M‎. ‎Abbas‎, ‎S‎. ‎H‎. ‎Khan‎, ‎T‎. ‎Nazir‎, ‎Common fixed points‎ of R-weakly commuting maps in generalized metric spaces‎,‎ Fixed Point Theory‎ Appl. 2011‎, ‎2011:41‎, ‎11 pp‎. 2. M. Abbas, G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl. 341 (2008) 416-420. 3. M. Abbas, T Nazir, D. Ðjorić, Common fixed point of mappings satisfying (E:A) property in generalized metric spaces, Appl. Math. Comput. 218 (2012) 7665-7670. 4. M. Abbas, T. Nazir, S. Radenović, Common fixed point of generalized weakly contractive maps in partially ordered G-metric spaces, Appl. Math. Comput. 218 (2012) 9383 - 9395. 5. M‎. ‎Abbas‎, ‎B‎. ‎E‎. ‎Rhoades‎, ‎Common fixed point results‎ for noncommuting mappings without‎‎ continuity in generalized metric spaces‎, ‎Appl‎. ‎Math‎. ‎Comput. 215 (2009) 262-269‎. 6. A. H. Ansari, Note on ’ - -contractive type mappings and related fixed point, in Proceedings of the 2nd Regional Conference on Mathematics and Applications, pp. 377-380, Payame Noor University, Tonekabon, Iran, 2014. 7. H. Aydi, A common fixed point of integral type contraction in generalized metric spaces, J. Adv. Math. Stud. 5 (2012) 111-117. 8. H‎. ‎Aydi‎, ‎S‎. ‎Chauhan‎, ‎S‎. ‎Radenovic, ‎Fixed point‎ of weakly compatible mappings in G-metric spaces‎ satisfying common limit‎ ‎range property‎,‎ Facta Univ‎. ‎Ser‎. ‎Math‎. ‎Inform. 28 (2013) 197-210‎. 9. A‎. ‎Branciari‎, ‎A fixed point theorem for mappings‎ satisfying a general contractive condition of integral type‎, Int‎. ‎J‎. ‎Math‎. Math‎. ‎Sci. 29 (2002) 531-536‎. 10. F. Gu, Common fixed point theorems for six mappings in generalized metric spaces, Abstr. Appl. Anal. 2012, Art. ID 379212, 21 pp. 11. F‎. ‎Gu‎, ‎W‎. ‎Shatanawi‎, ‎Common fixed point for‎ generalized weakly G-contraction mappings satisfying common‎ (E.A‎‎) property in \$G\$-metric spaces‎, ‎‎ Fixed Point Theory Appl. 2013‎, ‎2013:309‎, ‎15 pp‎. 12. F‎. ‎Gu‎, ‎Y‎. ‎Yin‎, ‎Common fixed point for three pairs of‎ self-maps satisfying common (E.A) property in generalized metric‎ spaces‎, ‎Abstr‎. ‎Appl‎. ‎Anal. 2013, ‎Art‎. ‎ID 808092‎, ‎11 pp‎. 13. A‎. ‎Kaewcharoen‎, ‎Common fixed points for four mappings‎ in G-metric spaces‎, ‎ Int‎. ‎J‎. ‎Math‎. ‎Anal. 6 (2012) 2345-2356. 14. M. S. Khan, M. Swaleh, S. Sessa, Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc. 30 (1984) 1-9. 15. W. Long, M. Abbas, T. Nazir, S. Radenović, Common fixed point for two pairs of mappings satisfying (E.A) property in generalized metric spaces, Abstr. Appl. Anal. 2012, Art. ID 394830, 15 pp. 16.Z. Mustafa, Common fixed points of weakly compatible mappings in G -metric spaces, Appl. Math. Sci. 6 (2012) 4589-4600. 17. Z. Mustafa, Some new common fixed point theorems under strict contractive conditions in G-metric spaces, J. Appl. Math. 2012, Art. ID 248937, 21 pp. 18. Z. Mustafa, H. Aydi, E. Karapinar, On common fixed points in G-metric spaces using (E:A) property, Comput. Math. Appl. 64 (2012) 1944 - 1956. 19. Z. Mustafa, M. Khandagji, W. Shatanawi, Fixed point results on complete G-metric spaces, Studia Sci. Math. Hungar. 48 (2011) 304 - 319. 20. Z. Mustafa, H. Obiedat, F. Awawdeh, Some fixed point theorems for mappings on complete G-metric space, Fixed Point Theory Appl. 2008, Art. ID 189870, 12 pp. 21. Z. Mustafa, W. Shatanawi, M. Bataineh, Existence of fixed points results in G-metric spaces, Int. J. Math. Math. Sci. 2009, Art. ID 283028, 10 pp. 22. Z‎. ‎Mustafa‎, ‎B‎. ‎Sims‎, ‎A new approach to generalized‎ metric spaces‎, J‎. ‎Nonlinear Convex Anal. 7 (2006) 289-297‎. 23. Z‎. ‎Mustafa‎, ‎B‎. ‎Sims‎, ‎Fixed point theorems for‎ contractive mappings in complete G-metric spaces‎, ‎Fixed Point‎ Theory Appl. 2009‎, ‎Art‎. ‎ID 917175‎, ‎10 pp‎. 24. V. Popa, A. M. Patriciu, A general fixed point theorem for pairs of weakly compatible mappings in G-metric spaces, J. Nonlinear Sci. Appl. 5 (2012) 151 - 160. 25. S. Radenović, Remarks on some recent coupled coincidence point results in symmetric G-metric spaces, Journal of Operators 2013, Article ID 290525, 8 pp. 26. S. Radenović, S. Pantelić, P. Salimi, J. Vujaković, A note on some tripled coincidence point results i G-metric spaces, Int. J. Math. Sci. Eng. Appl. 6 (2012) 23-38. 27. W. Shatanawi, S. Chauhan, M. Postolache, M. Abbas, S. Radenović, Common fixed points for contractive mappings of integral type in G-metric spaces, J. Adv. Math. Stud. 6 (2013) 53-72. 28. W. Sintunavarat, P. Kumam, Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces, J. Appl. Math. 2011 Art. ID 637958, 14 pp.
Math. Interdisc. Res. University of Kashan Mathematics Interdisciplinary Research 2538-3639 University of Kashan 106 10.22052/mir.2016.15512 Applicable Analysis Mathematical Engineering Unconditionally Stable Difference Scheme for the Numerical Solution of Nonlinear Rosenau-KdV Equation Unconditionally stable difference scheme for the numerical solution of nonlinear Rosenau-KdV equation Mohebbi Akbar University of Kashan Faraz Zahra University of Kashan 01 07 2016 1 2 291 304 10 10 2015 26 03 2016 Copyright © 2016, University of Kashan. 2016 http://mir.kashanu.ac.ir/article_15512.html

In this paper we investigate a nonlinear evolution model described by the Rosenau-KdV equation. We propose a three-level average implicit finite difference scheme for its numerical solutions and prove that this scheme is stable and convergent in the order of O(τ2 + h2). Furthermore we show the existence and uniqueness of numerical solutions. Comparing the numerical results with other methods in the literature show the efficiency and high accuracy of the proposed method. In this paper we investigate a nonlinear evolution model described by the Rosenau-KdV equation. We propose a three-level average implicit finite difference scheme for its numerical solutions and prove that this scheme is stable and convergent in the order of O(τ2 + h2). Furthermore we show the existence and uniqueness of numerical solutions. Comparing the numerical results with other methods in the literature show the efficiency and high accuracy of the proposed method. In this paper we investigate a nonlinear evolution model described by the Rosenau-KdV equation. We propose a three-level average implicit finite difference scheme for its numerical solutions and prove that this scheme is stable and convergent in the order of O(τ2 + h2). Furthermore we show the existence and uniqueness of numerical solutions. Comparing the numerical results with other methods in the literature show the efficiency and high accuracy of the proposed method.

Finite difference scheme solvability unconditional stability Convergence
1. S. K. Chung, S. N. Ha, Finite element Galerkin solutions for the Rosenau equation, Appl. Anal. 54 (1994) 39-56. 2. G. Ebadi, A. Mojaver, H. Triki, A. Yildirim, A. Biswas, Topological solitons and other solutions of the Rosenau-KdV equation with power law nonlinearity, Romanian J. Phys. 58 (2013) 3-14. 3. A. Esfahani, Solitary wave solutions for generalized Rosenau-KdV equation, Commun. Theor. Phys. 55 (2011) 396-398. 4. J. Hu, Y. Xu, B. Hu, Conservative linear difference scheme for Rosenau-KdV equation, Adv. Math. Phys. 2013, Art. ID 423718, 7 pp. 5. Y. D. Kim, H. Y. Lee, The convergence of finite element Galerkin solution for the Rosenau equation, Korean J. Comput. Appl. Math. 5 (1998) 171-180. 6. P. Rosenau, A quasi-continuous description of a nonlinear transmission line, Phys. Scr. 34 (1986) 827-829. 7. P. Rosenau, Dynamics of dense discrete systems, Progr. Theoret. Phys. 79 (1988) 1028-1042. 8. Z. Z. Sun, D. D. Zhao, On the L1 convergence of a difference scheme for coupled nonlinear Schrödinger equations, Comput. Math. Appl. 59 (2010) 3286-3300. 9. J‎. ‎M‎. ‎Zuo‎, ‎Solitons and periodic solutions ‎ for the Rosenau-KdV and Rosenau-Kawahara equations‎, ‎ Appl‎. ‎Math‎. ‎Comput. 215 (2009) 835-840.
Math. Interdisc. Res. University of Kashan Mathematics Interdisciplinary Research 2538-3639 University of Kashan 106 10.22052/mir.2016.34109 Applied Algebra Wiener Polarity Index of Tensor Product of Graphs Wiener Polarity Index of Tensor Product of Graphs Mogharrab Mojgan Persian Gulf University Sharafdini Reza Persian Gulf University Musavi Somayeh Mathematics House of Bushehr 01 07 2016 1 2 305 316 04 07 2016 05 09 2016 Copyright © 2016, University of Kashan. 2016 http://mir.kashanu.ac.ir/article_34109.html

Mathematical chemistry is a branch of theoretical chemistry for discussion and prediction of the molecular structure using mathematical methods without necessarily referring to quantum mechanics. In theoretical chemistry, distance-based molecular structure descriptors are used for modeling physical, pharmacologic, biological and other properties of chemical compounds. The Wiener Polarity index of a graph G is denoted by W_P (G) is the number of unordered pairs of vertices of distance 3. The Wiener polarity index is used to demonstrate quantitative structure-property relationships in a series of acyclic and cycle-containing hydrocarbons. Let G,H be two simple connected graphs. Then the tensor product of them is denoted by G⨂H whose vertex set is V(G⨂H)=V(G)×V(H) and edge set is E(G⨂H)={(a,b)(c,d)| ac∈E(G) ,bd∈E(H) }. In this paper, we aim to compute the Wiener polarity index of G⨂H which was computed wrongly in [J. Ma, Y. Shi and J. Yue, The Wiener Polarity Index of Graph Products, Ars Combin., 116 (2014) 235-244].

topological index Wiener polarity index tensor product Graph Distance
1. A. R. Ashrafi, T. Dehghanzade, R. Sharafdini, The maximum Wiener polarity index in the class of bicyclic graphs, unpublished paper. 2. A. Behmaram, H. Yousefi-azari, Further results on Wiener polarity index of graphs, Iranian J. Math. Chem. 2(1) (2011) 67–70. 3. A. Behmaram, H. Yousefi-Azari, A. R. Ashrafi, Wiener polarity index of fullerenes and hexagonal systems, Appl. Math. Lett. 25(10) (2012) 1510–1513. 4. A. Bottreau, Y. Métivier, Some remarks on the Kronecker product of graphs, Inform. Process. Lett. 68(2) (1998) 55–61. 5. N. Chen, W. Du, Y. Fan, On Wiener polarity index of cactus graphs, Math. Appl. 26(4) (2013) 798–802. 6. H. Deng, H. Xiao, The Wiener polarity index of molecular graphs of alkanes with a given number of methyl group, J. Serb. Chem. Soc. 75(10) (2010) 1405–1412. 7. H. Deng, H. Xiao, F. Tang, On the extremal Wiener polarity index of trees with a given diameter, MATCH Commun. Math. Comput. Chem. 63(1) (2010) 257–264. 8. H. Deng, On the extremal Wiener polarity index of chemical trees, MATCH Commun. Math. Comput. Chem. 66(1) (2011) 305–314. 9. H‎. ‎Deng‎, ‎H‎. ‎Xiao‎, ‎The maximum Wiener ‎polarity index of trees with k pendants‎,‎ Appl‎. ‎Math‎. ‎Lett. 23(6) (2010) 710-715‎. 10. W. Du, X. Li, Y. Shi, Algorithms and extremal problem on Wiener polarity index, MATCH Commun. Math. Comput. Chem. 62(1) (2009) 235–244. 11. I. Gutman, A property of the Wiener number and its modifications, Indian J. Chem. 36A (1997) 128–132. 12. I. Gutman, A. A. Dobrynin, S. Klavžar, L. Pavlović, Wiener-type invariants of trees and their relation, Bull. Inst. Combin. Appl. 40 (2004) 23–30. 13. R. Hammack, W. Imrich, S. Klavžar, Handbook of product graphs, Second edition. With a foreword by Peter Winkler. Discrete Mathematics and its Applications (Boca Raton). CRC Press, Boca Raton, FL, 2011. 14. H. Hosoya, Mathematical and chemical analysis of Wiener’s polarity number, in: D. H. Rouvray and R. B. King (Eds.), Topology in Chemistry–Discrete Mathematics of Molecules, Horwood, Chichester, 2002. 15. H. Hou, B. Liu, Y. Huang, The maximum Wiener polarity index of unicyclic graphs, Appl. Math. Comput. 218(20) (2012) 10149–10157. 16. A. Ilić, M. Ilić, Generalizations of Wiener polarity index and terminal Wiener index, Graphs Combin. 29(5) (2013) 1403–1416. 17. W. Imrich, S. Klavžar, Product graphs. Structure and recognition. With a foreword by Peter Winkler, Wiley Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York, 2000. 18. B‎. ‎Liu‎, ‎H‎. ‎Hou‎, ‎Y‎. ‎Huang‎,‎ On the Wiener polarity index of trees with ‎ maximum degree or given numbers of leaves‎, ‎ Comput‎. ‎Math‎. ‎Appl. 60(7) (2010) 2053-2057‎. 19. M. Liu, B. Liu, On the Wiener polarity index, MATCH Commun. Math. Comput. Chem. 66(1) (2011) 293-304. 20. I. Lukovits, W. Linert, Polarity-numbers of cycle-containing structures, J. Chem. Inf. Comput. Sci. 38(4) (1998) 715-719. 21. J. Ma, Y. Shi, J. Yue, The Wiener polarity index of graph products, Ars Combin. 116 (2014) 235–244. 22. S. Moradi, A note on tensor product of graphs, Iran. J. Math. Sci. Inform. 7(1) (2012) 73–81. 23. J. Ou, X. Feng, S. Liu, On minimum Wiener polarity index of unicyclic graphs with prescribed maximum degree, J. Appl. Math. 2014 (2014) Art. ID 316108, 9 pp. 24. E. Sampathkumar, On tensor product graphs, J. Austral. Math. Soc. 20(3) (1975) 268–273. 25. P. M. Weichsel, The Kronecker product of graphs, Proc. Amer. Math. Soc. 13 (1962) 47–52. 26. A. N. Whitehead, B. Russell, Principia Mathematica, Cambridge University Press, (1927). 27. H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69(1) (1947) 17-20.
Math. Interdisc. Res. University of Kashan Mathematics Interdisciplinary Research 2538-3639 University of Kashan 106 10.22052/mir.2016.34107 Mathematical Engineering Diameter Two Graphs of Minimum Order with Given Degree Set Diameter Two Graphs of Minimum Order with Given Degree Set Abrishami Gholamreza Ferdowsi University of Mashhad Rahbarnia Freydoon Ferdowsi University of Mashhad Rezaee Irandokht Ferdowsi University of Mashhad 01 07 2016 1 2 317 323 02 05 2016 12 06 2016 Copyright © 2016, University of Kashan. 2016 http://mir.kashanu.ac.ir/article_34107.html

The degree set of a graph is the set of its degrees. Kapoor et al. [Degree sets for graphs, Fund. Math. 95 (1977) 189-194] proved that for every set of positive integers, there exists a graph of diameter at most two and radius one with that degree set. Furthermore, the minimum order of such a graph is determined. A graph is 2-self- centered if its radius and diameter are two. In this paper for a given set of natural numbers greater than one, we determine the minimum order of a 2-self-centered graph with that degree set.

Degree set self-centered graph radius diameter
1. J. Akiyama, K. Ando and D. Avis, Miscellaneous properties of equi-eccentric graphs, Ann. Discrete Math. 20 (1984) 13-23. 2. F. Buckley, The central ratio of a graph, Discrete Math. 38 (1982) 17-21. 3. F. Göbel, H. J. Veldman, Even graphs, J. Graph Theory 10 (1986) 225-239. 4. S. F. Kapoor, A. D. Polimeni, C. E. Wall, Degree sets for graphs, Fund. Math. 95 (1977) 189-194. 5. Z. Stanić, Some notes on minimal self-centered graphs, AKCE J. Graphs. Comb. 7 (2010) 97-102. 6. I. E. Zverovich, On a problem of Lesniak, Polimeni and Vanderjagt. Rend. Mat. Appl. (7)26 (2006) 211-220.
Math. Interdisc. Res. University of Kashan Mathematics Interdisciplinary Research 2538-3639 University of Kashan 106 10.22052/mir.2016.33850 Mathematical Physics Eigenfunction Expansions for Second-Order Boundary Value Problems with Separated Boundary Conditions Eigenfunction Expansions for Second-Order Boundary Value Problems Mosazadeh Seyfollah University of Kashan 01 07 2016 1 2 325 334 07 05 2016 04 10 2016 Copyright © 2016, University of Kashan. 2016 http://mir.kashanu.ac.ir/article_33850.html

In this paper, we investigate some properties of eigenvalues and eigenfunctions of boundary value problems with separated boundary conditions. Also, we obtain formal series solutions for some partial differential equations associated with the second order differential equation, and study necessary and sufficient conditions for the negative and positive eigenvalues of the boundary value problem. Finally, by the sequence of orthogonal eigenfunctions, we provide the eigenfunction expansions for twice continuously differentiable functions.

Boundary value problem Eigenvalue eigenfunction completeness eigenfunction expansion
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