On Total Edge Irregularity Strength of Staircase Graphs and Related Graphs 2 2 Let G=(V(G),E(G)) be a connected simple undirected graph with non empty vertex set V(G) and edge set E(G). For a positive integer k, by an edge irregular total k-labeling we mean a function f : V(G)UE(G) --> {1,2,...,k} such that for each two edges ab and cd, it follows that f(a)+f(ab)+f(b) is different from f(c)+f(cd)+f(d), i.e. every two edges have distinct weights. The minimum k for which G has an edge irregular total k-labeling is called the total edge irregularity strength of graph G and denoted by tes(G). In this paper, we determine the exact value of total edge irregularity strength for staircase graphs, double staircase graphs and mirror-staircase graphs. 1 13 2017/07/26 1396/5/4 2019/05/9 1398/2/19 Y. Susanti Y. Susanti Dept. of Mathematics Universitas Gadjah Mada Indonesia inielsusan@yahoo.com Y. I. Puspitasari Y. I. Puspitasari Surakarta Indonesia Indonesia yuliaindahp.mail@gmail.com H. Khotimah H. Khotimah Department of Mathematics Universitas Muhammadiyah Pringsewu Lampung Indonesia Indonesia husnul.khotimah18@mail.ugm.ac.id Total edge irregularity strength Staircase graphs Double staircase graphs Mirror-staircase graphs [1] Ahmad, A., On the total edge irregularity strength of zigzag graphs, Australasian Journal of Combinatorics, Volume 54, Pages 141-149 (2012).##[2] Bau{c}a, M., Jendrol, S., Miller, M., Ryan, J. On irregular total labeling. Discrete Math, 307, 1378-1388 (2007).##[3] Chartrand, G., Jacobson, M.S., Lehel, J., Oellermann, O.R., Ruiz, S., Saba, F., Irregular networks, Congr. Numer. 64 pp. 355-374, (1988).##[4] Gallian, J.A. A dynamic survey of graph labeling, The Electronic Journal of Combinatorics,18, 247-252 (2015).##[5] Ivanco, J., Jendrol, S., The total edge irregularity strength of trees, Discuss. Math. Graph Theory, 26 pp. 449-456 (2006).##[6] Jendrol, S., Miskuf, J., Sotak, R., Total edge irregularity strength of complete graphs and complete bipartite graphs, Discrete Mathematics, Volume 310, Issue 3, Pages 400-407 (2010).##[7] Putra, R.W., Susanti, Y., On total edge irregularity strength of centralized uniform theta graphs, AKCE International Journal of Graphs and Combinatorics, volume 15 issue 1 page 7-13. (2018).##[8] Putra, R.W., Susanti, Y., The total edge irregularity strength of uniform theta graphs, IOPScience Journal of Physics: Conference Series, 2018 J. Phys.: Conf. Ser. 1097 012069 (2018).##[9] Ratnasari L., Susanti, Y., Total edge irregularity strength of ladder related graphs, Asian-European Journal of Mathematics, doi:10.1142/S1793557120500722.##[10] L. Ratnasari, S. Wahyuni, Y. Susanti, D. Junia Eksi Palupi and B. Surodjo, Total edge irregularity strength of arithmetic book graphs, Journal of Physics: Conference Series 1306(1), 012032 (2019).##[11] Solairaju, A., Arockiasamy, A. M. Graceful mirror-staircase graphs, Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 49, 2433 - 2441 (2010).##[12] Wallis, W.D. Magic graphs. Boston: Birkhauser (2011).## ##

On the Diophantine Equation x^6+ky^3=z^6+kw^3 2 2 Given the positive integers m,n, solving the well known symmetric Diophantine equation xm+kyn=zm+kwn, where k is a rational number, is a challenge. By computer calculations, we show that for all integers k from 1 to 500, the Diophantine equation x6+ky3=z6+kw3 has infinitely many nontrivial (y≠w) rational solutions. Clearly, the same result holds for positive integers k whose cube-free part is not greater than 500. We exhibit a collection of (probably infinitely many) rational numbers k for which this Diophantine equation is satisfied. Finally, appealing these observations, we conjecture that the above result is true for all rational numbers k. 15 21 2017/07/262017/01/7 1395/10/18 2019/05/92020/03/4 1398/12/14 H. Shabani-Solt. H. Shabani-Solt. Urmia University Iran h.shabani.solt@gmail.com N. Yusefnejad N. Yusefnejad Urmia University Iran yusefnejadnazanin@yahoo.com A. S. Janfada A. S. Janfada Urmia University Iran asjanfada@gmail.com Diophantine equation Elliptic curve. A. Choudhry, Symmetric Diophantine equations, Rocky Mountain J. Math., 34(4), (2004), 1281-1248.##H. Inose, On certain Kummer surface which can be realized as non-singular quartic surfaces in P^3, J. Fac. Sci. Univ. Tokyo, 23, (1476), 545-560.##A. S. Janfada and A. Abbaspoor, On Diophantine equations X^6+6Z^3=Y^6 ± 6W^3, Int. J. Pure and App. Math., 105(4), (2015), 709-713.##M. Kuwata, Elliptic fibrations on quartic K3 surfaces with large Picard numbers, Pacific J. Math., 171(1), (1995) 231-243.##T. N. Shorey and R. Tijdeman, Exponential Diophantine equations, Cambridge University Press, 1986.##L. C. Washington, Elliptic curves: Number theory and cryptography, Second edition, Taylor & Francis Group LLC, 2008.## ##

Sums of Strongly z-Ideals and Prime Ideals in ${mathcal{R}} L$ 2 2 It is well-known that the sum of two $z$-ideals in $C(X)$ is either $C(X)$ or a $z$-ideal. The main aim of this paper is to study the sum of strongly $z$-ideals in ${mathcal{R}} L$, the ring of real-valued continuous functions on a frame $L$. For every ideal $I$ in ${mathcal{R}} L$, we introduce the biggest strongly $z$-ideal included in $I$ and the smallest strongly $z$-ideal containing $I$, denoted by $I^{sz}$ and $I_{sz}$, respectively. We study some properties of $I^{sz}$ and $I_{sz}$. Also, it is observed that the sum of any family of minimal prime ideals in the ring ${mathcal{R}} L$ is either ${mathcal{R}} L$ or a prime strongly $z$-ideal in ${mathcal{R}} L$. In particular, we show that the sum of two prime ideals in ${mathcal{R}} L$ such that are not a chain, is a prime strongly $z$-ideal. 23 34 2017/07/262017/01/72017/02/12 1395/11/24 2019/05/92020/03/42019/11/7 1398/8/16 A. A. Estaji A. A. Estaji Hakim Sabzevari University Iran aaestaji@gmail.com A. Karimi Feizabadi A. Karimi Feizabadi Islamic Azad University Iran akarimi@gorganiau.ac.ir M. Robat Sarpoushi M. Robat Sarpoushi Hakim Sabzevari University Iran M.sarpooshi@yahoo.com Frame Ring of real-valued continuous functions z-Ideal Strongly z-ideal. R.N. Ball, and J. Walters-Wayland, $C$ and $C^*$ quotients on pointfree topology, textit{ Dissertations Mathematicae (Rozprawy Mat.)}, textit{412}, Warszawa (2002).##B. Banaschewski, The real numbers in pointfree topology, Textos de Mathematica (Series B), 12, University of Coimbra, Departmento de Mathematica, Coimbra, 1997.##T. Dube, Some algebraic characterizations of $F$-frames, textit{Algebra Universalis}, textit{62}, (2009), 273--288.##T. Dube, Concerning $P$-frames, essential $P$-frames and strongly zero-dimensional frames, Algebra Universalis, 61, (2009), 115--138.##M.M. Ebrahimi, and A. Karimi Feizabadi, Pointfree prime representation of real Riesz maps, Algebra Univers., textit{54}, (2005), 291--299.##A.A. Estaji, z-weak ideals and prime weak ideals, Iranian Journal of Mathematical Sciences and Informatics, 7(2), (2012), 53{62.##A.A. Estaji, A. Karimi Feizabadi, and M. Abedi, Zero sets in pointfree topology and strongly z-ideals, Bull. Iranian Math. Soc., 41, (2015), 1071--1084.##A.A. Estaji, A. Karimi Feizabadi, and M. Abedi, Strongly fixed ideals in $C (L)$ and compact frames, Archivum Mathematics, 51, (2015), 1--12.##A.A. Estaji, A. Karimi Feizabadi, and M. Abedi, Intersection of essential ideals in the ring of real-valued continuous functions on a frame, Journal of Algebraic Systems, 5, (2017), pp 149--161##L. Gillman, and M. Jerison, Rings of continuous functions, Springer-Verlag, 1976##M. Henriksen and F.A. Smith, Sums of z-ideals and semiprime ideals, General Topology and Its Relations to Modern Analysis and Algebra, 5, (1982), 272--278.##O. Ighedo, Concerning ideals of pointfree function rings, Ph.D. Thesis, University of South Africa, 2013.##R.Y. Sharp, Steps in commutative algebra, Cambridge Univ. press (2000).##P.T. Johnstone, Stone spaces, Cambridge Univ. Press (Cambridge 1982).##A. Karimi Feizabadi, A. A. Estaji and M. Abedi, On minimal ideals in the ring of real-valued continuous functions on a frame, Archivum Mathematics, 54, (2018), No. 1, 1--13.##C. Kohls, Ideals in rings of continuous functions, Fund. Math., 45, (1957), 28--50.##G. Mason, $z$-ideals and prime ideals, J. Algebra, textit{26}, (1973), 280--297.##J. Picado and A. Pultr, Frames and Locales: Topology without Points, Frontiers in Mathematics, Birkh"{a}user/Springer, Basel AG, Basel (2012).##A. Rezaei Aliabad and M. Parsinia, zR-Ideals and zR-Ideals in subrings of RX,Iranian Journal of Mathematical Sciences and Informatics, 14(1), (2019), 55-67.##D. Rudd, On two sum theorems for ideals of $C(X)$, Mich. Math. J., 17, (1970), 139--141.## ##

Characterization of $mathrm{PSL}(5,q)$ by its Order and One Conjugacy Class Size 2 2 Let $p=(q^4+q^3+q^2+q+1)/(5,q-1)$ be a prime number, where $q$ is a prime power. In this paper, we will show $Gcong mathrm{PSL}(5,q)$ if and only if $|G|=|mathrm{PSL}(5,q)|$, and $G$ has a conjugacy class size $frac{| mathrm{PSL}(5,q)|}{p}$. Further, the validity of a conjecture of J. G. Thompson is generalized to the groups under consideration by a new way. 35 40 2017/07/262017/01/72017/02/122017/04/9 1396/1/20 2019/05/92020/03/42019/11/72017/08/14 1396/5/23 A.R. Khalili Asboei A.R. Khalili Asboei Department of Mathematics, Farhangian University Iran khaliliasbo@yahoo.com Conjugacy class size Prime graph Thompson's conjecture. N. Ahanjideh, On Thompson's conjecture for some nite simple groups, J. Algebra, 344,(2011), 205-228.##S. S. Amiri, A. K. Asboei, Characterization of some nite groups by order and length of one conjugacy class, Sib. Math. J, 57(2), (2016), 185-189.##A. K. Asboei, New characterization of symmetric groups of prime degree, Acta Univ.Sapientiae Math, 9(1), (2017), 5-12.##A. K. Asboei, A new characterization of PSL(3; q), Jordan J. Math. Stat, 10(4), (2017),307-317.##A. K. Asboei, R. Mohammadyari, M. Rahimi, New characterization of some linear groups,Int. J. Industrial Mathematics, 8(2), (2016), 165-170.##A. K. Asboei, R. Mohammadyari, Recognizing alternating groups by their order and one conjugacy class length, J. Algebra. Appl, 15(2), (2016), 1650021.##A. K. Asboei, R. Mohammadyari, Characterization of the alternating groups by their order and one conjugacy class length, Czechoslovak Math. J, 66(141), (2016), 63-70.##A. K. Asboei, R. Mohammadyari, M. R. Darafsheh, The in uence of order and conjugacy class length on the structure of nite groups, Hokkaido Math. J, 47, (2018), 25-32.##G. Y. Chen, On Frobenius and 2-Frobenius group, J. Southwest China Normal Univ, 20, (1995), 485-487. (in Chinese).##G. Y. Chen, On Thompson's conjecture, J. Algebra, 185(1), (1996), 184-193.##G. Y. Chen, Further rections on Thompson's conjecture, J. Algebra, 218, (1999),276-285.##G. Y. Chen, A new characterization of sporadic simple groups, Algebra Colloq, 3(1),(1996), 49-58.##Y. Chen, G. Y. Chen, Recognizing PSL(2; p) by its order and one special conjugacy class size, J. Inequal. Appl, (2012), 310.##Y. H. Chen, G. Y. Chen, Recognization of Alt10 and PSL(4; 4) by two special conjugacy class size, Ital. J. Pure Appl. Math, 29, (2012), 387-394.##J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Wilson, Atlas of nite groups, Clarendon, Oxford, 1985.##M. Foroudi ghasemabadi, N. Ahanjideh, Characterization of the simple groups Dn(3) by prime graph and spectrum, Iran. J. Math. Sci. Inform, 7(1), (2012), 91-106.##A. Iranmanesh, S. H. Alavi, A characterization of simple group PSL(5; q), Bull. Austral. Math. Soc, 65, (2002), 211-222.##N. Iiyori, H. Yamaki, Prime graph components of the simple groups of Lie type over the eld of even characteristic, J. Algebra, 155(2), (1993), 335-343.##E. I. Khukhro, V. D. Mazurov, Unsolved Problems in Group Theory, The Kourovka Notebook, 17th edition, Sobolev Institute of Mathematics, Novosibirsk, 2010.##A. S. Kondtratev, V. D. Mazurov, Recognition of Alternating groups of prime degree from their element orders, Sib. Math. J, 41(2), (2000), 294-302.##J. B. Li, Finite groups with special conjugacy class sizes or generalized permutable subgroups, (2012), (Chongqing: Southwest University).##G. R. Rezaeezadeh, M. R. Darafsheh, M. Bibak, M. Sajjadi, OD-characterization of Almost Simple Groups Related to D4(4), Iran. J. Math. Sci. Inform, 10(1), (2015), 23-43.##J. S. Williams, Prime graph components of nite groups, J. Algebra, 69(2), (1981),487-513.## ##

The Banach Type Contraction for Mappings on Algebraic Cone Metric Spaces Associated with An Algebraic Distance and Endowed with a Graph 2 2 In this work, we define the notion of an algebraic distance in algebraic cone metric spaces defined by Niknam et al. [A. Niknam, S. Shamsi Gamchi and M. Janfada, Some results on TVS-cone normed spaces and algebraic cone metric spaces, Iranian J. Math. Sci. Infor. 9 (1) (2014), 71--80] and introduce some its elementary properties. Then we prove the existence and uniqueness of fixed point for a Banach contractive type mapping in algebraic cone metric spaces associated with an algebraic distance and endowed with a graph. 41 52 2017/07/262017/01/72017/02/122017/04/92017/04/14 1396/1/25 2019/05/92020/03/42019/11/72017/08/142019/05/8 1398/2/18 K. Fallahi K. Fallahi Department of Mathematics, Payame Noor University, P.O. Box 19395-4697, Tehran, Iran Iran fallahi1361@gmail.com Gh. Soleimani Rad Gh. Soleimani Rad Department of Mathematics, Payame Noor University, P.O. Box 19395-4697, Tehran, Iran Iran gh.soleimani2008@gmail.com Algebraic cone metric space Algebraic distance Banach contraction Orbitally G-continuous mapping F. Bojor, Fixed Point Theorems for Reich Type Contractions on Metric Spaces with a Graph, Nonlinear Analysis (TMA), 75(1), (2012), 1359-1373.##J. A. Bondy, U. S. R. Murty, Graph Theory, Springer, New York, 2008.##Y. J. Cho, R. Saadati, S. H. Wang, Common Fixed Point Theorems on Generalized Distance in Ordered Cone Metric Spaces, Computers & Mathematics with Applications, 61, (2011), 1254-1260.##P. Cholamjiak, Fixed Point Theorems for Banach Type Contarction on TVS-Cone Metric Spaces Endowed with a Graph, Journal of Computational Analysis and Applications, 16(2), (2011), 338-345.##Lj. Ciric, H. Lakzian, V. Rakocevic, Fixed Point Theorems for w-Cone Distance Contraction Mappings in tvs-Cone Metric Spaces, Fixed Point Theory and Applications, 2012, 2012:3.##M. Dordevic, D. Doric, Z. Kadelburg, S. Radenovic, D. Spasic, Fixed Point Results under c-Distance in tvs-Cone Metric Spaces, Fixed Point Theory and Applications, 2011, 2011:29.##L. G. Huang, X. Zhang, Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings, Journal of Mathematical Analysis and Applications, 332, (2007), 1467-1475.##J. Jachymski, The Contraction Principle for Mappings on a Metric Space with a Graph, Proceedings of the American Mathematical Society, 136(4), (2008), 1359-1373.##A. Nicolae, D. O'Regan, A. Petru¸sel, Fixed Point Theorems for Singlevalued and Multivalued Generalized Contractionsin Metric Spaces Endowed with a Graph, Georgian Mathematical Journal, 18, (2011), 307-327.##J. J. Nieto, R. Rodrıguez-Lopez, Contractive Mapping Theorems in Partially Ordered Sets and Applications to Ordinary Differential Equations, Order, 22(3), (2005), 223-239.##A. Niknam, S. Shamsi Gamchi, M. Janfada, Some Results on T V S-Cone Normed Spaces and Algebraic Cone Metric Spaces, Iranian Journal of Mathematical Sciences and Informatics, 9(1), (2014), 71-80.##A. Petrusel, I. A. Rus, Fixed Point Theorems in Ordered L-Spaces, Proceedings of the American Mathematical Society, 134(2), (2006), 411-418.##H. Rahimi, G. Soleimani Rad, Common Fixed-Point Theorems and c-Distance in Ordered Cone Metric Spaces, Ukrainian Mathematical Journal, 65(12), (2014), 1845-1861.##H. Rahimi, G. Soleimani Rad, S. Radenovic, Algebraic Cone b-Metric Spaces and its Equivalence, Miskolc Mathematical Notes, 17(1), (2016), 553-560.## ##

On the Prime Spectrum of Torsion Modules 2 2 The paper uses a new approach to investigate prime submodules and minimal prime submodules of certain modules such as Artinian and torsion modules. In particular, we introduce a concrete formula for the radical of submodules of Artinian modules. 53 63 2017/07/262017/01/72017/02/122017/04/92017/04/142017/04/19 1396/1/30 2019/05/92020/03/42019/11/72017/08/142019/05/82018/07/30 1397/5/8 D. Hassanzadeh-lelekaami D. Hassanzadeh-lelekaami Arak University of Technology Iran lelekaami@gmail.com Torsion modules Artinian module Prime submodules. A. Abbasi and D. Hassanzadeh-Lelekaami, Modules and spectral spaces, Comm. Algebra,40(11), (2012), 4111-4129.##S. Abu-Saymeh, On dimensions of finitely generated modules, Comm. Algebra, 23(3),(1995), 1131-1144.##D. D. Anderson, A note on minimal prime ideals, Proc. Amer. Math. Soc., 122, (1994),13-14.##A. Azizi, Weak multiplication modules, Czechoslovak Math. J., 53(128), (2003), 529-534.##A. Azizi, Strongly irreducible ideals, J. Aust. Math. Soc., 84 (2008), 145-154.##A. Azizi, Prime submodules of Artinian modules, Taiwanese J. Math., 13(6B), (2009),2011-2020.##M. Behboodi, A generalization of the classical krull dimension for modules, J. Algebra, 305, (2006), 1128-1148.##M. P. Brodmann, R. Y. Sharp, Local cohomology: An algebraic introduction with geometric applications, Cambridge University Press, 1998.##J. Dauns, Prime modules, J. Rein. Ang. Math., 298, (1978), 165-181.##Z. A. El-Bast, P. F. Smith, Multiplication modules, Comm. Algebra, 16(4), (1988), 755-779.##E. H. Feller and E. W. Swokowski, Prime modules, Canad. J. Math., 17, (1965), 1041-1052.##D. Hassanzadeh-Lelekaami and H. Roshan-Shekalgourabi, Prime submodules and a sheaf on the prime spectra of modules, Comm. Algebra, 42(7), (2014), 3063-3077.##D. Hassanzadeh-Lelekaami and H. Roshan-Shekalgourabi, On regular modules over commutative rings, Bull. Malays. Math. Sci. Soc (2017), ##https://doi.org/10.1007/s40840-017-0501-0##H. Koohy, On finiteness of multiplication modules, Acta Math. Hungar., 118(1-2), (2008),1-7.##Chin-Pi Lu, Prime submodules of modules, Comment. Math. Univ. St. Pauli, 33(1),(1984), 61-69.##Chin-Pi Lu, M-radicals of submodules in modules, Math. Japonica, 34(2), (1989), 211-219.##Chin-Pi Lu, Spectra of modules, Comm. Algebra 23(10), (1995), 3741-3752.##Chin-Pi Lu, The Zariski topology on the prime spectrum of a module,Houston J. Math.,25(3), (1999), 417-432.##Chin-Pi Lu, Saturations of submodules, Comm. Algebra, 31(6), (2003), 2655-2673.##Chin-Pi Lu, A module whose prime spectrum has the surjective natural map, Houston J. Math., 33(1), (2007),125-143.##Chin-Pi Lu, Modules with Noetherian spectrum, Comm. Algebra, 38(3), (2010), 807-828.##S. H. Man, On commutative rings which satisfy the generalized radical formula, Comm.Algebra, 27(8), (1999), 4075-4088.##R. L. McCasland, M. E. Moore, On radicals of submodules of finitely generated modules,Comm. Algebra, 29(1), (1986), 37-39.##R. L. McCasland, M. E. Moore, On radicals of submodules, Comm. Algebra, 19, (1991),1327-1341.##R. L. McCasland, M. E. Moore, Prime submodules, Comm. Algebra, 20(6), (1992),1803-1817.##R. L. McCasland, M. E. Moore, and P. F. Smith, On the spectrum of a module over a commutative ring, Comm. Algebra, 25(1), (1997), 79-103.##R. L. McCasland, P. F. Smith, Prime submodules of noetherian modules, Rocky Mtn. J.Math., 23(3), (1993), 1041-1062.##M. E. Moore, S. J. Smith, Prime and radical submodules of modules over commutative rings, Comm. Algebra, 30(10), (2002), 5037-5064.##S. Namazi, Y. Sharifi, Catenary modules, Acta Math. Hungar., 85(3), (1999), 211-218.##D. Pusat-Yilmaz, P. F. Smith, Radicals of submodules of free modules, Comm. Algebra,27(5), (1999), 2253-2266.##D. Pusat-Yilmaz, P. F. Smith, Modules which satisfy the radical formula, Acta Math.Hungar, 95(1-2), (2002), 155-167.##Y. Sharp, Steps in commutative algebra, second ed., Cambridge University Press, 2000.##P. F. Smith, Concerning a theorem of i. s. cohen, Analele stiintifice ale Universitatii Ovidius Constanta, XIth National Conference of Algebra (Constanta, 1994), National Conference of Algebra, 1994, pp. 160-167.##Y. Tiras, M. Alkan, Prime modules and submodules, Comm. Algebra, 31(11), (2003), 5253-5261.##O. Zariski, P. Samuel, Commutative algebra, vol. I, Princeton, New Jersey: D. Van Nostrand Co., Inc., 1958.## ##

Harmonicity and Minimality of Vector Fields on Lorentzian Lie Groups 2 2 ‎We consider four-dimensional lie groups equipped with‎ ‎left-invariant Lorentzian Einstein metrics‎, ‎and determine the harmonicity properties ‎of vector fields on these spaces‎. ‎In some cases‎, ‎all these vector fields are critical points for the energy functional ‎restricted to vector fields‎. ‎We also classify vector fields defining harmonic maps‎, ‎and calculate explicitly the energy of these vector ‎fields‎. ‎Then we study the minimality of critical points for the energy functional‎. 65 78 2017/07/262017/01/72017/02/122017/04/92017/04/142017/04/192015/11/30 1394/9/9 2019/05/92020/03/42019/11/72017/08/142019/05/82018/07/302019/10/20 1398/7/28 Y. Aryanejad Y. Aryanejad Payame noor University Iran y.keshavarzi@pnu.ac.ir Harmonic vector fields‎ ‎Harmonic maps‎ ‎Einstein metrics‎ ‎Lie group‎ ‎Pseudo-Riemannian homogeneous spaces. A. R. Ashrafi, M. R. Ahmadi, Symmetry of fullerene C60, Iranian Journal of Mathematical Sciences and Informatics, 1(1), (2006) 1-13.##L. Bérard-Bérgery, Homogeneous Riemannian spaces of dimension four, Seminar A.Besse, Four-dimensional Riemannian geometry, (1985).##G. Calvaruso, Harmonicity properties of invariant vector fields on three-dimensional Lorentzian Lie groups, J.Geom. Phys. 61 (2011), 498-515.##G. Calvaruso and A. Zaeim, Four-dimensional Lorentzian Lie groups, Differ. Geom. Appl. 31 (2013), 496-509.##M. Chaichi, E. Garcia-Rio, Y. Matsushita, Curvature properties of four-dimensional Walker metrics, Classical Quantum Gravity, 22(3), (2005), 559-577.##M. Chaichi, E. Garcia-Rio, Y. Matsushita, Three-dimensional Lorentz manifolds admitting a parallel null vector field, J. Phys. A, 38(4), (2005), 841-850.##M. Dabirian, A. Iranmanesh, The molecular symmetry group theory of trimethylamine-BH3 addend (BH3 free of rotation), Iranian Journal of Mathematical Sciences and Informatics, 1(1), (2006) 15-26.##S. Dragomir, D. Perrone, Harmonic Vector Fields: Variational Principles and Differential Geometry, Elsevier, Science Ltd, (2011).##O. Gil-Medrano, Relationship between volume and energy of vector fields, Diff. Geom. Appl. 15 (2001), 137-152.##T. Ishihara, Harmonic sections of tangent bundles, J. Math. Tokushima Univ. 13, (1979), 23-27.##O. Nouhaud, Applications harmoniques d'une variété riemannienne dans son fibré tangent, Généralisation, C. R. Acad. Sci. Paris Sér. A-B, 284(14), (1977), A815-A818.##A. Zaeim, M. Chaichi, Y. Aryanejad, On Lorentzian Two-Symmetric Manifolds of Dimension-Four, Iranian Journal of Mathematical Sciences and Informatics, 12(4), (2017) 81-94.## ##

New Integral Inequalities Through the phi-Preinvexity 2 2 Abstract. In this note, we give some estimates of the generalized quadrature formula of Gauss-Jacobi type for phi-preinvex functions. 79 83 2017/07/262017/01/72017/02/122017/04/92017/04/142017/04/192015/11/302017/04/2 1396/1/13 2019/05/92020/03/42019/11/72017/08/142019/05/82018/07/302019/10/202018/03/11 1396/12/20 B. Meftah B. Meftah Laboratoire des Télécommunications, Faculté des Sciences et de la Technologie, University of 8 May 1945 Guelma, P.O. Box 401, 24000 Guelma, Algeria. Algeria badrimeftah@yahoo.fr Integral inequality $varphi $-preinvex function H ̈older inequality power mean inequality I. Ahmad, Integral inequalities under beta function and preinvex type functions. SpringerPlus,5(1), (2016), 1-6.##P. S. Bullen, Handbook of means and their inequalities. Revised from the 1988 original [P.S. Bullen, D. S. Mitrinovic and P. M. Vasic, Means and their inequalities, Reidel, Dordrecht MR0947142]. Mathematics and its Applications, 560. Kluwer Academic Publishers Group, Dordrecht, 2003.##M. E. Gordji, M. R. Delavar and M. De La Sen, On '-convex functions, J. Math. Inequal. 10(1),(2016), 173-183.##I. Iscan, M. Aydin and S. Dikmenoglu, New integral inequalities via harmonically convex functions. Mathematics and Statistics 3(5), (2015), 134-140.##W. Liu, New integral inequalities via (, m)-convexity and quasi-convexity. Hacet. J.Math. Stat. 42(3), (2013), 289-297.##W. Liu, New integral inequalities involving beta function via P -convexity. Miskolc Math.Notes 15(2), (2014), 585-591.##D. S. Mitrinovi c, J. E. Pecarc and A. M. Fink, Classical and new inequalities in analysis. Mathematics and its Applications (East European Series), 61. Kluwer Academic Publishers Group, Dordrecht, 1993.##M. Muddassar, A. Ali, New integral inequalities through generalized convex functions. Punjab Univ. J. Math. (Lahore) 46(2), (2014), 47-51.##M. A. Noor, K. I. Noor, S. Iftikhar and M. U. Awan, Strongly generalized harmonic convex functions and integral inequalities. Journal of Mathematical Analysis. 7(3), (2016), 66-77.##M. E. Ozdemir, E. Set, M. Alomari, Integral inequalities via several kinds of convexity. Creat. Math. Inform. 20(1), (2011), 62-73.##B. G. Pachpatte, Analytic inequalities. Recent advances. Atlantis Studies in Mathematics, 3. Atlantis Press, Paris, 2012.##D. D. Stancu, G. Coman and P. Plaga, Analiza numerica si teoria aproximarii. Vol.II. (Romanian) [Numerical analysis and approximation theory. Vol. II] Presa Universitara Clujeana, Cluj-Napoca, 2002.## ##

Quotient G-systems and Green's Relations 2 2 In this paper, we first introduce the concepts of G-systems, quotient G-systems and isomorphism theorems on G-systems of n-ary semihypergroups . Also we consider the Green's equivalences on G-systems and further in-vestigate some of their properties. A number of n-ary semihypergroups are constructed and presented as examples in this paper. 85 97 2017/07/262017/01/72017/02/122017/04/92017/04/142017/04/192015/11/302017/04/22017/05/10 1396/2/20 2019/05/92020/03/42019/11/72017/08/142019/05/82018/07/302019/10/202018/03/112017/09/8 1396/6/17 S. Ostadhadi-Dehkordi S. Ostadhadi-Dehkordi Department of Mathematics, Hormozgan University, Bandar Abbas, Iran. Iran ostadhadi-dehkordi@hotmail.com K. P. Shum K. P. Shum Institute of Mathematics,Yunnan University, Kunming,650091, P.R. China China kpshum@ynu.edu.cn n-ary Semihypergroup G-system Greens relations. R. Chinram and P. Siammai, On Green s relations for Gamma-semigroups and reductive Gamma-semigroups, International Journal of Algebra, 2(4), (2008), 187-195.##P. Corsini, Prolegomena of hypergroup theory, Second Edition, Aviani Editore (1993).##P. Corsini and V. Leoreanu, Applications of hyperstructure theory, Advances in Mathematics, Kluwer Academic Publishers, Dordrecht, (2003).##I. Cristea and M. Stefanescu, Hypergroups and n-ary relations, European Journal of Combinatorics, 31(3), (2010), 780-789.##B. Davvaz and V. Leoreanu-Fotea, Hyperring theory and applications, International Academic Press USA (2007).##B. Davvaz and T. Vougiouklis, n-ary hypergroups, Iranian Journal of Science and. Technology, Transaction A, 30 (A2), (2006), 165-174.##W. Dornte, Unterschungen uber einen verallgemeinerten gruppenbegriff, Mathematische Zeitschrift, 29, (1929), 1-19.##M. Farshi and B. Davvaz, Relations and homomorphisms of n-hypergroups, European Journal of Combinatorics, 44 (2015), 218-230.##J. A. Green, On the structure of semigroups. Annals of Mathematics, 54, (1951), 163-172.##K. Hila , B. Davvaz and K. Naka, On hyperideal structure of ternary semihypergroup, Iranian Journal of Mathematical Sciences and Informatics, 9(1), (2014), 81-98.##V. Leoreanu-Fotea and B. Davvaz, n-hypergroups and binary relations, European Journal of Combinatorics, 29 (5), (2008), 1207-1218.##V. Leoreanu-Fotea, Ivo Rosenberg, B. Davvaz and T. Vougiouklis, A new class of n-ary hyperoperations, European Journal of Combinatorics, 44 (2015), 265-273.##F. Marty, Sur une generalization de la notion de group, 8 th Congres mathematics Scandinaves (1934), 45-49.##S. Mirvakili and B. Davvaz, Relations on Krasner (m; n)-hyperrings, European Journal of Combinatorics, 31(3), (2010), 79-802.##S. Ostadhadi-Dehkordi m-Ary Hypervector Space: Convergent Sequence and Bundle Sub-sets, Iranian Journal of Mathematical Sciences and Informatics, 11(2), (2016), 23-41.##F. M. Sioson, A note on congruences, Proceedings of the Japan Academy, 43 (1967), 103-107## ##

Comparing Model-based Versus K-means Clustering for the Planar Shapes 2 2 ‎In some fields‎, ‎there is an interest in distinguishing different geometrical objects from each other‎. ‎A field of research that studies the objects from a statistical point of view‎, ‎provided they are‎ ‎invariant under translation‎, ‎rotation and scaling effects‎, ‎is known as the statistical shape analysis‎. ‎Having some objects that are registered using key points on the outline of the objects‎, ‎the main purpose‎ ‎of this paper is to compare two popular clustering procedures to cluster objects‎. ‎We also use some indexes‎ ‎to evaluate our clustering application‎. ‎The proposed methods are applied to the real life data. 99 109 2017/07/262017/01/72017/02/122017/04/92017/04/142017/04/192015/11/302017/04/22017/05/102017/05/9 1396/2/19 2019/05/92020/03/42019/11/72017/08/142019/05/82018/07/302019/10/202018/03/112017/09/82018/03/6 1396/12/15 M. Golalizadeh M. Golalizadeh Department of Statistics, Faculty of Mathematical Sciences, Tarbiat Modares University Iran golalizadeh@modares.ac.ir H. Jafari H. Jafari Department of Statistics, Faculty of Mathematical Sciences, Tarbiat Modares University Iran ‎hamed.jafari@modares.ac.ir Shape‎ ‎Clustering‎ ‎K-means‎ ‎Model-based‎ ‎Landmark‎. ‎F‎. ‎L‎. ‎Bookstein‎, ‎Size and Shape Spaces for Landmark Data in Two Dimensions‎, Statistical Science‎, 10‎, ‎1986‎, ‎181-222‎.##‎A‎. ‎P‎. ‎Dempster‎, ‎N‎. ‎M‎. ‎Laird‎, ‎D‎. ‎B‎. ‎Rubin‎, ‎Maximum Likelihood from Incomplete Data via the EM Algorithm‎, Journal of the Royal Statistical Society‎, ‎Series B‎, 39‎, ‎1977‎, ‎1-38‎.##‎I‎. ‎L‎. ‎Dryden‎, ‎K‎. ‎V‎. ‎Mardia‎, ‎ Statistical Shape Analysis‎, ‎Wiley‎, ‎Chichester‎, ‎1998‎.##‎I‎. ‎L‎. ‎Dryden‎, ‎K‎. ‎V‎. ‎Mardia‎, ‎ Statistical Shape Analysis‎: With Application in R, ‎Wiley‎, ‎Chichester‎, ‎2016‎.##‎J‎. ‎A‎. ‎Hartigan‎, ‎M‎. ‎A‎. ‎Wong‎, ‎Algorithm AS 136‎: ‎A k-means Clustering Algorithm‎, ‎Applied Statistics, ‎ 28, ‎1979‎, ‎100-108‎.##‎M‎. ‎Halkidi‎, ‎Y‎. ‎Batistakis‎, ‎M‎. ‎Vazirgiannis‎, ‎On Clustering Validation Techniques‎, Journal of Intelligent Information Systems‎, 17‎, ‎2001‎, ‎107-145‎.##‎C‎. ‎Huang‎, , ‎C‎. ‎Styner‎, ‎H‎. ‎Zhu‎, ‎Clustering High-Dimensional Landmark-based Two-Dimensional Shape Data‎, Journal of the American Statistical Association‎, 110‎, ‎2015‎, ‎946-961‎.##‎D‎. ‎G‎. ‎Kendall‎, ‎The Diffusion of Shape‎, Advances in Applied Probability‎, 9‎, ‎1977‎, ‎428-430‎.##‎D‎. ‎G‎. ‎Kendall‎, ‎Shape Manifolds‎, ‎Procrustean Metrics‎, ‎and Complex Projective Spaces‎, Bulletin of the London Mathematical Society‎, 16‎, ‎1984‎, ‎81-121‎.##‎P‎. ‎O'Higgins‎, A Morphometric Study of Cranial Shape in the Hominoidea‎, ‎Phd Dissertation‎, ‎University of Leeds‎, ‎Leeds‎, ‎1989‎.##‎R‎. ‎Willink‎, ‎Normal Moments and Hermite Polynomials‎. ‎ Statistics and Probability Letters‎, 73‎, ‎2005‎, ‎271-275‎.## ##

Extensions of Some Fixed Point Theorems for Weak-Contraction Mappings in Partially Ordered Modular Metric Spaces 2 2 The purpose of this paper is to establish fixed point results for a single mapping in a partially ordered modular metric space, and to prove a common fixed point theorem for two self-maps satisfying some weak contractive inequalities. 111 124 2017/07/262017/01/72017/02/122017/04/92017/04/142017/04/192015/11/302017/04/22017/05/102017/05/92017/06/12 1396/3/22 2019/05/92020/03/42019/11/72017/08/142019/05/82018/07/302019/10/202018/03/112017/09/82018/03/62019/05/29 1398/3/8 K. Chaira K. Chaira University of Casablanca Morocoo chaira_karim@yahoo.fr A. Eladraoui A. Eladraoui University of Casablanca Morocoo a.adraoui@live.fr M. Kabil M. Kabil University of Casablanca Morocoo kabilfstm@gmail.com Fixed point Weak contraction Partially ordered space Modular metric space. A.Abkar, B.S.Choudhury, Fixed point results in partially ordered metric spaces using weak contractive inequalities, Facta Universitatis(NIS), Ser. Math. Inform. Vol. 27, No1 (2012), 1-11.##A.A.N.Abdou, Some xed point theorems in modular metric spaces, J. Nonlinear Sci. Appl. 9 (2016), 4381-4387.##A.A.N.Abdou, M.A.Khamsi, Fixed point results of pointwise contractions in modular metric spaces, Fixed Point Theory and Applications 2013, 2013:163.##M.Abbas, S.Ali, P.Kumam, Common xed points in partially ordered modular function spaces, Journal of Inequalities and Applications 2014, 2014:78.##Y.I.Alber, S.Guerre-Delabriere, Principles of weakly contractive maps in Hilbert spaces, new results in operator theory, In: Gohberg, I, Lyubich, Yu (eds.) Advances and Appl., vol. 98, pp. 7{22. Birkhauser Verlag, Basel (1997).##M. Beygmohammadi and A. Razani, Two xed-point theorems for mappings satisfying a general contractive condition of integral type in the modular space, International Journal of Mathematics and Mathematical Sciences, Article ID 317107 (2010), 10 pages.##N. Cakic, Z.Kadelbur, S.Radenovic and A. Razani, Common xed point results in cone metric spaces for a family of weakly compatible maps, Advances and Applications in Mathematical Sciences, Vol. 1 Issue 1 (2009), 183-207.##V.V.Chistyakov, Metric Modular Spaces-Theory and Applications, Springer International Publishing Switzerland 2015.##V.V.Chistyakov, Modular metric spaces, I: Basic concepts, Nonlinear Anal., 72 (2010), 1-14.##L.Ciric, A. Razan, S. Radivic and J.S. Ume, Common xed point theorems for families of weakly compatible maps, Computers and Mathematics with Applications, 55 (2008), 2533-2543.##P.N.Dutta, B.S.Choudhury, A Generalisation of Contraction Principle in Metric Spaces, Fixed Point Theory and Applications. Volume 2008, Article ID 406368, 8 pages.##M.B. Ghaemi and A. Razani, Fixed and periodic points in the probabilistic normed and metric spaces, Chaos, Solitons and Fractals, 28 (2006), 1181-1187.##J.Harjani, K.Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary di erential equations, Nonlinear Analysis 72 (2010) 11881197.##M. Jleli, E. Karapinar and B. Samet, Best Proximity Point Result in Modular Spaces with the Fatou Property, Abstract and Applied Analysis, Volume 2013, Article ID 329451, 4 pages http://dx.doi.org/10.1155/2013/329451.##M.A. Khamsi, Quasicontraction Mappings in Modular Spaces without 2Condition, Fixed Point Theory and Applications, Volume 2008, Article ID 916187, 6 pages doi:10.1155/2008/916187.##M.A.Khamsi, W.M.Kozlowski, Fixed Point Theory in Modular Function Spaces, DOI 10.1007/978-3-319-14051-3, Springer International Publishing Switzerland 2015.##C.Mongkolkeha, P.Kumam, Some xed point Results for Generalised Weak Contraction Mappings in Modular Spaces, International Journal of Analysis. Volume 2013, Articlle ID 247378, 6 pages.##M.Ozturk, M.Abbas, E.Girgin, Common xed point results of a pair of generalized contraction mappings in modular spaces, Fixed Point Theory and Applications (2016) 2016:19.##A.Padcharoen, D.Gopal, P.Chaipunya and P.Kumam, Fixed point and periodic point results for type F-contractions in modular metric spaces, Fixed Point Theory and Applications (2016), 2016:39 DOI 10.1186/s13663-016-0525-4.##A. Razani, A xed point theorem in the Menger probabilistic metric space, New Zealand J. Math., 35 (2006), 109-114.##A. Razani, Results in Fixed Point Theory, Andisheh Zarin publisher, Qazvin, August 2010.##A. Razani and M. Shiradaryazdi, Some results on xed points in the fuzzy metric space, J. Appl. Math. Comput., 20 (2006), 401-408.##A. Razani and S. Homaeipour, Viscosity approximation to common xed points of families of nonexpansive mappings with weakly contractive mappings, Fixed Point Theory Applications, Article ID 476913 (2010), 8 pages.##A. Razani and R. Moradi, Common xed point theorems of integral type in modular spaces, Bulletin of the Iranian Mathematical Society, Vol. 35 No. 2 (2009), 11-24.##A. Razani and R. Moradi, Double sequence iterations for a strongly contractive mapping in the modular space, Iranian Journal of Mathematical Sciences and Informatics, 11(2016), No. 2, 119-130.##A. Razani and V. Parvaneh, Some xed point theorems for weakly T-Chatterjea and weakly T-Kannan-contractive mappings in complete metric spaces, Russian Mathematics (Iz. VUZ), 57 (2013), No.3, 38-45.##A. Razani and V. Parvaneh, On generalized weakly G-contractive mappings in partially ordered G-metric spaces, Abstract and Applied Analysis, Article ID 701910 (2012), 18 pages.##A. Razani and M. Samanipour, Common xed point theorems for families of weakly compatible maps in a 2-metric space, Applied Mathematics and Information Sciences, 2(3) (2008), 275-289.##H. R. Sahebi, A. Razani, An explicit viscosity iterative algorithm for nding xed points of two noncommutative nonexpansive mappings, Iranian Journal of Mathematical Sciences and Informatics, 11 (2016), No. 1, 69-83.## ##

On Identities with Additive Mappings in Rings 2 2 begin{abstract} If $F,D:Rto R$ are additive mappings which satisfy $F(x^{n}y^{n})=x^nF(y^{n})+y^nD(x^{n})$ for all $x,yin R$. Then, $F$ is a generalized left derivation with associated Jordan left derivation $D$ on $R$. Similar type of result has been done for the other identity forcing to generalized derivation and at last an example has given in support of the theorems. end{abstract} 125 133 2017/07/262017/01/72017/02/122017/04/92017/04/142017/04/192015/11/302017/04/22017/05/102017/05/92017/06/122017/03/22 1396/1/2 2019/05/92020/03/42019/11/72017/08/142019/05/82018/07/302019/10/202018/03/112017/09/82018/03/62019/05/292017/09/25 1396/7/3 A. Z. ansari A. Z. ansari Department of Mathematics, Faculty of Science, Islamic University of Madinah, K.S.A K.S.A. ansari.abuzaid@gmail.com Prime (Semiprime) ring Additive mappings Generalized (Jordan) left derivations Generalized (Jordan) derivations (Jordan)Centralizers. S. Ali, On generalized left derivations in rings and Banach algebras, Aequat. Math., 81, (2011), 209-226.##A. Z. Ansari, F. Shujat, Additive mappings satisfying algebraic conditions in rings Rendiconti del Circolo Matematico di Palermo, 63(2), (2014), 211-219.##M. Ashraf, S. Ali, On generalized Jordan left derivations in rings, Bull. Korean Math. Soc. 45(2), (2008), 253-261.##M. Ashraf, N. Rehman, and A. Z. Ansari, An additive mapping satisfying an algebraic condition in rings with identity, Journal of Advanced Research in Pure Mathematics, 5(2), (2013), 38-45.##B. Dhara, R. K. Sharma, On addtive mappings in rings with identity elements, Interenational Mathematical Forum, 4(15), 2009, 727-732.##I. N. Herstein, Derivations in prime rings, Proc. Amer. Math. Soc., 8, (1957), 1104-1110.##I. N. Herstein, Topics in ring theory, Univ. Chicago Press, Chicago, 1969.##B. E. Johnson, and A.M. Sinklair, Continuity of derivations and a problem of Kaplansky, Amer. J. Math., 90, (1968), 1068-1073.##E. C. Posner, Derivations in prime rings, Proc. amer. Math. Soc., (1957), 1093-1100.##I. M. Singer, and J. Wermer, Derivations on commutative normes spaces, Math. Ann., 129, (1995) 435-460.##M.P. Thomos, The image of a derivation is contained in the radical, Annals of Math., 128, (1988), 435-460.##Vukman, J. Jordan left derivations on semiprime rings, Math. J. Okayama Univ., 39, 1-6 (1997).##Vukman, J. On left Jordan derivations on rings and Banach algebras, Aequationes Math, 75, (2008), 260-266.##Zalar, B. On centralizers of semiprime rings, Comment. Math. Univ. carolin., 32(4), (1991) 609-614.## ##

The Dynamical Analysis of a Delayed Prey-Predator Model with a Refuge-Stage Structure Prey Population 2 2 A mathematical model describing the dynamics of a delayed stage structure prey - predator system with prey refuge is considered. The existence, uniqueness and bounded- ness of the solution are discussed. All the feasibl e equilibrium points are determined. The stability analysis of them are investigated. By employ ing the time delay as the bifurcation parameter, we observed the existence of Hopf bifurcation at the positive equilibrium. The stability and direction of the Hopf bifurcation are determined by utilizing the normal form method and the center manifold reduction. Numerical simulations are given to support the analytic results. 135 159 2017/07/262017/01/72017/02/122017/04/92017/04/142017/04/192015/11/302017/04/22017/05/102017/05/92017/06/122017/03/222017/07/14 1396/4/23 2019/05/92020/03/42019/11/72017/08/142019/05/82018/07/302019/10/202018/03/112017/09/82018/03/62019/05/292017/09/252018/07/7 1397/4/16 R. Naji R. Naji Baghdad university Iraq rknaji@gmail.com S. Majeed S. Majeed Thi-Qar University Iraq sm.salammajeed@yahoo.com Delayed Prey - Predator System Stage- Structure Refuge Stability Hop f Bifurcation. H.I. Freedman, P. Waltman,, Persistence in models of three interacting predator prey populations, Math. Biosci., 68, (1984), 213-231.##Y. Xiao, L. Chen, Modelling and analysis of predator- prey with disease in the prey,Mathematical Biosciences, 171(1), (2001), 59-82.##B. Dubey, R.K. Upadhyay, Persistence and Extinction of One-Prey and Two-Predators System, Nonlinear Analysis: Modelling and Control, 9(4),307-329. (2004).##K.P. Das, A Mathematical Study of a predator-prey Dynamics with Disease in Predator, International Scholarly Research Network ISRN Applied Mathematics, (2011), 669-684.##S. Sharma, G. P. Samanta, A ratio-dependent predator-prey model with Allee eect and disease in prey, J. Appl. Math. Comput., 47, (2015), 345-364.##R. K. Naji and R. M. Hussien, The Dynamics of Epidemic Model with Two Types of Infectious Diseases and Vertical Transmission, Journal of applied mathematics, 2016, (2016).##A. J. Lotka, Elements of physical biology, Williams & Wilkins, Princeton, N. J., 1925.##V. Volterra, Variazioni e uttuazioni del numero dindividui in specie conviventi, Mem.Accad. Lincei Roma, 2,(1926), 31-113.##W. Wang, L. Chen, A predator-prey system with stage-structure for predator, Computers Math., Applic., 33(8), 83-91, (1997).##J. A. Cui, Y. Takeuchi, A predator-prey system with a stage structure for the prey, Mathematical and Computer Modelling, 11, (2006), 26-1132.##X. Song, L. Chen, Optimal harvesting and stability for a two-species competitive system with stage structure, Mathematical Biosciences, 170, (2001), 173-186.##T.K. Kar, U.K. Pahari, Modelling and analysis of a prey-predator system with stage structure and harvesting, Nonlinear Analysis: Real World Applications, 8, (2007), 601609.##R. Shi and L. Chen, The study of a ratio-dependent predator-prey model with stage structure in the prey, Nonlinear Dyn, 58, (2009), 443-451.##S. Xu., Dynamics of a general prey-predator model with prey-stage structure and diusive effects, Computers & Mathematics with Applications, 68(3),(2014), 405-423.##P. Hao, J. Wei and D. Fan, Analysis of dynamics in an eco-epidemiological model with stage structure, Advances in Difference Equations 2016(1),(2016).##Raid K. Naji and Salam J. Majeed, The Dynamical Analysis of a Prey-Predator Model with a Refuge-Stage Structure Prey Population, International Journal of Differential Equations, 2016, (2016).##W.G. Aiello and H. I. Freedman,A time-delay model of single-species growth with stage structure, Mathematical Biosciences, 101, (1990), 139-153.##Y. Kuang, Delay differential equations with applications in population dynamics, Boston: Academic Press , 1993.##K. Ye, X. Song, Predator-prey system with stage structure and delay, Appl. Math. J. Chinese Univ. Ser. B, 18(2), (2003), 143-150.##M. Bandyopadhyaya and S. Banerjee,A stage-structured prey-predator model with discrete time delay, Applied Mathematics and Computation, 182,1385-1398.(2006).##Y. Chen and S. Changming, Stability and Hopf bifurcation analysis in a prey-predator system with stage-structure for prey and time delay, Chaos, Solitons and Fractals, 38,1104-1114.(2008).##L. Wang, R. Xu, G. Feng, A stage-structured prey-predator system with time delay, J. Appl. Math. Comput., 33, (2010), 267-281.##H. L. Smith, An introduction to delay di erential equations with applications to the life sciences, Texts in Applied Mathematics. New York: Springer, 2011.##X. K. Sun, H. F, Huo and X. B. Zhang, A Predator-Prey Model with Functional Response and Stage Structure for Prey, Abstract and Applied Analysis, 2012, (2012).##Q. Gao and Z. Jin, A Time Delay Predator-Prey System with Three-Stage-Structure, The Scienti c World Journal,2014, (2014).##J.F.M. Al-Omari, A stage-structured predator-prey model with distributed maturation delay and harvesting, journal of biological dynamics, 9(1),278-287.(2015).##B. Hassard, N. Kazarino and Y. H. Wan, Theory and applications of Hopf bifurcation. London mathematical society lecture note series, Cambridge University Press, 41.(1981).## ##

ABSTRACTS IN PERSIAN Vol.15, No.1 2 2 Please see the full text contains the Pesian abstracts for this volume. 161 174 2017/07/262017/01/72017/02/122017/04/92017/04/142017/04/192015/11/302017/04/22017/05/102017/05/92017/06/122017/03/222017/07/142020/08/21 1399/5/31 2019/05/92020/03/42019/11/72017/08/142019/05/82018/07/302019/10/202018/03/112017/09/82018/03/62019/05/292017/09/252018/07/72020/08/21 1399/5/31 IJMSI IJMSI IJMSI IJMSI Academic Center for Education, Culture and Research (ACECR) Tarbiat Modares University (TMU) Iran ABSTRACTS PERSIAN Vol. 15 No. 1 ## ##