Deterministic Fuzzy Automaton on Subclasses of Fuzzy Regular ω-Languages 2 2 In formal language theory, we are mainly interested in the natural language computational aspects of ω-languages. Therefore in this respect it is convenient to consider fuzzy ω-languages. In this paper, we introduce two subclasses of fuzzy regular ω-languages called fuzzy n-local ω-languages and Buchi fuzzy n-local ω-languages, and give some closure properties for those subclasses. We define a deterministic fuzzy automaton acceptance conditions on fuzzy ω-languages and fuzzy n-local automaton. The relationship between deterministic fuzzy n-local automaton and two subclasses of fuzzy regular ω-languages are established and proved that every fuzzy ω-language accepted by a deterministic fuzzy automaton in 2-mode is a projection of a Buchi fuzzy 2-local ω-language. 1 11 2014/03/28 1393/1/8 2018/09/3 1397/6/12 R. Arulprakasam R. Arulprakasam SRM University India r.arulprakasam@yahoo.com V. R. Dare V. R. Dare Madras Christian College, India rajkumardare@yahoo.com S. Gnanasekara S. Gnanasekara Periyar Arts College, Cuddalore-607 001, Tamil nadu, India India sargunam.g.sekaran@gmail.com Fuzzy set Local ω-language Deterministic fuzzy automaton Fuzzy regular ω-languages. R. Arulprakasam, V.R. Dare, S. Gnanasekaran, Fuzzy local ω-systems, Journal of Discrete Algorithms, 23, (2013), 157-164.##M.P. B´eal, Codes circulaires, automates locaux et entropie, Theoretical Compute Science,57, (1988) 283-302.##J. Berstel and J.-E. Pin, Local languages and the Berry-Sethi algorithm, Theoretical Computer Science, 155, (1996), 439-446.##P. Caron, Families of locally testable languages, Theoretical Computer Science, 242,(2000), 361-376.##M. Ciric, J. Ignjatovic, N. Damljanovc, M. Basic, Bisimulations for fuzzy automata, Fuzzy Sets and Systems, 186(1), (2012), 100-139.##S. Gnanasekaran, Fuzzy local languages, International Mathematical Forum, 5(44),(2010), 2149-2155.##J. Ignjatovic, M.Ciric, Formal power series and regular operations on fuzzy languages,Information Sciences, 180(7), (2010), 1104-1120.##Z. Jancic, M. Ciric, Brzozowski type determinization for fuzzy automata, Fuzzy Sets and Systems, 249, (2014), 73-82.##J. Jin, Q. Li, Y. Li, Algebraic properties of L-fuzzy finite automata, Information Sciences,234, (2013), 182-202.##Kamala Krithivasan, K. Sharda, Fuzzy ω-automata, Information Sciences, 138(1),(2001), 257-281.##Y. Li, D. Manfred, L. Lei, Model checking of linear-time properties in multi-valued systems, Information Sciences, 377, (2017), 51-74.##Y. Li, Q. Wang, The universal fuzzy automaton, Fuzzy Sets and Systems, 249, (2014), 27-48.##D.S. Malik, John N. Mordeson, On Fuzzy regular languages, Information Sciences, 88(1-4), (1996), 263-273.##J.N. Mordeson and D.S. Malik, Fuzzy automata and languages, Chapman and Hall,CRC, 2002.##D. Perrin and J.-E. Pin, Infinite Words, Automata, Pure and Applied Mathematics, Elsevier, 141, 2004.##S.P. Tiwari, Anupam K. Singh, Shambhu Sharan and Viajy K. Yadav, Bifuzzy core of fuzzy automata, Iranian Journal of Fuzzy Systems, 12(2), (2015), 63-73.##S.P. Tiwari and Anupam Kumar Singh, On minimal realization of fuzzy behaviour and associated categories, Journal of Applied Mathematics and Computing, 45(1-2), (2014), 223-234.##W.G. Wee and K.S. Fu, A Formation of Fuzzy Automata and its application as a model of learning system, IEEE Transactions on In Systems Science and Cybernetics, 5(3),(1969), 215-223.##X.Wei, Y Li, Fuzzy alternating automata over distributive lattices, Information Sciences,425, (2018), 34-47.##X.Wei, Y Li, Fuzzy alternating Buchi automata over distributive lattices, International Journal of Approximate Reasoning, 90, (2017), 144-162.##L.A. Zadeh, Fuzzy sets, Information and Control, 8, (1965), 338-353.## ##

Best Coapproximation in Quotient Spaces 2 2 As a counterpart to best approximation, a new kind of approximation, called best coapproximation was introduced in normed linear spaces by C. Franchetti and M. Furi. In this paper, we use this coapproximation to prove some results on the existence and uniqueness of best coapproximation in quotient spaces when the underlying spaces are metric linear spaces. We shall also see how coproximinality can be transmitted to and from quotient spaces. 13 20 2014/03/282014/11/26 1393/9/5 2018/09/32018/10/2 1397/7/10 S. Gupta S. Gupta Guru Nanak Dev University India sahilmath@yahoo.in T. Narang T. Narang Guru Nanak Dev University India tdnarang1948@yahoo.co.in Best coapproximation co-Proximinal set co-Chebyshev set Boundedly compact set Pseudo co-Chebyshev set. G. Albinus: Approximation in Metric Linear Spaces, Approximation Theory, Banach Center Publications, Vol. 4, PWN- Polish Scientific Publishers, Warsaw, 1979.##E.W. Cheney, D.E. Wulbert, The Existence and Uniqueness of Best Approximation, Math. Scand., 24, (1969), 113-140.##C. Franchetti, M. Furi: Some Characteristic Properties of Real Hilbert Spaces, Rev. Roumaine Math. Pures Appl., 17(1972), 1045-1048.##G. G. Lorentz: Operations in Linear Metric Spaces, Duke Math. J., 15, (1948), 755-761.##H. Mazheri, S.M.S. Modarres: Some Results Concerning Proximinality and co-Proximinality, Nonlinear Anal., 62, (2005), 1123-1126.##H. Mazheri: Best Coapproximation in Quotient Spaces, Nonlinear Anal., 68, (2008),3122-3126.##T.D. Narang: On Some Approximation Problems in Metric Linear Spaces, Indian J. Pure Appl. Math., 14, (1983), 253-256.##T.D. Narang: Best Coapproximation in Metric Spaces, Publications de lInstitut Mathematique, 51, (1992), 71-76.##T.D. Narang, S.P. Singh, Best Coapproximation in Metric Linear Spaces, Tamkang J. Math., 30, (1999), 241-252.##T.D. Narang, S. Gupta, On Best Approximation and Best Coapproximation, Thai J. Math., 14, (2016), 505-516.##G. Pantelidis, Approximationstheorie fur Metrische Lineare Rame, Math. Ann., 184,(1969), 30-48.##P.L. Papini, I. Singer, Best Coapproximation in Normed Linear Spaces, Mh. Math., 88, (1979), 27-44.##G.S. Rao, R. Sarvanan, Some Results Concerning Uniform Best Coapproximation, J. Inequal. Pure Appl. Math., 3, (2002),1-13.##S. Rolewicz, Metric Linear Spaces, Monografie Matematyczne 56 (Warszawa 1972) ##W. Rudin, Functional Analysis, McGraw-Hill, Inc., 1973.##K. Schnatz, Nonlinear Duality and Best Approximation Metric Linear Spaces J. Approx.Theory, 49, (1987), 201-218.##I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Springer-Verlag, New York, 1970.##I. Singer, The Theory of Best Approximation and Functional Analysis, SIAM Philadelphia, Pennslvania, 1974.## A. I. Vasilev, Bounded Compactness of Sets in Linear Metric Spaces, Math. Notes, 11,(1972), 396-401.## ##

Isotropic Constant Dimension Subspace Codes 2 2 In network code setting, a constant dimension code is a set of k-dimensional subspaces of F nq . If F_q n is a nondegenerated symlectic vector space with bilinear form f, an isotropic subspace U of F^n_q is a subspace that for all x, y ∈ U, f(x, y) = 0. We introduce isotropic subspace codes simply as a set of isotropic subspaces and show how the isotropic property use in decoding process, then we show that for suitable parameters there are isotropic spread codes. 21 34 2014/03/282014/11/262016/02/6 1394/11/17 2018/09/32018/10/22017/01/25 1395/11/6 F. Bardestani F. Bardestani Mathematics and Informatics Research Group, ACECR at Tarbiat Modares University Iran fatemh.bardestani@gmail.com S. R. Adhami S. R. Adhami Faculty of Mathematical Sciences, Tarbiat Modares University Iran Isotropic subspace Constant dimension subspace code Spread Codes. S. Ball, J. Bamberg, M. Lavrauw, T. Penttila, Symplectic spreads, Designs, Codes and Cryptography, 32(1), 9–14, (2004).##R. W. Carter, Simple Groups of Lie Type, John Wiley and sons, London, (1972).##R. H. Dye, Partitions and their stabilizers for line complexes and quadrics, Annali diMatematica pura ed applicata, 114(1) 173–194, (1977).##T. Etzion, Problems on q-analogs in coding theory, arXiv:13056126 [cs.IT], (2013).##T. Etzion, A. Vardy, Error-correcting codes in projective geometry, IEEE Trans. Inform. Theory, IT-57, 1165–1173, (2011).##The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.7.8; 2015. (http://www.gap-system.org)##H. Gluesing-Luerssen, K. Morrison, C. Troha, Cyclic orbit codes and stabilizer subfields,arXiv:1403.1218, (2014).##N. L. Johnson, O. Vega, Symplectic spreads and symplectically paired spreads, Note di Matematica, 26(2), 119–134, (2006).##R. Kotter , F.R Kschischang, Coding for errors and erasures in random network coding,IEEE Trans. Inf. Theor., 54(8), 3579–3591, (2008).##W. J. Martin, X. J. Zhu, Anticodes for the Grassman and bilinear forms graphs, Designs,Codes and Cryptography, 6, 73–79 (1995).##F. Manganiello, A.-L. Trautmann, J. Rosenthal, On conjugacy classes of subgroups of the general linear group and cyclic orbit codes, In Proceedings of the 2011 IEEE International Symposium on Information Theory, 31(5) 1916–1920, (2011).##J. Rosenthal, A.-L. Trautman, A complete characterization of irreducible cyclic orbit codes and their Plucker embedding, Des. Codes Cryptogr., 275–289, (2013).##M. Schwartz, T. Etzion, Codes and anticodes in the Grassman graph, J. Combinatorial Theory, Series A, 97, 27-42, (2002).##A. L. Trautmann, F. Manganiello, M. Braun, J. Rosenthal, Cyclic orbit codes, IEEE Trans. Inf. Theor., IT-59, 7386-7404, (2013).## ##

Domination and Signed Domination Number of Cayley Graphs 2 2 In this paper, we investigate domination number as well as signed domination numbers of Cay(G : S) for all cyclic group G of order n, where n in {p^m; pq} and S = { a^i : i in B(1; n)}. We also introduce some families of connected regular graphs gamma such that gamma_S(Gamma) in {2,3,4,5 }. 35 42 2014/03/282014/11/262016/02/62016/04/20 1395/2/1 2018/09/32018/10/22017/01/252017/01/18 1395/10/29 F. Ramezani F. Ramezani Imam Khomeini International University Iran f.ramezani988@gmail.com E. Vatandoost E. Vatandoost Imam Khomeini International University Iran vatandoost@sci.ikiu.ac.ir Cayley graph cyclic group Domination number Signed domination number. S. Alikhani, On the Domination Polynomials of non P4-Free Graphs, Iranian Journal of Mathematical Sciences and Informatics, 8(2), ( 2013), 49–55.##J. E. Dunbar, S. T. Hedetniemi, M. A. Henning, P. J. Slater, Signed Domination in Graphs, Graph Theory, Combinatorics, and Applications, 1, (1995), 311-322.##O. Favaron, Signed Domination in Regular Graphs. Discrete Mathematics, 158(1), (1996), 287-293.##R. Haas, T. B. Wexler, Bounds on the Signed Domination Number of a Graph, Electron. Notes Discrete Math., 11, (2002), 742–750.##R. Haas, T. B. Wexler, Signed Domination Numbers of a Graph and its Complement, Discrete mathematics, 283(1), (2004), 87–92.##T. W. Haynes, S. Hedetniemi, P. Slater, Fundamentals of Domination in Graphs, CRC Press; 1998 Jan 5.##M. A. Henning, P. J. Slater, Inequalities Relating Domination Parameters in Cubic Graphs, Discrete Mathematics. 158(1), (1996), 87–98.##S. Klavzar, G. Kosmrlj, S. Schmidt, On the Computational Complexity of the Domination Game, Iranian Journal of Mathematical Sciences and Informatics, 10( 2), (2015),115–122.##A. Meir, J. Moon, Relations Between Packing and Covering Numbers of a Tree, Pacific Journal of Mathematics. 61(1), (1975) , 225–233.##P. Pavlic, J. Zerovnik, A Note on the Domination Number of the Cartesian Product s of Paths and Cycles, Kragujevac Journal of Mathematics,37(2), (2013), 275–285.##L. Volkmann, B. Zelinka, Signed Domatic Number of a Graph, Discrete applied mathematics, 150(1), (2005), 261–267.##B. Zelinka, Some Remarks on Domination in Cubic Graphs, Discrete Mathematics,158(1), (1996) , 249–255.##B. Zelinka, Signed and Minus Domination in Bipartite Graphs, Czechoslovak Mathematical Journal, 56(2), (2006), 587-590.## ##

A Modified Degenerate Kernel Method for the System of Fredholm Integral Equations of the Second Kind 2 2 In this paper, the system of Fredholm integral equations of the second kind is investigated by using a modified degenerate kernel method (MDKM). To construct a MDKM the source function is approximated by the same way of producing degenerate kernel. The interpolation is used to make the needed approximations. Lagrange polynomials are adopted for the interpolation. The equivalency of proposed method and Lagrange-collocation method is shown. The error and convergence of the algorithm are given strictly. The efficiency of the approach will be shown by applying the procedure on some prototype examples. 43 53 2014/03/282014/11/262016/02/62016/04/202016/04/24 1395/2/5 2018/09/32018/10/22017/01/252017/01/182017/02/20 1395/12/2 A. Molabahrami A. Molabahrami Ilam University Iran a.molabahrami@ilam.ac.ir A System of Fredholm Integral Equations of the Second Kind Degenerate Kernel Method A Modified Degenerate Kernel Method Lagrange Interpolation Method Lagrange-Collocation Method. K. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, UK, 1997.##K. Atkinson, W. Han, Theoretical Numerical Analysis (A Functional Analysis Framework), Springer-Verlag, New York, 2009, Third Edition.##A. Hashemi Borzabadi, M. Heidari, A Successive Numerical Scheme for Some Classes of Volterra-Fredholm Integral Equations, Iranian Journal of Mathematical Sciences and Informatics, 10, (2015), 1-10.##E. Hetmaniok, D. Slota, T. Trawinski and R. Witula, Usage of the Homotopy Analysis Method for Solving the Nonlinear and Linear Integral Equations of the Second Kind, Numer. Algor. 67, (2014), 163-185.##S. Hosseini, S. Shahmorad, F. Talati, A matrix based method for two dimensional nonlinear Volterra-Fredholm integral equations. Numer. Algor., 68, (2015), 511-529.##H. Jafari, M. Alipour, M. Ghorbani, T-Stability Approach to the Homotopy Perturbation Method for Solving Fredholm Integral Equations, Iranian Journal of Mathematical Sciences and Informatics, 8, (2013), 49-58.##K. Maleknejad, M. Shahrezaee, H. Khatami, Numerical solution of integral equations system of the second kind by Block-Pulse functions, Applied Mathematics and Computation, 166, (2005), 15-24.##K. Maleknejad, N. Aghazadeh, M. Rabbani, Numerical solution of second kind Fredholm integral equations system by using a Taylor-series expansion method, Applied Mathematics and Computation, 175, (2006), 1229-1234.##K. Maleknejad, F. Mirzaee, Numerical solution of linear Fredholm integral equations system by rationalized Haar functions method, Int. J. Comput. Math., 80 (11), (2003), 1397-1405.##A. Molabahrami, An algorithm based on the regularization and integral mean value methods for the Fredholm integral equation of the first kind, Appl. Math. Modelling, 37, (2013), 9634-9642.##A. Molabahrami, Direct computation method for solving a general nonlinear Fredholm integro-differential equation under the mixed conditions: Degenerate and nondegenerate kernels, Journal of Computational and Applied Mathematics, 282, (2015),34-43.##A. Molabahrami, The relationship of degenerate kernel and projection methods on Fredholm integral equations of the second kind, Commun. Numer. Anal. 1, (2017), 1-6.##J. Rashidinia, M. Zarebnia, Convergence of approximate solution of system of Fredholm integral equations, J. Math. Anal. Appl., 333, (2007), 1216-1227.##A. Shidfar, A. Molabahrami, Solving a system of integral equations by an analytic method, Math. Comput. Model., 54 (2011) 828-835.##A. Tari, M. Rahimi, S. Shahmorad, F. Talati, Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the differential transform method. J. Comput. Appl. Math., 228, (2009), 70-76.##A.-M. Wazwaz, Linear and Nonlinear Integral Equations: Methods and Applications,Higher Education Press, Beijin, 2011.## ##

z_R-Ideals and z^0_R-Ideals in Subrings of R^X 2 2 Let X be a topological space and R be a subring of RX. By determining some special topologies on X associated with the subring R, characterizations of maximal fixxed and maximal growing ideals in R of the form Mx(R) are given. Moreover, the classes of zR-ideals and z0R-ideals are introduced in R which are topological generalizations of z-ideals and z0-ideals of C(X), respectively. Various characterizations of these ideals are established, also, coincidence of zR-ideals with z-ideals and zR-ideals with z-ideals in R are investigated. It turns out that some fundamental statements in the context of C(X) are extended to the subrings of RX 55 67 2014/03/282014/11/262016/02/62016/04/202016/04/242016/05/12 1395/2/23 2018/09/32018/10/22017/01/252017/01/182017/02/202017/01/16 1395/10/27 A. Rezaei Aliabad A. Rezaei Aliabad Shahid Chamran University of Ahvaz Iran M. parsinia M. parsinia Shahid Chamran University of Ahvaz Iran parsiniaemhdi@gmail.com Z(R)-topology Coz(R)-topology Growing ideal z_R- ideal z^0_R-ideal Invertible subring. A.R. Aliabad, M. Parsinia, Remarks on Subrings of C(X) of the Form I+C^{∗}(X), Quaest. Math., 40(1), (2017), 63-73.##F. Azarpanah, O.A.S. Karamzadeh, A. Rezaei Aliabad, On Ideals Consisting Entirely of Zerodivisor, Commun. Algebra, 28(2), (2000), 1061-1073.##F. Azarpanah, O.A.S. Karamzadeh, A. Rezaei Aliabad, On z^{◦}-Ideals in C(X), Fund. Math., 160, (1999), 15-25.##F. Azarpanah, M. Karavan, On Nonregular Ideals and z^{◦}-Ideals in C(X), Cech. Math., 55, (130), (2005), 397-407.##F. Azarpanah, R. Mohamadian, radical{z}-Ideals and radical{z}^◦-Ideals in C(X), Acta. Math. Sinica, English Series, 23 (2007), 989-1006.##F. Azarpanah, M. Parsinia, On the Sum of z-Ideals in Subring of C(X), J. Commut. Algebra, to appear.##H.L. Byun, S. Watson, Prime and Maximal Ideals in Subrings of C(X), Topology. Appl., 40, (1991), 45-62.##J.M. Dominguez, J. Gomez Perez, M.A. Mulero, Intermediate Algebras between C^{∗}(X) and C(X) as Rings of Fractions of C^{∗}(X), Topology Appl., 77, (1997), 115-130.##M. Ghadermazi, O.A.S. Karamzadeh, M. Namdari, On the Functionally Countable Subalgebra of C(X), Rend. Semin. Math.Univ. Padova, 129, (2013), 47-69.##L. Gillman, M. Jerison, Rings of Continuous Functions, Springer-Verlag, New York., 1978.##M. Parsinia, Remarks on LBI-Subalgebras of C(X), Comment. Math. Univ. Carolin., 57, (2016), 261-270.##D.Plank, On a Class of Subalgebras of C(X) with Application to βX X, Fund. Math., 64, (1969), 41-54.##D. Rudd, On Structure Spaces of Ideals in Rings of Continuous Functions, Trans. Amer. Math. Soc., 190, (1974), 393-403.##A.B. Sidlovskij, N. Koblitz., Transcendental Numbers, Walter de Gruyter. Berlin. New York, 1989.## ##

On the Lie-Santilli Admissibility 2 2 The largest class of hyperstructures is the one which satisfies the weak properties. We connect the theory of P-hopes, a large class of hyperoperations, with the Lie-Santilli admissibility used in Hardonic Mechanics. This can be achieved by a kind of Ree, sandwich hyperoperation. 69 79 2014/03/282014/11/262016/02/62016/04/202016/04/242016/05/122016/06/6 1395/3/17 2018/09/32018/10/22017/01/252017/01/182017/02/202017/01/162017/11/22 1396/9/1 R. Mahjoob R. Mahjoob Department of Mathematics, Semnan University, Semnan, Iran Iran mahjoob@profs.semnan.ac.ir T. Vougioklis T. Vougioklis Democritus University of Thrace, School of Education,681 00 Alexandroupolis, Greece Greece tvougiou@eled.duth.gr H_v-structures Hopes. B. Davvaz, Groups in Polygroups, Iranian Journal of Mathematical Science and Informatics, 1(1), (2006), 25-31.##B. Davvaz,Polygroup Theory and Related Systems, World Scientific, 2013.##B. Davvaz, R. M. Santilli, T. Vougiouklis, Multi-valued hypermathematics for characterization of matter and antimatter systems, J. Computational Methods Sc. Eng., 13, (2013), 37-50.##B. Davvaz, R. M. Santilli, T. Vougiouklis, Mathematical prediction of Ying’s twin universes, American Journal of Modern Physics, 4(3), (2015), 5-9.##A. Dramalidis, R. Mahjoob, T. Vougiouklis, p-Hopes on non-Square Matrices for LieSantilli Admissibility, Clifford analysis, Clifford algebras and their applications, 4(3), (2015), 361-372.##R. Mahjoob, The e-theta Hopes, Iranian Journal of Mathematical Science and Informatics, 13(1), (2018), 39-50.##R. Mahjoob, T. Vougiouklis, Applications of the Uniting Elements Method, J. of pure and applied math., 36, (2016), 23-34.##P. Nikolaidou, T. Vougiouklis, The Lie-Santilli admissible hyperalgebras of type An, Ratio Mathematica, 26, (2014), 113-128.##R. M. Santilli, Hadronic Mathematics, Mechanics and Chemistry, Vol. I, II, III, IV and V, Int. Academic Press, USA, 2008.##R. M. Santilli, T. Vougiouklis, Isotopies, Genotopies, Hyperstructures and their Appl., New Frontiers Hyperstructures Related Algebras, Hadronic, 1996, 177-188##R.M. Santilli, T. Vougiouklis, Lie-admissible hyperalgebras, Italian J. Pure Applied Mathematics, 31, (2013), 239-254.##T. Vougiouklis, Generalization of P-hypergroups, Rendiconti Circolo Matematico di Palermo, S.II, 36, (1987), 114-121.##T. Vougiouklis, The Fundamental Relation in Hyperrings. The General Hyperfield, 4th AHA Congress, World Scientific, 1991, 203-211.##T. Vougiouklis, Hyperstructures and their Representations, Monographs Math., Hadronic Press, 1994.##T. Vougiouklis, Some Remarks on Hyperstructures, Contemporary Mathematics, Amer. Math. Society, 184, (1995), 427-431.##T. Vougiouklis, On Hv-rings and Hv-representations, Discrete Math., 208/209, (1999), 615-620.##T. Vougiouklis, On a Matrix Hyperalgebra, Journal of Basic Science, 3, (2006), 43-47. ##T. Vougiouklis, ∂-operations and Hv-fields, Acta Math. Sinica, (Engl. Ser.), 24(7), (2008), 1067-1078.##T. Vougiouklis, The Lie-Hyperalgebras and their Fundamental Relations, Southeast Asian Bull. Maths., 37(4), (2013), 601-614.## ##

Computation of Minimum Hamming Weight for Linear Codes 2 2 In this paper, we consider the minimum Hamming weight for linear codes over special finite quasi-Frobenius rings. Furthermore, we obtain minimal free $R$-submodules of a finite quasi-Frobenius ring $R$ which contain a linear code and derive the relation between their minimum Hamming weights. Finally, we suggest an algorithm that computes this weight using the Grobner basis and we show that under certain conditions a linear code takes the maximum of minimum Hamming weight. 81 93 2014/03/282014/11/262016/02/62016/04/202016/04/242016/05/122016/06/62016/08/13 1395/5/23 2018/09/32018/10/22017/01/252017/01/182017/02/202017/01/162017/11/222018/01/17 1396/10/27 E. Rostami E. Rostami Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman Iran e_rostami@uk.ac.ir R. Nekooei R. Nekooei Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman Iran Algebraic coding theory Linear codes Quasi-Frobenius rings Grobner basis SPAP-rings. W. Adams, P. Loustaunau, An introduction to Grobner basis, Graduate Studies in Math. 3, Amer. Math. Soc, Providence, 1994.## D. D. Anderson, M. Bataineh, Generalization of Prime Ideals, Comm. in Algebra, 36, 686- 696, 2008.##I. Constantinescu, W. Heise, T. Honold, Monomial extensions of isometries between codes over $Z_m$, Proceedings of the 5th International Workshop on Algebraic and Combinatorial Coding Theory (ACCT 96), Unicorn Shumen, 98-104, 1998.##D. Cox, J. Little, D. O'Shea, Ideals, varieties, and algorithms, Springer, New York, 1992.## M. Greferath, S. Schmidt, Finite-ring combinatorics and MacWilliams equivalence theorem,. J. of Combin. Theory, Ser. A. 92, 17-28, 2000.##J. H. Griesmer, A bound for error-correcting codes, IBM J. Res. Dev. 4, 532-542, 1960.##J. A. Hammons, P. Kumar, A. Calderbank, N. Sloane,P. Sol The $Z_4$ linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory 40, 301-319, 1994.##T. Honold, I. Landjev, Linear codes over finite chain rings. Electronic J. of Combinatorics, $7:R11$, 2000.##T. Honold, I. Landjev, Projective Hjelmslev geometries, Proceedings of the Second International Workshop on Optimal Codes, 97 -115, 1998.##H. Horimoto, K. Shiromoto, A Singleton bound for linear codes over quasi-Frobenius rings, Proceedings of the 13th AAECC Symposium On Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, Hawaii, 51 -52, 1999.## H. Horimoto, K. Shiromoto, MDS codes over finite quasi-Frobenius rings, preprint.##M. Kreuzer, L. Robbiano, Computational Commutative Algebra 1 and 2, Springer-Verlag, 2000.##A. S. Kuzmin, V. L. Kurakin, V. T. Markov, A. V. Mikhalev, A. A. Nechaev, Linear codes and polylinear recurrences over finite rings and modules (Survey), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. Proc. of the 13-th. Int Symp. AAECC-13. LNCS, Springer, 1999.## T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, Vol. 189, Springer, New York, 1998.## F. MacWilliams, N. Sloane, The Theory of Error-Correcting Codes, Amsterdam, The Netherlands: North-Holland, 1997.## G. Nebe, E. Rains, N. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.##A.A. Nechaev, Linear codes and polylinear recurrences over finite rings and quasi-Frobenius modules, Dokl. Akad. Nauk.(in Russian), 345(1), 229-254, 1995.## A. A. Nechaev, Linear codes over modules and over spaces, MacWilliams identity, Proceedings of the 1996 IEEE Int. Symp. Inf. Theory and Appl., Victoria B. C., Canada, 35-38, 1996.##A. A. Nechaev, The Kerdock code in a cyclic form. Diskret. Mat. 1, 123-139, English translation in Discrete Math. Appl., 1, 365-384, 1991.##A. A. Nechaev, A. S. Kuzmin, V. T. Markov, Linear codes over finite rings and modules, Fundamentalnaja i prikladnaja matematika(in Russian), 2(3), 195-254, 1996. English translation: CNIT of Mosc. State Univ. preprint N 1995-6-1, http://www.math.msu.su/~markov.##R. Nekooei, E. Rostami, On SPAP-rings, Bull. Iranian Math. Soc., 41 , 907-921, 2015.##G. Norton , A.Sualuagean, On the Hamming distance of linear codes over a finite chain ring. IEEE Trans. Inform. Theory, 46, 1060-1067, 2000.##E. Rostami, A Graphical Characterization for SPAP-Rings, Iran. J. Math. Sci. Inform., 13(1), (2018), 67–73.##K. Shiromoto, L. Storme, A Griesmer bound for linear codes over finite quasi-Frobenius rings, Discrete Applied Mathematics, 128,263-274, 2003. ##A. Tehranian, H. R. Maimani, A study of the Total graph, Iran. J. Math. Sci. Inform., 6(2), (2011), 75–80.##J. Wood, Duality for modules over finite rings and applications to coding theory. Amer. J. Math. 121, 555-575, 1999.## ##

Renormalized Solutions for Strongly Nonlinear Elliptic Problems with Lower Order Terms and Measure Data in Orlicz-Sobolev Spaces 2 2 The purpose of this paper is to prove the existence of a renormalized solution of perturbed elliptic problems$ -operatorname{div}Big(a(x,u,nabla u)+Phi(u) Big)+ g(x,u,nabla u) = mumbox{ in }Omega, $ in the framework of Orlicz-Sobolev spaces without any restriction on the $M$ N-function of the Orlicz spaces, where $-operatorname{div}Big(a(x,u,nabla u)Big)$ is a Leray-Lions operator defined from $W^{1}_{0}L_{M}(Omega)$ into its dual, $Phi in C^{0}(mathbb{R},mathbb{R}^{N})$. The function $g(x,u,nabla u)$ is a non linear lower order term with natural growth with respect to $|nabla u|$, satisfying the sign condition and the datum $mu$ is assumed belong to $L^1(Omega)+W^{-1}E_{overline{M}}(Omega)$. 95 119 2014/03/282014/11/262016/02/62016/04/202016/04/242016/05/122016/06/62016/08/132016/07/30 1395/5/9 2018/09/32018/10/22017/01/252017/01/182017/02/202017/01/162017/11/222018/01/172017/08/20 1396/5/29 M. El Moumni M. El Moumni Department of Mathematics, Faculty of Sciences El Jadida, University Chouaib Doukkali Morocoo mostafaelmoumni@gmail.com Elliptic equation Orlicz-Sobolev spaces Renormalized solution. R. Adams, Sobolev Spaces, Ac. Press, New york, 1975.##L. Aharouch, J. Bennouna and A. Touzani, Existence of Renormalized Solution of Some Elliptic Problems in Orlicz Spaces, Rev. Mat. Complut. 22(1), (2009), 91--110.##A. Aissaoui Fqayeh, A. Benkirane, M. El Moumni and A. Youssfi, Existence of renormalized solutions for some strongly nonlinear elliptic equations in Orlicz spaces, Georgian Math. J. 22(3), (2015), 305--321.##P. Benilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierre and J. L. Vazquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22(2), (1995), 240--273.##A. Benkirane and J. Bennouna, Existence of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms in Orlicz spaces, Partial differential equations, Lecture Notes in Pure and Appl. Math., vol. 229, (2002), 125--138. 11, 2 ,(1989), 729--758.##A. Benkirane, A. Elmahi, A Strongly Nonlinear Elliptic Equation Having Natural Growth Terms and L1Data, Nonlinear Analysis, 39, (2000), 403–411.##A. Benkirane, A. Elmahi, Almost Everywhere Convergence of the Gradients of Solutions to Elliptic Equations in Orlicz Spaces and Application, Nonlinear Anal. T. M. A., 11(28), (1997), 1769–1784.##A. Benkirane, A. Elmahi, An Existence Theorem for a Strongly Nonlinear Elliptic Problem in Orlicz Spaces, Nonlinear Anal., 36,(1999), 11–24.##M. F. Betta, A. Mercaldo, F. Murat, M. M. Porzio, Existence and Uniqueness Results for Nonlinear Elliptic Problems with a Lower Order Term and Measure Datum, C. R. Math. Acad. Sci., Paris 334 (9), (2002), 757–762.## M. F. Betta, A. Mercaldo, F. Murat, M. M. Porzio, Existence of Renormalized Solutions to Nonlinear Elliptic Equations with a Lower-order Term and Right-hand Side a Measure, J. Math. Pures Appl., 82(9), (2003), 90–124.##M. F. Betta, O. Guibe, A. Mercaldo, Neumann Problems for Nonlinear Elliptic Equations with L1 Data, J. Diﬀerential Equations 259, (2015), 898–924.## L. Boccardo, D. Giachetti, J.I. Diaz, F. Murat, Existence and Regularity of Renormalized Solutions for Some Elliptic Problems Involving Derivatives of Nonlinear Terms, J.Diﬀerential Equations, 106(2), (1993), 215–237.##G. Dal Maso, F. Murat, L. Orsina, A. Prignet, Renormalized Solutions of Elliptic Equations with General Measure Data, Ann. Scuola Norm. Pisa Cl. Sci., 28(4), (1999), 741–808.## A. Dall Aglio, Approximated Solutions of Equations with L1 Data. Application to the H-convergence of Quasi-Linear Parabolic Equations, Ann. Mat. Pura Appl., 170(4), (1996),207–240.##R. J. DiPerna, P.-L. Lions, On the Cauchy Problem for Boltzmannn Equations: Global Existence and Weak Stability, Ann. Mat., 130, (1989), 321–366.##R.J. DiPerna, P.-L. Lions, Global Existence for the Fokker-Planck-Boltzman Equations,Comm. Pure Appl. Math., 11(2) ,(1989), 729–758.##S. Djebali , O. Kavian, T. Moussaoui, Qualitative Properties and Existence of Solutions for a Generalized Fisher-like Equation, Iranian Journal of Mathematical Sciences and Informatics, 4(2), (2009), 65–81.##T. K. Donaldson, N. S. Trudinger, Orlicz-Sobolev Spaces and Imbedding Theorems, J.Funct. Anal., 8, (1971), 52–75.## J. Gossez, A Strongly Nonlinear Elliptic Problem in Orlicz-Sobolev Spaces, Proc. A.M.S. Sympos. Pure Math., 45, (1986), 455–462.##J. Gossez, Nonlinear Elliptic Boundary Value Problems for Equations with Rapidly (or slowly) Increasing Coeﬃcients, Trans. Amer. Math. Soc., 190, (1974), 163–205.##J. Gossez, Some Approximation Properties in Orlicz-Sobolev Spaces, Studia Math., 74,(1982), 17–24.##J.-P. Gossez, V. Mustonen, Variational Inequality in Orlicz-Sobolev Spaces, Nonlinear Anal. Theory Appl., 11, (1987), 379–392.##O. Guibe, A. Mercaldo, Existence of Renormalized Solutions to Nonlinear Elliptic Equations with two Lower Order Terms and Measure Data, Trans. Amer. Math. Soc., 360(2),(2008), 643–669.##O. Guibe, A. Mercaldo, Existence and Stability Results for Renormalized Solutions to Noncoercive Nonlinear Elliptic Equations with Measure Data, Potential Anal., 25(3),(2006), 223–258.##B. Hazarika, Strongly Almost Ideal Convergent Sequences in a Locally Convex Space Deﬁned by Musielak-Orlicz Function, Iranian Journal of Mathematical Sciences and Informatics, 9(2), (2014), 15–35.##E. Hewitt, K. Stromberg, Real and Abstract Analysis, Springer-Verlag, Berlin Heidelberg, New York, 1965.##M.A. Krasnoselskii,Y.B. Rutickii, Convex Functions and Orlicz Space, Noordhoﬀ, Groningen, 1961.##F. Murat, Soluciones Renormalizadas de EDP Ellipticas no Lineales, Cours a lUniversite de Seville, Mars 1992. Publication 93023 du Laboratoire d Analyse Numerique de l Universite Paris VI, 1993.##F. Murat, Equations Elliptiques non lineaires avec Second Membre L1emeou Mesure, 26 Congres National d Analyse Numerique. Les Karellis, Juin, (1994), 12–24.##J.M. Rakotoson, Uniqueness of Renormalized Solutions in a T -set for the L1-Data and the Link between Various Formulations, Indiana Univ. Math. J., 43(2), (1994), 685–702.##A. Youssﬁ, A. Benkirane, M. El Moumni, Bounded Solutions of Unilateral Problems for Strongly Nonlinear Equations in Orlicz Spaces, E. J. Qualitative Theory of Diﬀ. Equ.,21, (2013), 1–25.## ##

On the 2-Adjointable Operators and Superstability of Them Between 2-Pre Hilbert $C^*$-module Spaces 2 2 In this paper, first, we introduce the new concept of 2-inner product on Banach modules over a $C^*$-algebra. Next, we present the concept of 2-linear operators over a $C^*$-algebra. Our result improve the main result of the paper Z. Lewandowska. In the final of this paper, we define the notions 2-adjointable mappings between 2-pre Hilbert C*-modules and prove supperstability of them in the spirit of Hyers-Ulam-Rassias. 121 126 2014/03/282014/11/262016/02/62016/04/202016/04/242016/05/122016/06/62016/08/132016/07/302016/08/13 1395/5/23 2018/09/32018/10/22017/01/252017/01/182017/02/202017/01/162017/11/222018/01/172017/08/202017/09/27 1396/7/5 M. Ramezani M. Ramezani University of Bojnord Iran mar.ram.math@gmail.com H. Baghani H. Baghani University of Sistan and Baluchestan Iran h.baghani@gmail.com $C^*$-algebra 2-Adjointable mapping Supperstability. Y.J. Cho, P.C.S. Lin, S.S. Kim, A. Misiak, Theory of 2-Inner Product Spaces, Nova Science Publishers, Inc., New York, 2001.##S.S. Dragomir, Y.J. Cho, S.S. Kim, A. Sofo, Some Boas-Bellman Type Inequalitys in 2-Inner Product Space, JIPAM, 6(2),(2005), article 55.##M. Frank, P. Gavruta, M.S. Moslehian, Superstability of Adjointable Mappings on Hilbert C- Modules, Appl. Anal. Discrete Math., No3, (2009), 39-45.##Z. Lewandowska, On 2-Normed Sets, Glasnik Mat., 38(58), (2003), 99-110.##Z. Lewandowska, Bounded 2-Linear Operators on 2-Normed Sets, Glas. Mat. Ser. III, 39(59)(2), (2004), 301-312.##A, Ashyani, H, Mohammadinejad, O, RabieiMotlagh, Stability Analysis of Mathematical Model of Virus Therapy for Cancer, Iranian Journal of Mathematical Sciences and Informatics, 11( 2), (2016), 97-110.##H, Sadeghi, Generalized Approximate Amenability of Direct Sum of Banach Algebras, Iranian Journal of Mathematical Sciences and Informatics, 13(1), (2018), 75-87.##M.E. Gordji, M. Ramezani, Approximate inner products on Hilbert C-modules; A Fixed Point Approach, Operators and Matrices, 6(4), (2012), 757-766.##M.E. Gordji, M. Ramezani, Y.J. Cho, H. Baghani, Approximate Lie brackets: A Fixed Point Approach, Journal of Inequalities and Applications, (2012), doi:10.1186/1029242X-2012-125.## ##

Linear Resolutions of Powers of Generalized Mixed Product Ideals 2 2 Let L be the generalized mixed product ideal induced by a monomial ideal I. In this paper we compute powers of the genearlized mixed product ideals and show that Lk have a linear resolution if and only if Ik have a linear resolution for all k. We also introduce the generalized mixed polymatroidal ideals and prove that powers and monomial localizations of a generalized mixed polymatroidal ideal is again generalized mixed polymatroidal ideal. 127 134 2014/03/282014/11/262016/02/62016/04/202016/04/242016/05/122016/06/62016/08/132016/07/302016/08/132016/08/13 1395/5/23 2018/09/32018/10/22017/01/252017/01/182017/02/202017/01/162017/11/222018/01/172017/08/202017/09/272017/02/6 1395/11/18 A. Tehranian A. Tehranian Department of mathematics, Science and Research branch, Islamic azad university, Tehran Iran Iran tehranian@srbiau.ac.ir R. Moghimipor R. Moghimipor Department of mathematics, Science and Research branch, Islamic azad university, Tehran Iran Iran roya_moghimipour@yahoo.com Free resolutoins Graded betti numbers Monomial ideals. S. Bayati, J. Herzog, Expansion of Monomial Ideals and Multigraded Modules, Rocky Mountain J. Math., to appear.##A. Conca, E. De Negri. M-sequences, Graph Ideals, and Ladder Ideals of Linear Type. J. Algebra, 211, (1999), 599-624.##J. Herzog, T. Hibi, Monomial Ideals. GTM 260, Springer 2010.##J. Herzog, R. Moghimipor, S. Yassemi, Generalized Mixed Product Ideals, Arch. Math,103, (2014), 39-51.##J. Herzog, A. Rauf, M. Vladoiu, The Stable Set of Associated Prime Ideals of a Polymatroidal Ideal, J. Algebraic Combinatorics, DOI 10.1007/s10801-012-0367-z.##T. Hoa, N. Tam, On Some Invariants of a Mixed Product of Ideals, Arch. Math, 94,(2010), 327-337.##C. Ionescu, G. Rinaldo, Some Algebraic Invariants Related to Mixed Product Ideals,Arch. Math, 91, (2008), 20-30.##G. Restuccia, R. Villarreal, On the Normality of Monomial Ideals of Mixed products,Commun. Algebra, 29(2001), 3571-3580.##G. Rinaldo, Betti Numbers of Mixed Product Ideals, Arch. Math, 91, (2008), 416-426.##G. Rinaldo, Sequentially Cohen-Macaulay Mixed Product Ideals, Algebra Colloquium,22, (2015), 223-232.## ##

On Skew Cyclic Codes over a Finite Ring 2 2 In this paper, we classify the skew cyclic codes over Fp + vF_p + v^2F_p, where p is a prime number and v^3 = v. Each skew cyclic code is a F_p+vF_p+v^2F_p-submodule of the (F_p+vF_p+v^2F_p)[x;alpha], where v^3 = v and alpha(v) = -v. Also, we give an explicit forms for the generator of these codes. Moreover, an algorithm of encoding and decoding for these codes is presented. 135 145 2014/03/282014/11/262016/02/62016/04/202016/04/242016/05/122016/06/62016/08/132016/07/302016/08/132016/08/132016/09/4 1395/6/14 2018/09/32018/10/22017/01/252017/01/182017/02/202017/01/162017/11/222018/01/172017/08/202017/09/272017/02/62017/10/8 1396/7/16 H. Mousavi H. Mousavi Department of Mathematics, Tarbiat Modares University, Tehran, Iran. Iran h.moosavi@modares.ac.ir R. Mohammadi R. Mohammadi Department of Mathematics, Faculty of Mathematical Sciences University of Mazandaran, Babolsar, Iran. Iran mohamadi.rasul@yahoo.com S. Rahimi S. Rahimi Department of Information Technology, Imam Hossein University, Tehran, Iran. Iran s.rahimi@sharif.edu Skew Cycilc Codes Skew Polynomial Rings Hamming Distance. T. Abuarub, A. Ghrayeb, N. Aydin, I. Siap, On the Construction of Skew Quasi-cyclic Codes, IEEE Transactions on Information Theory, 56(5), (2010), 2081-2090.##T. Blackford, Negacyclic Codes over Z4 of Even Length, IEEE Transactions on Information Theory, 49(6),(2003), 1417-1424.##A. Bonnecaze and U. Parampalli, Cyclic Codes and Self-dual Codes over F2 + uF2, IEEE Transactions on Information Theory, 45##(4), (1999), 1250-1255.##D. Boucher, W. Geiselmann, F. Ulmer, Skew-cyclic Codes, Applicable Algebra in Engineering,Communication and Computing, 18(4), (2007), 379-389.##D. Boucher, F. Ulmer, A Note on the Dual Codes of Module Skew Codes, Cryptography and Coding Springer Berlin Heidelberg, (2011), 230-243.##D. Boucher, P. Sole, F. Ulmer, Skew Constacyclic Codes over Galois Rings, Advances in Mathematics of Communications, 23(3), (2008), 273292.## A. R. Calderbank and J. S. Neil, Modular and p-adic Cyclic Codes, Designs Codes and Cryptography, 6(1), (1995), 21-35.##Cengellenmis, Yasemin, On the Cyclic Codes over F3 + vF3, International Journal of Algebra, 4(6), (2010), 253-259.##S. T. Dougherty and H. P Young, On modular cyclic codes, Finite Fields and Their Applications, 13(1), (2007), 31-57.##J. Gao, Skew Cyclic Codes over Fp + vFp, J. Appl. Math. Inform, 31(3-4), (2013), 337-342.##J. Gao, Sh. Minjia, W. Tingting, F. Fang-Wei, On Double Cyclic Codes over Z4, Finite Fields and Their Applications 39, (2016), 233-250.##L. Jin, Skew Cyclic Codes over Ring Fp + vFp, Journal of Electronics(China), 31(3),(2014), 228-231.##S. Jitman, L. San, P. Udomkavanich, Skew Constacyclic Codes over Finite Chain Rings, arXiv preprint, 1008.0327(2010).##D. Mandelbaum, An Application of Cyclic Coding to Message Identiﬁcation. IEEE Transactions on Communication Technology, 17(1), (1969), 42–48.##C. Pierre-Louis, C. Christophe, N. Abdelkader, Quasi-cyclic Codes as Codes over Rings of Matrices, Finite Fields and Their Applications, 16(2), (2010), 100-115.##V. S. Pless, Q. Zhong, Cyclic Codes and Quadratic Residue Codes over Z4, IEEE Transactions on Information Theory, 42(5), (1996), 1594-1600.## ##

A Submodule-Based Zero Divisors Graph for Modules 2 2 ‎Let $R$ be commutative ring with identity and $M$ be an $R$-module‎. ‎The zero divisor graph of $M$ is denoted $Gamma{(M)}$‎. ‎In this study‎, ‎we are going to generalize the zero divisor graph $Gamma(M)$ to submodule-based zero divisor graph $Gamma(M‎, ‎N)$ by replacing elements whose product is zero with elements whose product is in some submodules $N$ of $M$‎. ‎The main objective of this paper is to study the interplay of the properties of submodule $N$ and‎ ‎the properties of $Gamma(M‎, ‎N)$‎. 147 157 2014/03/282014/11/262016/02/62016/04/202016/04/242016/05/122016/06/62016/08/132016/07/302016/08/132016/08/132016/09/42016/09/8 1395/6/18 2018/09/32018/10/22017/01/252017/01/182017/02/202017/01/162017/11/222018/01/172017/08/202017/09/272017/02/62017/10/82017/01/12 1395/10/23 S. Babaei S. Babaei Department of Mathematics,‎ Imam Khomeini International University Iran sbabaei@edu.ikiu.ac.ir Sh. Payrovi Sh. Payrovi Imam Khomeini International University Iran shpayrovi@sci.ikiu.ac.ir E. Sengelen Sevim E. Sengelen Sevim ‎Istanbul Bilgi University Turky esra.sengelen@bilgi.edu.tr Zero-divisor graph‎ ‎Submodule-based zero-divisor graph‎ ‎ Semi simple module. D. F. Anderson, R. Levy, J. Shapiro, The Zero-divisor Graphs von Neumann Reguler Rings and Boolean Algebras, J. Pure Appl. Algebra, 180, (2003),221-241.##D. F. Anderson, P. S. Livingstone, The Zero-divisor Graph of a Commutative Ring, J. Algebra, 217, (1999), 434-447.##I. Beck, Coloring of Commutative Rings, J. Algebra, 116, (1988), 208-226.##M. Behboodi, Zero Divisor Graph for Modules over Commutative Rings,J. Commut. Algebra, 4, (2012), 175-197.##B. Bollobas, Graph Theory: An Introuduction Course, Springer, New-York, 1979.##G. Chartrand, Graphs as Mathmatical Models, Prindle, Boston, 1977.##R. Diestel, Graph Theory, Springer, New-York, 1997.##A. Haghany, M. D. Vedadi, Endoprime Modules, Acta Math. Hungarca, 106, (2002), 89-99.##S. B. Mulay, Cycles and Symmetries of Zero-divisor, Comm. Algebra, 30,(2002), 3533-3558.##Sh. Payrovi, S. Babaei, On 2-absorbing Submodules, Algebra Colloquium, 19, (2012), 913-920.##Sh. Payrovi, S. Babaei, On the 2-Absorbing Submodules, Iranian Journal of Mathematical Sciences and Informatics, 10(1), (2015), 131-137.##S. P. Redmond, An Ideal-based Zero-divisor Graph of a Commutative Rings, Comm. Algebra, 31, (2003), 4425-4443.##R. Wisbauer, Modules and algebras: Bimodule Structure and Group Action on Algebras, Pitman Mono 81, Addison-Wesley-Longman, Chicago, 1996.## ##

Some Generalizations of Locally Closed Sets 2 2 Arenas et al. [1] introduced the notion of lambda-closed sets as a generalization of locally closed sets. In this paper, we introduce the notions of lambda locally closed sets, Lambda_lambda closed sets and lambda_g-closed sets and obtain some decompositions of closed sets and continuity in topological spaces. 159 165 2014/03/282014/11/262016/02/62016/04/202016/04/242016/05/122016/06/62016/08/132016/07/302016/08/132016/08/132016/09/42016/09/82016/11/23 1395/9/3 2018/09/32018/10/22017/01/252017/01/182017/02/202017/01/162017/11/222018/01/172017/08/202017/09/272017/02/62017/10/82017/01/122017/04/18 1396/1/29 Sh. Modak Sh. Modak Department of Mathematics, University of Gour Banga India spmodak2000@yahoo.co.in T. Noiri T. Noiri Kuvempu University, Shiokita - cho, Hinagu Japan t.noiri@nifty.com Lambda-open set Lambda-locally closed set Lambda_lambda-closed set Lambda_g-closed set Decompositions of continuity. F. G. Arenas, J. Dontchev, M. Ganster, On λ-Sets and the Dual of Generalized Continuity, Questions Answers General Topology, 15, (1997), 3–13.##R. A. Borzooei, G. R. Rezaei, N. Kouhestani, On (semi) Topological BL-Algebras, Iranian Journal of Mathematical Sciences and Informatics, 6(1), (2011), 59–77.##N. Bourbaki, General Topology, Chapters 1-4, Springer-Verlag, 1989.##J. Dontechev, On Superconnected Spaces, Serdica-Bulgaricae Mathematicae Publications, 20, (1994), 345–350.##A. A. Estaji, z-Weak Ideals and Prime Weak Ideals, Iranian Journal of Mathematical Sciences and Informatics, 7(2), (2012), 53–62.##M. Ganster, I. L. Reilly, Locally Closed Sets and LC-continuous Functions, Internat. J.Math. Math. Sci., 12(3), (1989), 417–424.##N, Levine, Semi-open Sets and Semi-continuity in Topological Spaces, Amer. Math.Monthly, 70, (1963), 36–41.##N. Levine, Generalized Closed Sets in Topology, Rend. Circ. Mat. Palermo (2), 19,(1970), 89–96.##H. Maki, Generalized Λ-Sets and the Associated Closure Operator, The Special Issue in Commemoration of Prof. Kazusada IKEDAs Retirement, (1986), 139–146.##A. S. Mashhour, M. E. Abd El-Monsef, S. N. El-Deeb, On Precontinuous and Week Precontinuous Mappings, Proc. Math. Phys. Soc. Egypt., 53, (1982), 47–53.##O. Njastad, On Some Classes of Nearly Open Sets, Paciﬁc J. Math., 15, (1965), 961–970.##T. Roudari, L. Torkzadeh, A Topology on BCK-Algebra via Left and Right Stabilizers,Iranian Journal of Mathematical Sciences and Informatics, 4(2), (2009), 1–8.## ##

Iterative Process for an α- Nonexpansive Mapping and a Mapping Satisfying Condition(C) in a Convex Metric Space 2 2 We construct one-step iterative process for an α- nonexpansive mapping and a mapping satisfying condition (C) in the framework of a convex metric space. We study -convergence and strong convergence of the iterative process to the common fixed point of the mappings. Our results are new and are valid in hyperbolic spaces, CAT(0) spaces, Banach spaces and Hilbert spaces, simultaneously. 167 179 2014/03/282014/11/262016/02/62016/04/202016/04/242016/05/122016/06/62016/08/132016/07/302016/08/132016/08/132016/09/42016/09/82016/11/232016/11/24 1395/9/4 2018/09/32018/10/22017/01/252017/01/182017/02/202017/01/162017/11/222018/01/172017/08/202017/09/272017/02/62017/10/82017/01/122017/04/182018/09/17 1397/6/26 H. Fukhar-ud-din H. Fukhar-ud-din Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals Saudi Arabia Convex metric space α-Nonexpansive mapping Condition(C) Common fixed point One-step iterative process Convergence. M. Abbas, S. H. Khan, Some 4− Convergence Theorems in CAT (0) Spaces, Hacet. J.Math. Stat., 40(4), (2011), 563-569.##K. Aoyama, F. Kohsaka, Fixed Point Theorem for α-Nonexpansive Mappings in Banach Spaces, Nonlinear Analysis, 74, (2011), 4387-4391.##M. Alimohammady , M. Ramazannejad, Z. Bagheric , R. J. Shahkoohid, Common Zero Points of Two Finite Families of Maximal Monotone Operators Via Iteration Methods, Iranian Journal of Mathematical Sciences and Informatics, 12(2), (2017), 73-99.##H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer-Verlag, New York, 2011.##S. Dhompongsa, W. Inthakon, W. Takahashi, A Weak Convergence Theorem for Common Fixed Points of Some Generalized Nonexpansive Mappings and non-Spreading Mappings in a Hilbert Space, Optimization: A Journal of Mathematical Programming and Operations Research, 60, (2011), 769-779.##H. Fukhar-ud-din, Existence and Approximation of Fixed Points in Convex Metric Spaces, Carpathian J. Math., 30, (2014), 175-185.##H. Fukhar-ud-din, One Step Iterative Scheme for a Pair of Nonexpansive Mappings in a Convex Metric Space, Hacet. J. Math.Stat., 44, (2015), 1023–1031.##H. Fukhar-ud-din, A. R. Khan, Approximation of Common Fixed Point of Two Quasinonexpansive Mappings in Convex Metric Spaces, Mediterr. J. Math., (2018), 15:77.##B. Gunduz, S. Akbulut, Strong and 4− Convergence Theorems in Hyperbolic Spaces, Miskolc Mathematical Notes, 14(3), (2013), 915-925.##M. A. Khamsi, Approximate Fixed Point Sequences of Nonlinear Semigroups in Metric Spaces, Canad. Math. Bull., 58(2), (2015), 297-305.##U. Kohlenbach, Some Logical Metatheorems with Applications in Functional Analysis, Trans. Am. Math. Soc., 357(1), (2005), 89-128.##W. R. Mann, Mean Value Methods in Iterations, Proc.Amer. Math. Soc., 4, (1953),506-510.##H. R. Sahebi, A. Razani, An Explicit Viscosity Iterative Algorithm for Finding Fixed Points of Two Noncommutative Nonexpansive Mappings,Iranian Journal of Mathematical Sciences and Informatics, 11(1) (2016), 69-83.##T. Shimizu, W. Takahashi, Fixed Points of Multivalued Mappings in Certain Convex Metric Spaces, Topol. Methods Nonlinear Anal., 8, (1996), 197-203.##T. Suzuki, Fixed Point Theorems and Convergence Theorems for Some Generalized Nonexpansive Mappings, J. Math. Anal. Appl., 341, (2008), 1088-1095.##W. Takahashi, A Convexity in Metric Spaces and Nonexpansive Mappings, Kodai Math.Sem. Rep., 22, (1970), 142-149.##W. Takahashi, T. Tamura, Convergence Theorems for a Pair of Nonexpansive Mappings, J. Convex Anal., 5, (1998), 45-56.##K. Wattanawitoon, Y. Khamlae, Weak and Strong Convergence Theorems for an α-Nonexpansive Mapping and a Generalized Nonexpansive Mapping in Hilbert Spaces,Thai J. Math., 11, (2013), 633-643.##Y. Yao, R. Chen, Weak and Strong Convergence of a Modiﬁed Mann Iteration for Asymptotically Nonexpansive Mappings, Nonlinear Funct. Anal. Appl., 12, (2007), 307-315.## ##

ABSTRACTS IN PERSIAN Vol.14, No.1 2 2 Please see the full text contains the Pesian abstracts for this volume. 181 196 2014/03/282014/11/262016/02/62016/04/202016/04/242016/05/122016/06/62016/08/132016/07/302016/08/132016/08/132016/09/42016/09/82016/11/232016/11/242019/07/20 1398/4/29 2018/09/32018/10/22017/01/252017/01/182017/02/202017/01/162017/11/222018/01/172017/08/202017/09/272017/02/62017/10/82017/01/122017/04/182018/09/172019/07/20 1398/4/29 The Name of Authors In This Volume The Name of Authors In This Volume All Affilliations Iran fatemeh.bardestani@gmail.com ABSTRACTS PERSIAN Vol. 14 No. 1 1-All references of this volume## ##