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Stabilization of a Type III Thermoelastic Bresse System with Distributed Delay-time 2 2 In this paper, we investigate a Bresse-type system of thermoelasticity of type III in the presence of a distributed delay. We prove the well-posedness of the problem. Furthermore, an exponential stability result will be shown without the usual assumption on the wave speeds. To achieve our goals, we make use of the semigroup method and the energy method. 1 18 2019/02/2 1397/11/13 2020/05/30 1399/3/10 Lamine Bouzettouta Lamine Bouzettouta University of 20 August 1955, Skikda, Algeria Algeria lami_750000@yahoo.fr Bresse system Delay terms Decay rate Lyaponov methode Thermoelastic. F. Alabau-Boussouira, J. E. Munôz Rivera, D. S. Almeida Junior, Stability to Weak Dissipative Bresse System, Journal of Mathematical Analysis and Applications, 374, (2011), 481-498.##T. A. Appalara, Well-posedness and Exponential Stability for a Linear Damped Timoshenko System with Second Sound and Internal Distributed Delay, Journal of Differential , 254, (2014), 1-15.##A. Benaissa, M. Miloudi, M. Mokhtari, Global Existence and Energy Decay of Solutions to a Bresse System with Delay Terms, Math. Univ. Carolin., 56(2), (2015), 169-186.##L. Bouzettouta, S. Zitouni, Kh. Zennir, A. Guessmia, Stability of Bresse System with Internal Distributed Delay, J. Math. Comput. Sci., 7(1), (2017), 92-118.##L. Bouzettouta, S. Zitouni, Kh. Zennir, H. Sissaoui, Well-posedness and Decay of Solutions to Bresse System with Internal Distributed Delay, Int. J. Appl. Math. Stat., 56(4), (2017), 153-168##J. A. C. Bresse, Cours de Méchanique Appliquée, Mallet Bachelier, Paris, 1859.##A. Fareh, S. A. Messaoudi, Stabilization of a Type III Thermoelastic Timoshenko System in the Presence of a Time-distributed Delay, Math. Nachr., (2016), 1-16.##A. A. Keddi, A. T. Apalara, A. S. Messaoudi, Exponential and Polynomial Decay in a Thermoelastic-Bresse System with Second Sound, Appl Math Optim, Springer- New York, 2016.##H. E. Khochemane, L. Bouzettouta, A. Guerouah, Exponential Decay and Well-posedness for a One-dimensional Porous-elastic System with Distributed Delay, Applicable Analysis, DOI: 10.1080/00036811.2019.1703958.##H. E. Khochemane, A. Djebabla, S. Zitouni, L. Bouzettouta, Well-posedness and General Decay of a Nonlinear Damping Porous-elastic System with Infinite Memory, J. Math. Phys., 61, 021505 (2020), https://doi.org/10.1063/1.5131031.##Z. Liu, B. Rao, Energy Decay Rate of the Thermoelastic Bresse System, Z. Angew. Math. Phys., 60, (2009), 54-69.##S. A. Messaoudi, M. Pokojovy, B. Said-Houari, Nonlinear Damped Timoshenko Systems with Second Sound: Global Existence and Exponential Stability, Math. Methods Appl. Sci., 32, (2009), 505-534 .##M. I. Mustafa, M. Kafini, Exponential Decay in Thermoelastic Systems with Internal Distributed Delay, Palestine J. Math., 2(2), (2013), 287-299.##M. I. Mustafa, A Uniform Stability Result for Thermoelasticity of Type III with Boundary Distributed Delay, J. Abstr. Diff. Equa. Appl., 2(1), (2014), 1-13.##S. Nicaise, C. Pignotti, Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks, SIAM J. Control Optim., 45(5), (2006), 1561-1585.##A. S. Nicaise, C. Pignotti, Stabilization of the Wave Equation with Boundary or Internal Distributed Delay, Diff. Int. Equs., 21(9-10), (2008), 935-958.##A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Math. Sciences, Springer-Verlag, New York, 44, 1983.## ##

Arithmetic Deformation Theory of Lie Algebras 2 2 This paper is devoted to deformation theory of graded Lie algebras over Z or Zl with finite dimensional graded pieces. Such deformation problems naturally appear in number theory. In the first part of the paper, we use Schlessinger criteria for functors on Artinian local rings in order to obtain universal deformation rings for deformations of graded Lie algebras and their graded representations. In the second part, we use a version of Schlessinger criteria for functors on the Artinian category of nilpotent Lie algebras which is formulated by Pridham, and explore arithmetic deformations using this technique. 19 32 2019/02/22017/12/13 1396/9/22 2020/05/302023/02/27 1401/12/8 Arash Rastegar Arash Rastegar Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran Iran rastegar1352@gmail.com Arithemtic Lie algebras Deforfation of Lie algebras Schlessinger criteria. M. Barr, A Cohomology Theory for Commutative Algebras I II, Proc. A.M.S., 16, (1965), 1379-1391.##A. Fialowski, Deformations of Lie Algebras, Math USSR Sbornik, 55, (1986), 467-473.##A. Fialowski, D. Fuchs, Construction of Miniversal Deformations of Lie Algebras, Funct. Anal., 161(1), (1999), 76-110.##A. Grothendieck, Revetement Etale et Groupe Fondamental (SGA I), LNM 224, Springer-Verlag, 1971.##D. K. Harrison, Commutative Algebras and Cohomology, Trans. A.M.S., 104, (1962), 191-204.##R. Hain, M. Matsumoto, Weighted Completion of Galois Groups and Galois Actions on the Fundamental Group of P^1-{1,0,∞}, Compositio Math., 139(2), (2003), 119-167.##J. P. Pridham, Deforming l-adic Representations of the Fundamental Group of a Smooth Variety., J. Algebraic Geom., 15(3), (2006), 415-442.##D. Quillen, On the (co)-homology of Commutative Rings, in Applications of Commutative Algebra, Proc. Sympos. Pure. Math., Amer. Math. Soc. New York, XVII, (1968), 65-87.##A. Rastegar, Deformation of Outer Representations of Galois Group I, Iran. J. Math. Sci. Inform., 6(1), 101, (2011), 33-52.##A. Rastegar, Deformation of Outer Representations of Galois Group II, Iran. J. Math. Sci. Inform., 6(2), 85, (2011), 33-41.##M. Schlessinger, Functors of Artin Rings, Trans. A.M.S., 130, (1968), 208-222.## ##

Coefficient Estimates for a New Subclasses of m-fold Symmetric Bi-Univalent Functions 2 2 The purpose of the present paper is to introduce two new subclasses of the function class ∑m  of bi-univalent functions which both f  and f-1  are m-fold symmetric analytic functions. Furthermore, we obtain estimates on the initial coefficients for functions in each of these new subclasses. Also we explain the relation between our results with earlier known results. 33 39 2019/02/22017/12/132018/08/13 1397/5/22 2020/05/302023/02/272023/02/27 1401/12/8 Aqeel Ketab AL-khafaji Aqeel Ketab AL-khafaji Department of Mathematics College of Education for Pure Sciences, University of Babylon, Iraq Iraq aqeelketab@gmail.com Analytic function Univalent function m-fold symmetric bi-univalent functions. A. Akgul, On the Coefficient Estimates of Analytic and Bi-Univalent m-Fold Symmetric Functions, Mathematica Aeterna, 7(3), (2017), 253 -260.##Altinkaya, S. Yaln, Coefficient Bounds for Two New Subclasses of m-fold Symmetric Bi-univalent Functions, Serdica Mathematical Journal, 42(2), (2016), 175-186.##Altinkaya, S. Yaln, On Some Subclasses of m-fold Symmetric Bi-Univalent Functions, Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics, 67(1), (2018), 29-36.##S. Azizi, A. Ebadian, Sh. Najafzadeh, Coefficient Estimates for a Subclass of Bi-Univalent Functions, Advanced Computational Science with Applications, 1(1),(2015), 41-44.##D. A. Brannan, T. S. Taha , On Some Classes of Bi-Univalent Functions, Studia Univ. Babes-Bolyai Math., 31(2), (1986), 70-77.##W. Koepf, Coefficient of Symmetric Functions of Bounded Boundary Rotations, Proc. Amer. Math. Soc., 10(5), (1989), 324-329.##E. Mazi, Altinkaya, On a New Subclass of m-fold Symmetric bi-Univalent Functions Equipped With Subordinate Conditions, Khayyam Journal of Mathematics, 4(2), (2018), 187-197.##C. Pommerenke, On The Coefficient of Closed-to-Convex Functions, Michigan Math. J., 9, (1962), 259-269.##C. Pommerenke , Univalent Functions, Vandenhoeck and Rupercht, Go Ttingen, 1975.##H. M., Srivastava, S. Gaboury, F. Ghanim, Initial Coefficient Estimates for Some Subclasses of m-fold Symmetric bi-univalent Functions, Acta Mathematica Scientia, 36B(3), (2016), 863-871.##H. M. Srivastava, A. K. Mishra, P. Gochhayat, Certain Subclasses of Analytic and biUnivalent Functions, Appl. Math. Lett., 23(10), (2010), 1188-1192.##H. M. Srivastava, S. Sivasubramanian, R. Sivakumer, Initial Coefficient Bounds for a Subclasses of m-fold Symmetric bi-univalent Functions, Tbilisi Mathematical J., 7(2), (2014), 1-10.## ##

The Hyper-Zagreb Index of Trees and Unicyclic Graphs 2 2 Topological indices are widely used as mathematical tools to analyze different types of graphs emerged in a broad range of applications. The Hyper-Zagreb index (HM) is an important tool because it integrates the first two Zagreb indices. In this paper, we characterize the trees and unicyclic graphs with the first four and first eight greatest HM-value, respectively. 41 54 2019/02/22017/12/132018/08/132018/07/6 1397/4/15 2020/05/302023/02/272023/02/272019/05/8 1398/2/18 Hassan Rezapour Hassan Rezapour Department of Mathematics, Faculty of Basic Sciences, University of Qom, Qom, Iran Iran hassan.rezapour@gmail.com Ramin Nasiri Ramin Nasiri Department of Mathematics, Faculty of Sciences, Imam Hossein Comprehensive University, Tehran, Iran Iran R.Nasiri@Shahabdanesh.ac.ir Seyedahmad Mousavi Seyedahmad Mousavi Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, MD 21250, U.S.A. U.S.A. smousav1@umbc.edu Hyper-Zagreb index Vertex degree Unicyclic graphs Trees. S. Baby, M. K. Shafiq, H. M. A. Siddiqui, On Zagreb Indices and Zagreb Polynomials of Single-walled Titania Nanotubes, Science International, 28(4), (2016).##B. Basavanagoud, S. Patil, The Hyper-Zagreb Index of Four Operations on Graphs, Math. Sci. Lett., 6(2), (2017), 193-198.##J. Braun, A. Kerber, M. Meringer, C. Rucker, Similarity of Molecular Descriptors: the Equivalence of Zagreb Indices and Walk Counts, MATCH Commun. Math. Comput. Chem., 54, (2005), 163-176.##A. Dobrynin, R. Entringer, I. Gutman, Wiener Index of Trees: Theory and Applications, Acta Appl. Math., 66(3), (2001), 211-249.##F. Falahati-Nezhad, M. Azari, Bounds on the Hyper-Zagreb Index, J. Appl. Math. Inform., 34, (2016), 319-330.##G. H. Fath-Tabar, I. Gutman, R. Nasiri, Extremely Irregular Trees, Bull. Acad. Serbe Sci. Arts (Cl. Sci. Math. Natur), 38, (2013), 1-8.##W. Gao, M. K. Jamil, A. Javed, M. R. Farahani, Sh. Wang, J. B. Liu, Sharp Bounds of the Hyper-Zagreb Index on Acyclic, Unicylic, and Bicyclic Graphs,Discrete Dyn. Nat. Soc., (2017).##W. Gao, M. K. Jamil, M. R. Farahani, The Hyper-Zagreb Index and Some Graph Operations, J. Appl. Math. Comput., 54(1-2), (2017), 263-275.##W. Gao, M. K. Siddiqui, Molecular Descriptors of Nanotube, Oxide, Silicate, and Triangulene Networks, J. Chem., Article ID 6540754.10 (2017).##W. Gao, W. Wang, M. R. Farahani, Topological Indices Study of Molecular Structure in Anticancer Drugs, J. Chem., (2016).##I. Gutman, K. Ch. Das, The First Zagreb Index 30 Years After, MATCH Commun. Math. Comput. Chem., 50, (2004), 83-92.##I. Gutman, W. Linert, I. Lukovits, A. Dobrynin, Trees with Extremal Hyper-Wiener Index: Mathematical Basis and Chemical Applications, J. Chem. Inf. Comput. Sci., 37(2), (1997), 349-354.##I. Gutman, N. Trinajstić, Graph Theory and Molecular Orbitals. Total-electron Energy of Alternant Hydrocarbons, Chem. Phys. Lett., 17(4), (1972), 535-538.##S. M. Kang, M. Munir, A. R. Nizami, Z. Shahzadi, W. Nazeer, Some Topological Invariants of the Mbius Ladder, Global J.Pure Appl. Math., 12, (2016), 5317-5327.##D. J. Klein, I. Lukovits, I. Gutman, On the Definition of the Hyper-Wiener Index for Cycle-containing Structures, J. Chem. Inf. Comput. Sci., 35(1), (1995), 50-52.##V. R. Kulli, B. Chaluvaraju, H. S. Boregouda, Some Degree Based Connectivity Indices of Kulli Cycle Windmill Graphs, South Asain J. Math., 6(6), (2016), 263-268.##R. Nasiri, H. R. Ellahi, A. Gholami, G. H. Fath-Tabar, The Irregularity and Total Irregularity of Eulerian Graphs, Iranian J. Math. Chem., 9(2), (2018), 101-111.##R. Nasiri, A. Gholami, G. H. Fath-Tabar, H. R. Ellahi, Extremely Irregular Unicyclic Graphs, Kragujevac J. Math., 43(2), (2019), 281-292.##R. Nasiri, G. H. Fath-Tabar, The Second Minimum of the Irregularity of Graphs, Electron. Notes Discret. Math., 45, (2014), 133-140.##R. Nasiri, H. Yousefi-Azari, M. R. Darafsheh, A. R. Ashrafi, Remarks on the Wiener Index of Unicyclic Graphs, J. Appl. Math. Comput., 41, (2013), 49-59.##S. Nikolić, G. Kovacević, A. Milicević, N. Trinajstić, The Zagreb Indices 30 Years After, Croat. Chem. Acta., 76(2), (2003), 113-124.##M. K. Siddiqui, M. Imran, A. Ahmad,. On Zagreb Indices, Zagreb Polynomials of Some Nanostar Dendrimers, Appl. Math. Comput., 280, (2016), 132-139.##G. H. Shirdel, H. Rezapour, A. M. Sayadi, The Hyper-Zagreb Index of Graph Operations, Iranian J. Math. Chem., 4(2), (2013), 213-220.##H. Wiener, Structural Determination of Paraffin Boiling Points, J. Am. Chem. Soc., 69(1), (1947), 17-20.##B. Zhou, Zagreb Indices, MATCH Commun. Math. Comput. Chem., 52, (2004), 113-118.##B. Zhou, I. Gutman, Relations Between Wiener, Hyper-Wiener and Zagreb Indices, Chem Phys Lett., 394(1), (2004), 93-95.##B. Zhou, I. Gutman, Further Properties of Zagreb Indices, MATCH Commun. Math. Comput. Chem., 54, (2005), 233-239.## ##

On Time Fractional Modifed Camassa-Holm and Degasperis-Procesi Equations by Using the Haar Wavelet Iteration Method 2 1   2 The Haar wavelet collocation with iteration technique is applied for solving a class of time-fractional physical equations. The approximate solutions obtained by two dimensional Haar wavelet with iteration technique are compared with those obtained by analytical methods such as Adomian decomposition method (ADM) and variational iteration method (VIM). The results show that the present scheme is effective and appropriate for obtaining the numerical solution of the timefractional Modified Camassa-Holm equation and Time fractional Modified Degasperis-Procesi equation.   55 71 2019/02/22017/12/132018/08/132018/07/62018/10/27 1397/8/5 2020/05/302023/02/272023/02/272019/05/82019/07/12 1398/4/21 N. Aghazadeh N. Aghazadeh Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran Iran aghazadeh@azaruniv.ac.ir Gh. Ahmadnezhad Gh. Ahmadnezhad Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran Iran sh.rezapour@azaruniv.ac.ir Sh. Rezapour Sh. Rezapour Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran Iran rezapourshahram@yahoo.ca Fractional differential equation Haar wavelet Operational matrices Iterative method Sylvester equation. J. He, Nonlinear Oscillation with Fractional Derivative and its Applications, Int. Conf. Vibr. Eng., 98, (1998), 288-291.##J. He, Some Applications of Nonlinear Fractional Differential Equations and their Approximations, Bull. Sci. Technol., 15(2), (1999), 86-90.##F. Mainardi, Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics, Fract. Fract. Calculus Contin. Mech., (1997), 291-348.##R. Panda, M. Dash, Fractional Generalized Splines and Signal Processing, Signal Proc., 86, (2006), 2340-2350.##G. Bohannan, Analog Fractional Order Controller in Temperature and Motor Control Applications, J. Vibr. 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Geometric Studies on Inequalities of Harmonic Functions in a Complex Field Based on ξ-Generalized Hurwitz-Lerch Zeta Function 2 2 Authors, define and establish a new subclass of harmonic regular schlicht functions (HSF) in the open unit disc through the use of the extended generalized Noor-type integral operator associated with the ξ-generalized Hurwitz-Lerch Zeta function (GHLZF). Furthermore, some geometric properties of this subclass are also studied. 73 95 2019/02/22017/12/132018/08/132018/07/62018/10/272019/02/1 1397/11/12 2020/05/302023/02/272023/02/272019/05/82019/07/122020/01/18 1398/10/28 Hiba F. Al-Janaby Hiba F. Al-Janaby Department of Mathematics, College of Science, University of Baghdad, Baghdad-Iraq Iraq fawzihiba@yahoo.com F. Ghanim F. Ghanim Department of Mathematics, College of Science, University of Sharjah, Sharjah, United Arab Emirates United Arab Emirates fgahmed@sharjah.ac.ae P. Agarwal P. Agarwal Department of Mathematics, Anand International College of Engineering, Jaipur-303012, India India goyalpraveen2011@gmail.com <goyal.praveen2011@gmail.com Harmonic function Regular function Schlicht function Noor integral operator ξ-generalized Hurwitz-Lerch zeta function. P. Agarwal, J. Choi, R. B. Paris, Extended Riemann-Liouville Fractional Derivative Operator and its Applications, J. Nonlinear Sci. and Appl., 8, (2015), 451-466.##J. W. Alexander, Functions which Map the Interior of the Unit Circle upon Simple Regions, Ann. of Math., 17(1), (1915), 12-22.##H. F. Al-Janaby, F. Ghanim, M. Z. Ahmad, Harmonic Multivalent Functions Associated with an Extended Generalized Linear Operator of Noor-type, Nonlinear Funct. Anal. Appl., 24(2), (2019), 269-292.##M. Aydo~gan, E. Y. Duman, Y. Polato~glu, Y. Kahramaner, Harmonic Function for which the Second Dilatation is α-spiral, J. Inequal. Appl., 262, (2012), 1-7.##S. D. Bernardi, Convex and Starlike Univalent Functions, Trans. Amer. 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Coefficient Estimates for a General Subclass of m-fold Symmetric Bi-univalent Functions by Using Faber Polynomials 2 2 In the present paper, we introduce a new subclass H∑m (λ,β)of the m-fold symmetric bi-univalent functions. Also, we find the estimates of the Taylor-Maclaurin initial coefficients |am+1| , |a2m+1| and general coefficients |amk+1| (k ≥ 2) for functions in this new subclass. The results presented in this paper would generalize and improve some recent works of several earlier authors. 97 108 2019/02/22017/12/132018/08/132018/07/62018/10/272019/02/12019/01/23 1397/11/3 2020/05/302023/02/272023/02/272019/05/82019/07/122020/01/182019/11/28 1398/9/7 Safa Salehian Safa Salehian Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran Iran s.salehian84@gmail.com Ahmad Motamednezhad Ahmad Motamednezhad Faculty of Mathematical Sciences, Shahrood University of Technology, P.O.Box 316-36155, Shahrood, Iran Iran ‎a.motamedne@gmail.com ‎Nanjundan Magesh ‎Nanjundan Magesh Post-Graduate and Research Department of Mathematics, Government Arts College for Men, Krishnagiri 635001, Tamilnadu, India India ‎ ‎nmagi_2000@yahoo.co.in Bi-univalent functions m-fold symmetric bi-univalent functions Coefficient estimates Faber polynomials. H. Airault, J. Ren, An Algebra of Differential Operators and Generating Functions on the Set of Univalent Functions, Bull. Sci. Math., 126(5), (2002), 343-367.##H. Airault, A. Bouali, Differential Calculus on the Faber Polynomials, Bull. Sci. Math., 130, (2006),179-222.##Ş. Altinkaya, S. Yal¸cin, On Some Subclass of m-fold Symmetric Bi-univalent Functions, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 67(1), (2018), 29-36.## Ş. Altinkaya, S. Yal¸cin, Coefficient Bounds for Certain Subclasses of m-fold Symmetric bi-univalent Functions, Journal of Mathematics, 2015, Article ID 241683, (2015).##D. A. Brannan, J. G. Clunie(Eds.), Aspect of Contemporary Complex Analysis, Advanced Study Institute Held at the University of Durham, Durham; July 1-20, (1979).##D. A. Brannan, T. S. Taha, On Some Classes of Bi-univalent Functions, Studia Univ. Babes-Bolyai Math., 31(2), (1986), 70-77.##D. Breaz, N. Breaz, H. M. Srivastava, An Extention of the Univalent Conditions for a Family of Integral Operators, Appl. 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M. Sakar, H. O. Güney, Faber Polynomial Coefficient Estimates for Subclasses of m-fold Symmetric Bi-univalent Functions Defined by Fractional Derivative, Malays. J. Math. Sci., 11(2), (2017), 275-287.##B. Senthil, B. S. Keerthi, Certain Subclass of m-fold Symmetric-sakaguchi Type Biunivalent Functions, Int. J. Pure Appl. Math., 109(10), (2016), 29-37.##H. M. Srivastava, D. Bansal, Coefficient Estimates for a Subclass of Analytic and Biunivalent Functions, J. Egyptian Math. Soc., 23, (2015), 242-246.##H. M. Srivastava, S. Gaboury, F. Ghanim, Initial Coefficient Estimates for Some Subclasses of m-fold Symmetric Bi-univalent Functions, Acta Math. Sci. Ser. B, 36(3), (2016), 863-871.##H. M. Srivastava, S. Gaboury, F. Ghanim, Coefficient Estimates for Some Subclasses of m-fold Symmetric Bi-univalent Functions, Acta Univ. Apulensis Math. Inform., 41, (2015), 153-164.##H. M. Srivastava, B. S. Joshi, S. Joshi, H. 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Wanas, A. H. Majeed, Certain New Subclasses of Analytic and m-fold Symmetric Bi-univalent Functions, Appl. Math. E-Notes, 18, (2018), 178-188.##X.-F. li, A.-P. Wang, Two New Subclasses of bi-univalent Functions, International Mathematical Forum, 7(30), (2012), 1495-1504.##Q.-H. Xu, Y.-C. Gui, H. M. Srivastava, Coefficient Estimates for a Certain Subclass of Analytic and bi-univalent Functions, Appl. Math. Lett., 25, (2012), 990-994.##Q.-H. Xu, H.-G. Xiao, H. M. Srivastava, A Certain General Subclass of Analytic and bi-univalent Functions and Associated Coefficient Estimate Problems, Appl. Math. Comput., 218, (2012), 11461-11465.##A. Zireh, E. Analouei Adegani, S. Bulut, Faber Polynomial Coefficient Estimates for a Comprehensive Subclass of Analytic bi-univalent Functions Defined by Subordination, Bull. Belg. Math. Soc. Simon Stevin., 23, (2016), 487-504.##A. Zireh, S. Salehian, On the Certain Subclass of Analytic and Bi-univalent Functions Defined by Convolution, Acta Univ. Apulensis Math. Inform., 44, (2015), 9-19.## ##

An Optimal Algorithm for the δ-ziti Method to Solve Some Mathematical Problems 2 2 The numerical approximation methods of the differential problems solution are numerous and various. Their classifications are based on several criteria: Consistency, precision, stability, convergence, dispersion, diffusion, speed and many others. For this reason a great interest must be given to the construction and the study of the associated algorithm: indeed the algorithm must be simple, robust, less expensive and fast. In this paper, after having recalled the δ-ziti method, we reformulat it to obtain an algorithm that does not require as many calculations as many nodes knowing that they are counted by thousands. We have, therefore, managed to optimize the number of iterations by passing for example from 103 at 10 iterations. 109 129 2019/02/22017/12/132018/08/132018/07/62018/10/272019/02/12019/01/232018/04/2 1397/1/13 2020/05/302023/02/272023/02/272019/05/82019/07/122020/01/182019/11/282021/04/8 1400/1/19 L. Bsiss L. Bsiss Department of Mathematics, University Moulay Ismaïl, Faculty of Sciences, BP 11201 Zitoune, Meknès 50000, Morocco Morocco lyrbi01@gmail.com C. Ziti C. Ziti Department of Mathematics, University Moulay Ismaïl, Faculty of Sciences, BP 11201 Zitoune, Meknès 50000, Morocco Morocco chziti@gmail.com Algorithm Meshing δ−ziti Optimal Operations number. L. Bsiss, C. Ziti, A New Approximation (Ziti’s δ-scheme) of the Entropic (Admissible) Solution of the Hyperbolic Problems in One and Several Dimensions, Applications to Convection, Burgers, Gas dynamics and Some Biological Problems, Turkish Journal of Analysis and Number Theory, 4(4), (2016), 98-108.##L. Bsiss, C. Ziti, The δ-ziti’s Method to Detect the Blow-up in Finite Time in Some Models of Chemotaxis, Ponte Journal, 73(2), (2017), 245-260.##L. Bsiss, C. Ziti, A New Numerical Method for the Integral Approximation and Solving the Differential Problems: Non-oscillating Scheme, Detecting the Singularity in one and Several Dimension, Ponte Journal, 73(6), (2017), 126-172.##L. Bsiss, C. Ziti, A New Entropic Riemann Solver of Conservation Law of Mixed Type Including Ziti’s δ-Method with some Experimental Tests, Applied and Computational Mathematics Journal, 6(5), (2017), 222-232.##G. Dhatt, G. Touzot, Une Présentation de la Méthode Deséléments Finis, MALOINE S.A Editeur Paris et les Presses de l’université Laval Quebec, 1981.##B. Meyer, C. Baudoin, Méthodes de Programmation, Collection de la direction des Etudes et Recherches d’Elictricit de France Edition Eyrolles, 1984.##B. Demidovitch, T. Maron, Eléments de Calcul Numérique, Edition MIR de MOSCOU, 1979.##M. Sibony, J-CI. Mardon, Analyse Numérique I et II, HERMANN, Editions des sciences et des Arts, 1984.## ##

Reduction of BL-general L-fuzzy Automata 2 1   2 In this paper, we show that for any BL-general L-fuzzy automaton (BL-GLFA) there exists a complete deterministic accessible reduced BL-general L-fuzzy automaton that recognizing the behavior of the BL-GLFA. Also, we prove that for any finite realization β, there exists a minimal complete deterministic BL-GLFA recognizing β. We prove any complete deterministic accessible reduced BL-GLFA is a minimal BLGLFA. After that, we show that for any given finite realization β, the minimal complete deterministic BL-GLFA recognizing β is isomorphic to any complete accessible deterministic reduced BL-GLFA recognizing β. Moreover, we give some examples to clarify these notions. Finally, by using these notions, we give some theorems and algorithms and obtain some related results. 131 144 2019/02/22017/12/132018/08/132018/07/62018/10/272019/02/12019/01/232018/04/22019/02/3 1397/11/14 2020/05/302023/02/272023/02/272019/05/82019/07/122020/01/182019/11/282021/04/82020/02/7 1398/11/18 Marzieh Shamsizadeh Marzieh Shamsizadeh Department of Mathematics, Behbahan Khatam Alanbia University of Technology, Behbahan, Iran Iran shamsizadeh.m@gmail.com Mohammad Mehdi Zahedi Mohammad Mehdi Zahedi Department of Mathematics, Graduate University of Advanced Technology, Kerman, Iran Iran zahedi_mm@kgut.ac.ir Khadijeh Abolpour Khadijeh Abolpour Department of Mathematics, Shiraz Branch, Islamic Azad University, Shiraz, Iran Iran abolpor_kh@yahoo.com BL-general fuzzy automata Minimal automata Reduction Deterministic automata. Kh. Abolpour, M. M. Zahedi, Isomorphism Between Two BL-General Fuzzy Automata, Soft Computing, 16, (2012), 729-736.##N. C. Basak, A. Gupta, On Quotient Machines of a Fuzzy Automata and the Minimal Machine, Fuzzy Sets and Systems, 125, (2002), 223-229.##W. Cheng, Z. W. Mo, Minimization Algorithm of Fuzzy Finite Automata, Fuzzy Sets and Systems, 141, (2004), 439-448.##M. Doostfatemeh, S. C. Kremer, New Directions in Fuzzy Automata, International Journal of Approximate Reasoning, 38, (2005), 175-214.##C. L. Giles, C. W. Omlin, K. K. Thornber, Equivalence in Knowledge Representation: Automata, Recurrent Neural Networks, and Dynamical Fuzzy Systems, Proceedings of IEEE, 87, (1999), 1623-1640.##M. M. Gupta, G. N. Saridis, B. R. Gaines, Fuzzy Automata and Decision Processes, North Holland, New York, 1977, 111-175.##P. Hájek, Metamathematics of Fuzzy Logic, Kluwer, Dordrecht, Boston, London, 1988.##E. T. Lee, L. A. Zadeh, Note on Fuzzy Languages, Information Sciences, 1, (1969), 421-434.##D. S. Malik, J. N. Mordeson, Fuzzy Automata and Languages: Theory and Applications, Chapman Hall, CRC Boca Raton, London, New York, Washington DC, 2002.##A. Mateescu, A. Salomaa, K. Salomaa, S. Yu, Lexical Analysis with a Simple Finite Fuzzy Automaton Model, Journal of Universal Computer Science, 1, (1995), 292-311.##C. W. Omlin, K. K. Thornber, C. L. Giles, Fuzzy Finite-State Automata Can be Deterministically Encoded in Recurrent Neural Networks, IEEE Transactions on Fuzzy Systems, 5, (1998), 76-89.##W. Pedrycz, A. Gacek, Learning of Fuzzy Automata, International Journal of Computational Intelligence and Applications, 1, (2001), 19-33.##K. Peeva, Behavior, Reduction and Minimization of Finite L-Automata, Fuzzy Sets and Systems, 28, (1988), 171-181.##K. Peeva, Equivalence, Reduction and Minimization of Finite Automata Over Semirings, Theoretical Computer Science, 88, (1991), 269-285.##A. K. Ray, B. Chatterjee, A. K. Majumdar, A Formal Power Series Approach to the Construction of Minimal Fuzzy Automata, Information Sciences, 55, (1991), 189-207.##E. S. Santos, Fuzzy Automata and Languages, Information Sciences, 10, (1976), 193-197.##E. S. Santos, Maxmin Automata, Information Control, 13, (1968), 363-377.##E. S. Santos, On Reduction of Max-Min Machines, Journal of Mathematical Analysis and Applications, 37, (1972), 677-686.##M. Shamsizadeh, M. M. Zahedi, Bisimulation of Type 2 for BL- General Fuzzy Automata, Soft Computing, 23, (2019), 9843-9852.##M. Shamsizadeh, M. M. Zahedi, Intuitionistic General Fuzzy Automata, Soft Computing, 20, (2016), 3505-3519.##M. Shamsizadeh, M. M. Zahedi, Minimal and Statewise Minimal Intuitionistic General L-Fuzzy Automata, Iranian Journal of Fuzzy Systems, 23, (2016), 131-152.##M. Shamsizadeh, M. M. Zahedi, Minimal Intutionistic General L-Fuzzy Automata, Italian Journal of Pure and Applied Mathematics, 35, (2015), 155-186.##M. Shamsizadeh, M. M. Zahedi, Kh. Abolpour, Admissible Partition for BL-General Fuzzy Automata, Iranian Journal of Fuzzy Systems, 15, (2018), 79-90.##M. Shamsizadeh, M. M. Zahedi, Kh. Abolpour, Bisimulation for BL-General Fuzzy Automata, Iranian Journal of Fuzzy Systems, 13, (2016), 35-50.##M. Shamsizadeh, M. M. Zahedi, Kh. Abolpour, Kleens Theorem for BL-General L-Fuzzy Automata, Journal of Mahani Mathematical Research, 10, (2021), 125-144.##V. Topencharov, K. Peeva, Equivalence, Reduction and Minimization of Finite Fuzzy Automata, Journal of Mathematical Analysis and Applications, 84, (1981), 270-281.##W. G. Wee, On Generalization of Adaptive Algorithm and Application of the Fuzzy Sets Concept to Pattern Classification, Ph.D. Thesis, Purdue University, Lafayette, IN, 1967.##W. G. Wee, K. S. Fu, A Formulation of Fuzzy Automata and its Application as a Model of Learning Systems, IEEE Transactions on Systems, Man and Cybernetics, 5, (1969), 215-223.##W. G. 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A Numerical Method for Solving Stochastic Volterra-Fredholm Integral Equation 2 1   2 In this paper, we propose a numerical method based on the generalized hat functions (GHFs) and improved hat functions (IHFs) to find numerical solutions for stochastic Volterra-Fredholm integral equation. To do so, all known and unknown functions are expanded in terms of basic functions and replaced in the original equation. The operational matrices of both basic functions are calculated and embeded in the equation to achieve a linear system of equations which give the expansion coefficients of the solution. We prove that the rate of the convergence is O(h2) and O(h4) for these two different bases under some conditions. Two examples are solved and the results are compared with those of block pulse functions method (BPFs) to show the accuracy and reliability of the methods. 145 164 2019/02/22017/12/132018/08/132018/07/62018/10/272019/02/12019/01/232018/04/22019/02/32018/10/14 1397/7/22 2020/05/302023/02/272023/02/272019/05/82019/07/122020/01/182019/11/282021/04/82020/02/72019/08/31 1398/6/9 N. Momenzade N. Momenzade Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran Iran neda_momenzadeh@yahoo.com A. R. Vahidi A. R. Vahidi Department of Mathematics, College of Science, Yadegar-e-Emam Khomeyni (RAH) Shahr-e-Rey Branch, Islamic Azad University, Tehran, Iran Iran alrevahidi@yahoo.com E. Babolian E. Babolian Department of Mathematics, College of Science, Yadegar-e-Emam Khomeyni (RAH) Shahr-e-Rey Branch, Islamic Azad University, Tehran, Iran Iran babolian@khu.ac.ir Generalized hat functions Improved hat functions Stochastic operational matrix Stochastic Volterra-Fredholm integral equation Brownian motion. M. Rosestolato, A. Swiech, Partial Regularity of Viscosity Solutions for a Class of Kolmogorov Equations Arising from Mathematical Finance, J. Differ. Equations, 262, (2017), 1897-1930.##M. P. Laurini, J. F. Caldeira, A Macro-Finance Term Structure Model with Multivariate Stochastic Volatility, Int. Rev. Econ. Financ., 44, (2016), 68-90.##E. Platen, N. Bruti-Liberati, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Springer, 2010.##T. Mello dos Reis, A. C. Neto, M. C. M. Campos, Gaussian Basis Sets for Atomic and Molecular Calculations Obtained from Stochastic Optimization, Comput. Theor. Chem., 1099, (2017), 133-139.##C. Zimmer, S. Sahle, Comparison of Approaches for Parameter Estimation on Stochastic Models: Generic Least Squares Versus Specialized Approaches, Comput. Biol. Chem., 61, (2016), 75-85.##T. Manninen, M- Leena Linne, K. Ruohonen, Developing It^o Stochastic Differential Equation Models for Neuronal Signal Transduction Pathways, Comput. Biol. Chem., 30, (2006), 280-291.##K. D. Elworthy, H. Z. Zhao, J. G. Gaines, The Propagation of Travelling Waves for Stochastic Generalized KPP Equations, Math. Comput. Model., 20, (1994), 131-166.##K. Dadzie, Comment on A Solution Algorithm for the Fluid Dynamic Equations Based on a Stochastic Model for Molecular Motion, Jenny et. al., Journal of computational Physics, 229 (2010), J. Comput. Phys., 231, (2012), 7011-7013.##D. Peavoy, C. L. E. Franzke, G. O. Roberts, Systematic Physics Constrained Parameter Estimation of Stochastic Differential Equations, Comput. Stat. Data Anal., 83, (2015), 182-199.##T. D. Frank, Kramers-Moyal Expansion for Stochastic Differential Equations with Single and Multiple Delays: Aplication to Financial Physics and Neurophysics, Phys. Lett. A, 360, (2007), 552-562.##M. Michta, On Connections Between Stochastic Differential Inclusions and Set-Valued Stochastic Differential Equations Driven by Semimartingales, J. Differ. Equ., 262, (2017), 2106-2134.##Y. Li, R. Wang, N. Yao, S. Zhang, A Moderate Deviation Principle for Stochastic Volterra Equation, Stat. Probab. Lett., 122, (2017), 79-85.##G. Zong, Anticipated Backward Stochastic Differential Equations Driven by the Teugels Martingales, J. Math. Anal. Appl., 412, (2014), 989-997.##C. Zimmer, S. Shhle, A Termination Criterion for Parameter Estimation in Stochastic Models in System Biology, Biosystems, 137, (2015), 55-63.##T. Szekely Jr., K. Burrage, Stochastic Simulation in Systems Biology, Comput. Struct. Biotechnol. J., 12, (2014), 14-25.##P. Amar, L. Paulev´e, HSIM: A Hybrid Stochastic Simulation System for System Biology, Electron. Notes Theor. Comput. Sci., 313, (2015), 3-21.##M. Khodabin, K. Maleknejad, M. Rostami, M. Nouri, Numerical Approach for Solving Stochastic Volterra-Fredholm Integral Equations by Stochastic Operational Matrix, Comput. Math. Appl., 64, (2012), 1903-1913.##K. Maleknejad, M. Khodabin, M. Rostami, Numerical Solution of Stochastic Volterra Integral Equations by a Stochastic Operational Matrix Based on Block Pulse Functions, Math. Comput. Model., 55, (2012), 791-800.##K. Maleknejad, M. Khodabin, M. Rostami, A Numerical Method for Solving m-Dimensional Stochastic It^o-Volterra Integral Equations by Stochastic Operational Matrix, Comput. Math. Appl., 63, (2012), 133-143.##B. K. Oksendal, Stochastic Differential Equations:an Introduction with Applications, 4th ed., Springer, 1995.##M. H. Heydari, M. R. Hooshmandasl, F. M. Maalek Ghaini, C. Cattani, A Computational Method for Solving Stochastic Ito-Volterra Integral Equations Based on Stochastic Operational Matrix for Generalized Hat Basis Functions, J. Comput. Phys., 270, (2014), 402-415.##F. Mirzaee, E. Hadadiyan, Approximation Solution of Nonlinear Stratonovich Volterra Integral Equations by Applying Modification of Hat Functions, J. Comput. Appl. Math., 302, (2016), 272-284.##F. Mirzaee, A. Hamzeh, A Computational Method for Solving Nonlinear Stochastic Volterra Integral Equations, J. Comput. Appl. Math., 306, (2016), 166-178.##M. Khodabin, K. Maleknejad, F. Hosseini Shekarabi, Application of Triangular Functions to Numerical Solution of Stochastic Volterra Integral Equations, Int. J. Appl. Math., 43, (2013), 1-9.##F. Mirzaee, E. Hadadiyan, A Collocation Technique for Solving Nonlinear Stochastic Itô-Volterra Integral Equations, Appl. Math. Comput., 247, (2014), 1011-1020.##K. Maleknejad, M. Tavassoli Kajani, Solving Second Kind Integral Equations by Galerkin Methods with Hybrid Legendre and Block Pulse Functions, Appl. Math. Comput., 145, (2003), 623-629.## ##

On the Zagreb and Eccentricity Coindices of Graph Products 2 2 The second Zagreb coindex is a well-known graph invariant defined as the total degree product of all non-adjacent vertex pairs in a graph. The second Zagreb eccentricity coindex is defined analogously to the second Zagreb coindex by replacing the vertex degrees with the vertex eccentricities. In this paper, we present exact expressions or sharp lower bounds for the second Zagreb eccentricity coindex of some graph products such as lexicographic product, generalized hierarchical product, and strong product. Results are applied to compute the values of this eccentricity-based invariant for some chemical graphs and nanostructures such as hexagonal chain, linear phenylene chain, and zig-zag polyhex nanotube. 165 178 2019/02/22017/12/132018/08/132018/07/62018/10/272019/02/12019/01/232018/04/22019/02/32018/10/142019/02/5 1397/11/16 2020/05/302023/02/272023/02/272019/05/82019/07/122020/01/182019/11/282021/04/82020/02/72019/08/312020/05/30 1399/3/10 Mahdieh Azari Mahdieh Azari Department of Mathematics, Kazerun Branch, Islamic Azad University, P. O. Box: 73135-168, Kazerun, Iran Iran mahdie.azari@gmail.com Eccentricity of a vertex Graph invariants Graph products Chemical graphs. A. Ali, I. Gutman, E. Milovanović, I. Milovanović, Sum of Powers of the Degrees of Graphs: Extremal Results and Bounds, MATCH Commun. Math. Comput. Chem., 80, (2018), 5-84.##Y. Alizadeh, Szeged Dimension and P Iv Dimension of Composite Graphs, Iran. J. Math. Sci. Inf., 13(2), (2018), 45-57.##Y. Alizadeh, M. Azari, T. Došlić, Computing the Eccentricity-related Invariants of Singledefect Carbon Nanocones, J. Comput. Theor. Nanosci., 10(6), (2013), 1297-1300.##A. R. Ashrafi, T. Došlić, A. Hamzeh, The Zagreb Coindices of Graph Operations, Discrete Appl. Math., 158, (2010), 1571-1578.##M. Azari, Further Results on Zagreb Eccentricity Coindices, Discrete Math. Algorithms Appl., 12(6), (2020), 2050075 (15 pages).##M. Azari, On Eccentricity Version of Zagreb Coindices, Math. Interdisc. Res., 6(2), (2021), 107-120.##M. Azari, A. Iranmanesh, M. V. Diudea, Vertex-eccentricity Descriptors in Dendrimers, Studia Univ. Babes Bolyai Chem., 62(1), (2017), 129-142.##L. Barrière, C. Dalfó, M. A. Fiol, M. Mitjana, The Generalized Hierarchical Product of Graphs, Discrete Math., 309(12), (2009), 3871-3881.##B. Borovićanin, K. C. Das, B. Furtula, I. Gutman, Bounds for Zagreb Indices, MATCH Commun. Math. Comput. Chem., 78, (2017), 17-100.##K. C. Das, D.-W. Lee, A. Graovac, Some Properties of the Zagreb Eccentricity Indices, Ars Math. Contemp., 6(1), (2013), 117-125.##M. V. Diudea, QSPR/QSAR Studies by Molecular Descriptors, NOVA, New York, 2001.##T. Došlić, Vertex-Weighted Wiener Polynomials for Composite Graphs, Ars Math. Contemp., 1, (2008), 66-80.##T. Došlić, A. Graovac, F. Cataldo, O. Ori, Notes on Some Distance-based Invariants for 2-dimensional Square and Comb Lattices, Iran. J. Math. Sci. Inf., 5(2), (2010), 61-68.##T. Došlić, M. Saheli, Eccentric Connectivity Index of Composite Graphs, Util. Math., 95, (2014), 3-22.##I. Gutman, B. Furtula, Z. Kovijanić Vukićević, G. Popivoda, On Zagreb Indices and ˇCoindices, MATCH Commun. Math. Comput. Chem., 74, (2015), 5-16.##I. Gutman, B. Ruščić, N. Trinajstić, C. F. Wilcox, Graph Theory and Molecular Orbitals. XII. Acyclic Polyenes, J. Chem. Phys., 62, (1975), 3399-3405.##I. Gutman, N. Trinajstić, Graph Theory and Molecular Orbitals. Total π-electron Energy of Alternant Hydrocarbons, Chem. Phys. Lett., 17, (1972), 535-538.##H. Hua, Z. Miao, The Total Eccentricity Sum of Non-adjacent Vertex Pairs in Graphs, Bull. Malays. Math. Sci. Soc., 42(3), (2019), 947-963.##H. Hua, S. Zhang, Relations Between Zagreb Coindices and Some Distance-based Topological Indices, MATCH Commun. Math. Comput. Chem., 68, (2012), 199-208.##A. Ilić, I. Gutman, Eccentric Connectivity Index of Chemical Trees, MATCH Commun. Math. Comput. Chem., 65, (2011), 731-744.##Z. Kovijanić Vukićević, G. Popivoda, Chemical Trees with Extreme Values of Zagreb Indices and Coindices, Iranian J. Math. Chem., 5(1), (2014), 19-29.##X. Qi, Z. Du, On Zagreb Eccentricity Indices of Trees, MATCH Commun. Math. Comput. Chem., 78, (2017), 241-256.##R. Rasi, S. M. Sheikholeslami, A. Behmaram, Trees with Extreme Values of Second Zagreb Index and Coindex, Math. Interdisc. Res., 4(2), (2019), 227-238.##V. Sharma, R. Goswami, A. K. Madan, Eccentric Connectivity Index: a Novel Highly Discriminating Topological Descriptor for Structure-property and Structure-activity Studies, J. Chem. Inf. Comput. Sci., 37, (1997), 273-282.##M. Tavakoli, F. Rahbarnia, A. R. Ashrafi, Note on Strong Product of Graphs, Kragujevac J. Math., 37(1), (2013), 187-193.##D. Vukičević, A. Graovac, Note on the Comparison of the First and Second Normalized Zagreb Eccentricity Indices, Acta Chim. Slov., 57, (2010), 524-538.## ##

(C; C')-Controlled g-Fusion Frames in Hilbert Spaces 2 2 Controlled frames in Hilbert spaces have been recently introduced by P. Balazs and etc. for improving the numerical efficiency of interactive algorithms for inverting the frame operator. In this paper we develop a theory based on g-fusion frames on Hilbert spaces, which provides exactly the frameworks not only to model new frames on Hilbert spaces but also for deriving robust operators. In particular, we can define analysis, synthesis and frame operators with representation space compatible for (C,C')-Controlled g-fusion frames, which even yield a reconstruction formula. Also, some useful concepts such as Q-dual and perturbation are introduced and investigated. 179 191 2019/02/22017/12/132018/08/132018/07/62018/10/272019/02/12019/01/232018/04/22019/02/32018/10/142019/02/52019/02/10 1397/11/21 2020/05/302023/02/272023/02/272019/05/82019/07/122020/01/182019/11/282021/04/82020/02/72019/08/312020/05/302021/04/8 1400/1/19 Habib shakoory Habib shakoory Department of Mathematics, Shabestar Branch, Islamic Azad University Shabestar, Iran Iran habibshakoory@yahoo.com Reza ahmadi Reza ahmadi Research Institute for Fundamental Sciences, University of Tabriz, Tabriz, Iran Iran rahmadi@tabrizu.ac.ir Gholamreza Rahimlou Gholamreza Rahimlou Department of Mathematics, Faculty of Tabriz Branch, Technical and Vocational University (TVU), East Azarbaijan, Iran Iran grahimlou@gmail.com Vahid Sadri Vahid Sadri Department of Mathematics, Faculty of Tabriz Branch, Technical and Vocational University (TVU), East Azarbaijan, Iran Iran vahidsadri57@gmail.com G-fusion frame Controlled fusion frame Controlled g-fusion frame Q-dual. P. Balazs, J. P. Antoine, A. Grybos, Weighted and Controlled Frames: Mutual Relationship and First Numerical Properties, Int. J. Wavelets Multiresolut. Inf. Process., 8(1), (2010), 109-132.##I. Bogdanova, P. Vandergheynst, J. P. Antoine, L. Jacques, M. Morvidone, Stereographic Wavelet Frames on the Sphere, Appl. Comput. Harmon. Anal., 19, (2005), 223-252.##P. G. Casazza, G. Kutyniok, Finite Frames, Theory and Applications, Applied and Numerical Harmonic Analysis, Boston, Birkhauser, 2013.##R. G. Douglas, On Majorization, Factorization and Range Inclusion of Operators on Hilbert Spaces, Proc Amer. Math. Soc., 17(2), (1996), 413-415.##R. J. Duffin, A. C. Schaeffer, A Class of Nonharmonic Fourier Series, Trans. Amer. Math. Soc., 72(1), (1952), 341-366.##P. Găvrut¸a, On the Duality of Fusion Frames, J. Math. Anal. Appl., 333, (2007), 871-879.##S. Heineken, E. Matusiak, V. Paternostro, Perturbed Frame Sequences: Canonical Dual Systems, Approximate Reconstructions and Applications, Int. J. Wavelets Multiresolut. Inf. Process., 12(2), (2014).##D. Hua, Y. Huang, Controlled K-g-frames in Hilbert Spaces, Results. Math., 72(3), (2016), 1227-1238.##A. Khosravi, K. Musazadeh, Controlled Fusion Frames, Methods Funct. Anal. Top., 18(3), (2012), 256-265.##K. Musazadeh, H. Khandani, Some Results on Controlled Frames in Hilbert Spaces, Acta Math. Sci., 36B(3), (2016), 655-665.##A. Rahimi, A. Fereydooni, Controlled G-frames and Their G-multipliers in Hilbert Spaces, An. St. Univ. Ovidius Constanta., 21(2), (2013), 223-236.##A. Rahimi, S. Najafzadeh, M. Nouri, Controlled K-frames and Their Invariance under Compact Perturbation, arXiv:1602.03982, (2016).##V. Sadri, Gh. Rahimlou, R. Ahmadi, R. Zarghami Farfar, Generalized Fusion Frames in Hilbert Spaces, Submitted, (2018).## ##

On Fejér Type Inequalities for (η1,η2)-Convex Functions 2 1 ‎ 2 In this paper we find a characterization type result for (η1,η2)-convex functions. The Fejér integral inequality related to (η1,η2)-convex functions is obtained as a generalization of Fejér inequality related to the preinvex and η-convex functions. Also some Fejér trapezoid and midpoint type inequalities are given in the case that the absolute value of the derivative of considered function is (η1,η2)-convex. 193 210 2019/02/22017/12/132018/08/132018/07/62018/10/272019/02/12019/01/232018/04/22019/02/32018/10/142019/02/52019/02/102019/02/13 1397/11/24 2020/05/302023/02/272023/02/272019/05/82019/07/122020/01/182019/11/282021/04/82020/02/72019/08/312020/05/302021/04/82021/07/3 1400/4/12 M. Rostamian Delavar M. Rostamian Delavar Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, Bojnord, Iran Iran m.rostamian@ub.ac.ir S. Mohammadi Aslani S. Mohammadi Aslani Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran Iran soraya.mohammadi.aslani@gmail.com S. M. Vaezpour S. M. Vaezpour Department of Mathematics and Computer Sciences, Amirkabir University of Technology, 242 Hafez Ave, Tehran, Iran Iran vaez@aut.ac.ir Generalized convex function Fejér inequality Trapezoid inequality Mid-point inequality. A. Ben-Israel, B. Mond, What is Invexity?, Jornal of Australian Mathematical Society, Ser. B., 28, (1986), 1-9.##M. Eshaghi Gordji, S. S. Dragomir, M. Rostamian Delavar, An Inequality Related to η-convex Functions (II), International Journal of Nonlinear Analysis and Applications, 6(2), (2016), 26-32.##M. Eshaghi Gordji, M. Rostamian Delavar, M. De La Sen, On φ-convex Functions, Journal of Mathematical Inequalities, 10(1), (2016), 173-183.##L. Fejér, Über die Fourierreihen, II, Math. Naturwise. Anz Ungar. Akad. Wiss., 24, (1906), 369-390.##C. Fulga, V. Preda, Nonlinear Programming with E-preinvex and Local E-preinvex Functions, European Journal of Operational Research, 192, (2009), 737-743.##S. Guo, Y-M. Chu, G. Farid, S. Mehmood, W. Nazeer, Fractional Hadamard and Fejér-Hadamard Inequalities Associated with Exponentially (s, m)-convex Functions, Journal of Function Spaces, 2020, (2020), Art. ID 2410385, 10 pp.##M. A. Hanson, B. Mond, Convex Transformable Programming Problems and Invexity, Journal of Information and Optimization Sciences, 8, (1987), 201-207.##Hwang, D-Y, Some Inequalities for Differentiable Convex Mapping with Application to Weighted Trapezoidal Formula and Higher Moments of Random Variables, Appl. Math. Comput., 217(23), (2011), 9598-9605.##W. Jeyakumar, Strong and Weak Invexity in Mathematical Programming, European Journal of Operational Research, 55, (1985), 109-125.##C. Y. Jung, M. Yussouf, Y-M. Chu, G. Farid, S. M. Kang, Generalized Fractional Hadamard and Fejér-Hadamard Inequalities for Generalized Harmonically Convex Functions, Journal of Mathematics, 2020, (2020), Article ID 8245324, 13 pp.##Y. Khurshid, M. Adil Khan, Y-M Chu, Conformable Integral Version of HermiteHadamard-Fejér Inequalities via η-convex Functions, AIMS Mathematics, 5(5), 2020, 5106-5120.##Y. Khurshid, M. Adil Khan, Y.-M. Chu, Z. A. Khan, Hermite-Hadamard-Fejér Inequalities for Conformable Fractional Integrals via Preinvex Functions, J. Funct. Spaces, 2019, (2019), Article ID 3146210, 9 pages.##M. A. Latif, S. S. Dragomir, New Inequalities of Hermite-Hadamard and Fejér Type via Preinvexity, Journal of Computational Analysis and Applications, 19(4), (2015), 725-739.##M. Mat loka, Inequalities for h-preinvex Functions, Applied Mathematics and Computation, 234, (2014), 52-57.##S. Mohammadi Aslani, M. Rostamian Delavar, S. M. Vaezpour, Inequalities of Fejér Type Related to Generalized Convex Functions with Applications, International Journal of Analysis and Applications, 16(1), (2018), 38-49.##S. R. Mohan, S. K. Neogy, On Invex Sets and Preinvex Function, J. Math. Anal. Appl., 189, (1995), 901-908.##M. Rostamian Delavar, On Fejérs Inequality: Generalizations and Applications, J Inequal Appl., 2023(42), (2023).##M. Rostamian Delavar, Manuel De La Sen, Difference Mappings Associated with Nonsymmetric Monotone Types of Fejér’s Inequality, Mathematics, 7(9), (2019), 1-11.##M. Rostamian Delavar, M. De La Sen, On Generalization of Fejér Type Inequalities, Communications in Mathematics and Applications, 8(1), (2017), 31-43.##M. Rostamian Delavar, M. De La Sen, Some Generalizations of Hermite-Hadamard Type Inequalities, SpringerPlus, 2016, 5:1661.##M. Rostamian Delavar, S. S. Dragomir, On η-convexity, Mathematical Inequalities and Applications, 20, (2017), 203-216.##M. Z. Sarikaya, On New Hermite Hadamard Fejér Type Integral Inequalities, Stud. Univ. Babes-Bolyai Math., 57(3), (2012), 377-386.## ##

Inverse and Reverse 2-facility Location Problems with Equality Measures on a Network 2 2 In this paper we consider the inverse and reverse network facility location problems with considering the equity on servers. The inverse facility location with equality measure deals with modifying the weights of vertices with minimum cost, such that the difference between the maximum and minimum weights of clients allocated to the given facilities is minimized. On the other hand, the reverse case of facility location problem with equality measure considers modifying the weights of vertices with a given budget constraint, such that the difference between the maximum and minimum weights of vertices allocated to the given facilities is reduced as much as possible. Two algorithms with time complexity O(nlogn) are presented for the inverse and reverse 2-facility location problems with equality measures. Computational results show their superiority with respect to the linear programming models. 211 225 2019/02/22017/12/132018/08/132018/07/62018/10/272019/02/12019/01/232018/04/22019/02/32018/10/142019/02/52019/02/102019/02/132019/02/19 1397/11/30 2020/05/302023/02/272023/02/272019/05/82019/07/122020/01/182019/11/282021/04/82020/02/72019/08/312020/05/302021/04/82021/07/32021/02/6 1399/11/18 Morteza Nazari Morteza Nazari Faculty of Mathematical Sciences, Shahrood University of Technology, University Blvd., Shahrood, Iran Iran mnazari_ms65@yahoo.com Jafar Fathali Jafar Fathali Faculty of Mathematical Sciences, Shahrood University of Technology, University Blvd., Shahrood, Iran Iran fathali@shahroodut.ac.ir Inverse facility location Reverse facility location Balanced allocation Equality measure. B. Alizadeh, R. E. Burkard, U. Pferschy, Inverse 1-center Location Problems with Edge Length Augmentation on Trees, Computing, 86, (2009), 331-343.##B. Alizadeh, R. E. Burkard, Combinatorial Algorithms for Inverse Absolute and Vertex 1-center Location Problems on Trees, Networks, 58, (2011), 190-200.##M. Barbati, C. Piccolo, Equality Measures Properties for Location Problems, Optimization Letters, 10, (2015), 903-920.##J. E. Beasley, OR-Library: Distributing Test Problems by Electronic Mail, Journal of the Operational Research Society, 41, (1990), 1069-1072.##O. Berman, Z. Drezner, A. Tamir, G. O. Wesolowsky, Optimal Location with Equitable Loads, Annals of Operations Research, 167, (2009), 307-325.##O. Berman, D. I. Ingco, A. Odoni, Improving the Location of Minisum Facilities through Network Modification, Annals of Operations Research, 40, (1992), 1-16.##O. Berman, D. I. Ingco, A. Odoni, Improving the Location of Minimax Facilities through Network Modification, Networks, 24, (1994), 31-41.##F. Baroughi Bonab, R. E. Burkard, E. Gassner, Inverse p-median Problems with Variable Edge Lengths, Mathematical Methods of Operations Research, 73, (2011), 263-280.##R. E. Burkard, E. Gassner, J. Hatzl, A Linear Time Algorithm for the Reverse 1-median Problem on a Cycle, Networks, 48, (2006), 16-23.##R. E. Burkard, E. Gassner, J. Hatzl, Reverse 2-median Problem on Trees, Discrete Applied Mathematics, 156, (2008), 1963-1976.##R. E. Burkard, C. Pleschiutschnig, J. Z. Zhang, The Inverse 1-median Problem on a Cycle, Discrete Optimization, 5, (2008), 242-253.##R. E. Burkard, C. Pleschiutschnig, J. Z. Zhang, Inverse Median Problems, Discrete Optimization, 1, (2004), 23-39.##M. C. Cai, X. G. Yang, J. Zhang, The Complexity Analysis of the Inverse Center Location Problem, Journal of Global Optimization, 15, (1999), 213-218.##H. A. Eiselt, G. Laporte, Objectives in Location Problems, In: Facility Location: A Survey of Applications and Methods. Ed.: Drezner Z. Springer, Berlin, (1995), 151-180.##J. Fathali, A Row Generation Method for the Inverse Continuous Facility Location Problem, Computers & Industrial Engineering, 171, (2022), 108482.##J. Fathali, M. Zaferanieh, The Balanced 2-median and 2-maxian Problems on a Tree, Journal of Combinatorial Optimization, 45, (2023), 69.##M. Galavii, The Inverse 1-median Problem on a Tree and on a Path, Electronic Notes in Discrete Mathematics, 36, (2010), 1241-1248.##M. Gavalec, O. Hudec, Balanced Location on a Graph, Optimization, 35, (1995), 367-372.##X. C. Guan, B. W. Zhang, Inverse 1-median Problem on Trees under Weighted Hamming Distance, Journal of Global Optimization, 54, (2012), 75-82.##M. Landete, A. Marin, Looking for Edge-equitable Spanning Trees, Computers & Operations Research, 41, (2014), 44-52.##M. A. Lejeune, S. Y. Prasad, Effectiveness-equity Models for Facility Location Problems on Tree Networks, Networks, 62, (2013), 243-254.##A. Marin, The Discrete Facility Location Problem with Balanced Allocation of Customers, European Journal of Operational Research, 210, (2011), 27-38.##M. T. Marsh, D. A. Schilling, Equity Measurement in Facility Location Analysis: a Review and Framework, European Journal of Operational Research, 74, (1994), 1-17.##M. Nazari, J. Fathali, Reverse Backup 2-median Problem with Variable Coordinate of Vertices, Journal of Operational Research and Its Applications, 15, (2018), 63-88.##M. Nazari, J. Fathali, M. Nazari, S. V. Koulaei, Inverse of Backup 2-median Problems with Variable Edge Lengths and Vertex Weight on Trees and Variable Coordinates on the Plane, Production and Operations Management, 9, (2018), 115-137.##K. T. Nguyen, Reverse 1-center Problem on Weighted Trees, Optimization, 654, (2016), 253-264.##S. Omidi, J. Fathali, M. Nazari, Inverse and Reverse Balanced Facility Location Problems with Variable Edge Lengths on Trees, OPSEARCH, 57, (2020), 261-273.##L. Wu, J. Lee, J. Zhang, Q. Wang, The Inverse 1-median Problem on Tree Networks with Variable Real Edge Lengths, Mathematical Problems in Engineering, Volume 2013, Article ID 313868, 6 pages, http://dx.doi.org/10.1155/2013/313868.##J. Zhang, Z. Liu, Z. Ma, Some Reverse Location Problems, European Journal of Operational Research, 124, (2000), 77-88.## ##

ABSTRACTS IN PERSIAN Vol. 18, No. 1 2 2 Please see the full text contains the Persian abstracts of this volume. 227 241 2019/02/22017/12/132018/08/132018/07/62018/10/272019/02/12019/01/232018/04/22019/02/32018/10/142019/02/52019/02/102019/02/132019/02/192023/05/2 1402/2/12 2020/05/302023/02/272023/02/272019/05/82019/07/122020/01/182019/11/282021/04/82020/02/72019/08/312020/05/302021/04/82021/07/32021/02/62023/05/2 1402/2/12 The Name of Authors in this Volume The Name of Authors in this Volume Academic Center for Education, Culture and Research (ACECR) Iran ma.hoseinzade@gmai.com ABSTRACTS PERSIAN Vol. 18 No. 1. All references of the papers in Vol 18, No 1## ##