A Topology Generated by ψ-Operation and Ideal Spaces 2 2 In this paper, we introduce the notion of ψ-operation on an m-space (X, m). By using the ψ-operation, for every subset A of X, we define the ψ-operation Aψ(I, m) with respect to an ideal I and an m-structure m and investigate their properties. Moreover, on the m we introduce and investigate the notion of ψ-compatibility with an ideal I. 1 11 2020/03/21 1399/1/2 2022/01/14 1400/10/24 Ahmad Al-Omari Ahmad Al-Omari Al al-Bayt University, Faculty of Sciences, Department of Mathematics P.O. Box 130095, Mafraq 25113, Jordan Jordan omarimutah1@yahoo.com Takashi Noiri Takashi Noiri 2949-1 Shiokita-cho, Hinagu, Yatsushiro-shi, Kumamoto-ken, 869-5142 Japan Japan t.noiri@nifty.com Ideal m-space ψ-operation ψ-open m-open m_{ψ}^{∗)-open ψ-compatible. A. Al-Omari, T. Noiri, On Ψ∗-operator in Ideal m-spaces, Bol. Soc. Paran. Mat. (3s.), 30(1), (2012), 53-66.##A. Al-Omari, T. Noiri, Local Closure Functions in Ideal Topological Spaces, Novi Sad J. Math., 43(2), (2013), 139-149.##A. Al-Omari, T. Noiri, On Operators in Ideal Minimal Spaces, Mathematica, 58(81)(1-2), (2016), 3- 13.##A. Al-Omari, T. Noiri, A Note on Topologies Generated by m-structures and ω-topologies, Commun. Fac. Sci. Univ. Ank. Series A1, 67(1), (2018), 141-146.##A. Al-Omari, H. Al-Saadi, A Topology via ω-local Functions in Ideal Spaces, Mathematica, 60(83)(2), (2018), 103- 110.##A. Al-Omari, T. Noiri, Operators in Minimal Spaces with Hereditary Classes, Mathematica, 61(84)(2), (2019), 101-110.##A. Al-Omari, T. Noiri, Weakly Φ-continuous Functions in Grill Topological Spaces, Hacettepe J. Math. Stat., 41(6), (2012), 785-793.##A. Al-Omari, T. Noiri, On ΨeG-sets in Grill Topological Spaces, Filomat, 25(1), (2011), 187 - 196.##D. Jankovic, T. R. Hamlett, New Topologies from Old via Ideals, Amer. Math. Monthly, 97(4), (1990), 295-310.##K. Kuratowski, Topology, Vol. I, Academic Press, New York, 1966.##V. Popa, T. Noiri, On M-continuous Functions, An. Univ. Dunarea de Jos Galati, Ser. Mat. Fiz. Mec. Teor. (2), 18(23), (2000), 31-41.## ##

An Advanced Numerical Approach To Solve Viscous Flow Via Modified Generalized Laguerre Functions 2 1 2 This paper presents an advanced numerical approach that applies the quasilinearization method (QLM) and collocation method (CM) based on modified generalized Laguerre functions (MGLFs) to solve a nonlinear system of ordinary differential equations governing viscous flow with heat transfer and magnetic fields on a semi-infinite domain. We demonstrate the effectiveness and accuracy of the proposed method by comparing it with previous well-known methods. The results show that the proposed method provides an effcient and accurate solution to the problem. 13 41 2020/03/212020/03/27 1399/1/8 2022/01/142024/06/13 1403/3/24 Zeinab Hajimohammadi Zeinab Hajimohammadi Department of Computer Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran Iran Z_Hajimohammadi@sbu.ac.ir Kourosh Parand Kourosh Parand Department of Computer Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran Iran k_parand@sbu.ac.ir Aida Pakniyat Aida Pakniyat Department of Computer Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran Iran aida.pakniyat17@gmail.com Viscous Flow Non-linear Stretching Modified Generalized Laguerre Functions Quasi Linearization Collocation Method System of Nonlinear Ordinary Differential Equations. P. Agarwal, F. Qi, M. Chand, Sh. Jain, Certain Integrals Involving the Generalized Hypergeometric Function and the Laguerre Polynomials, Journal of Computational and Applied Mathematics, 313, (2017), 307-317.##F. T. Akyildiz, H. Bellout, K. Vajravelu, Diffusion of Chemically Reactive Species in a Porous Medium Over a Stretching Sheet, Journal of mathematical analysis and applications, 320(1), (2006), 322-339.##U. M. Ascher, L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-algebraic Equations, Siam, 1998.##F. B. Babolghani, K. Parand, Comparison Between Hermite and Sinc Collocation Methods for Solving Steady Slow of a Third Grade Fluid in a Porous Half Space, Proceedings of the International Conference on Scientific Computing, (2013), 124.##A. H. Bhrawy, A. S. Alofi, A Jacobi-Gauss Collocation Method for Solving Nonlinear Lane-Emden Type Equations, Communications in Nonlinear Science and Numerical Simulation, 17(1), (2012), 62-70.##J. Boyd, Chebyshev and Fourier Spectral Methods, Dover Publications Inc., New York, 2000.##J. Boyd, C. Rangan, P. Bucksbaum, Pseudospectral Methods on a Semi-Infinite Interval With Application to the Hydrogen Atom: a Comparison of the Mapped Fourier-Sine Method With Laguerre Series and Rational Chebyshev Expansions, Journal of Computational Physics, 188(1), (2003), 56-74.##O. Cherroud, S. A. Yahiaoui, M. Bentaiba, Generalized Laguerre Polynomials With Position-Dependent Effective Mass Visualized via Wigner's Distribution Functions, Journal of Mathematical Physics, 58(6), (2017), 063503.##C. Christov, A Complete Orthonormal System of Functions in L^2(-∞,∞) Space, SIAM Journal on Applied Mathematics, 42(6), (1982), 1337-1344.##F. Comte, Ch. Cuenod, M. Pensky, Y. Rozenholc, Laplace Deconvolution on the Basis of Time Domain Data and Its Application to Dynamic Contrast-Enhanced Imaging, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79(1), (2017), 69-94.##O. R. 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Shen, Zh Wang, A Rational Approximation and Its Applications to Differential Equations on the Half Line, Journal of scientific computing, 15(2), (2000), 117-147.##B. Guo, J. Shen, C. Xu, Generalized Laguerre Approximation and Its Applications to Exterior Problems, Journal of computational Mathematics, (2005), 113-130.##E. Hairer, S. P. Nørsett, G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems, Springer Series in Computational Mathematics, 8, (1993).##Z. Hajimohammadi, F. Baharifard, A. Ghodsi, K. Parand, Fractional Chebyshev Deep Neural Network (FCDNN) for Solving Differential Models, Chaos, Solitons & Fractals, 153, (2021), 111530.##Z. Hajimohammadi, K. Parand, A New Numerical Solution for Solving the Flow of Eyring-Powell Fluid Problem via Fractional Rational Generalized Laguerre Polynomials, 48th Annual Iranian Mathematics Conference, 2017.##G. Hojjati, K. Parand, An Effcient Computational Algorithm for Solving the Nonlinear Lane-Emden Type Equations, Int. J. Math. Comp. Sci, 7(4), (2011), 182-187.##S. Kazem, J. A. Rad, K. Parand, M. Shaban, H. Saberi, The Numerical Study on the Unsteady Flow of Gas in a Semi-Infinite Porous Medium Using an RBF Collocation Method, International Journal of Computer Mathematics, 89(16), (2012), 2240-2258.##S. A. Kechil, I. Hashim, Series Solution of Flow Over Nonlinearly Stretching Sheet With Chemical Reaction and Magnetic Field, Physics Letters A, 372(13), (2008), 2258-2263.##Ch. Lin, Application of a V-belt Continuously Variable Transmission System by Using a Composite Recurrent Laguerre Orthogonal Polynomial Neural Network Control System and Modified Particle Swarm Optimization, Journal of Vibration and Control, 23(9), (2017), 1437-1462.##Ch. Lin, Hybrid Recurrent Laguerre-Orthogonal-Polynomials Neural Network Control With Modified Particle Swarm Optimization Application for V-belt Continuously Variable Transmission System, Neural Computing and Applications, 28(2), (2017), 245-264.##F. Liu, H. Li, Zh. Wang, Spectral Methods Using Generalized Laguerre Functions for Second and Fourth Order Problems, Numerical Algorithms, 75(4), (2017), 1005-1040.##F. J. Liu, Z. Q. Wang, H. Y. Li, A Fully Diagonalized Spectral Method Using Generalized Laguerre Functions on the Half Line, Advances in Computational Mathematics, 43(6), (2017), 1227-1259.##R. W. MacCormack, A Numerical Method for Solving the Equations of Compressible Viscous Flow, AIAA journal, 20(9), (1982), 1275-1281.##A. M. Makarenkov, E. V. Seregina, M. A. Stepovich, The Projection Galerkin Method for Solving the Time-Independent Differential Diffusion Equation in a Semi-Infinite Domain, Computational Mathematics and Mathematical Physics, 57(5), (2017), 802-814.##S. Mall, S. Chakraverty, Application of Legendre Neural Network for Solving Ordinary Differential Equations, Applied Soft Computing, 43, (2016), 347-356.##S. Mehrkanoon, T. Falck, J. A. K. Suykens, Approximate Solutions to Ordinary Differential Equations Using Least Squares Support Vector Machines, IEEE Trans. Neural Netw. Learn. Syst., 23(9), (2012), 1356-1367.##I. S. Oyelakin, R. Ghosh, S. Mondal, P. Sibanda, Entropy Generation in Casson Nanofluid Flow Past an Electromagnetic Stretching Riga Plate, Malaysian Journal of Mathematical Sciences, 15(3), (2021), 425-445.##I. S. Oyelakin, P. C. Lalramneihmawii, S. Mondal, P. Sibanda, Analysis of DoubleDiffusion Convection on Three-Dimensional MHD Stagnation Point Flow of a Tangent Hyperbolic Casson Nanofluid, International Journal of Ambient Energy, 43(1), (2020), 1854-1865.##I. S. Oyelakin, P. Mondal, S. Mondal, T. R. Mahapatra, P. Sibanda, Rheological Analysis of Suspended Single-Walled Carbon Nanotubes in a Walters' B Fluid, Nanoscience & Nanotechnology-Asia, 11(6), (2021), 87-95.##I. S. Oyelakin, PC. Lalramneihmawii, S. Mondal, S.K. Nandy, P. Sibanda, Thermophysical Analysis of Three‐Dimensional Magnetohydrodynamic Flow of a Tangent Hyperbolic Nanofluid, Engineering Reports, 2(4), (2017), e12144.##I. S. Oyelakin, S. Mondal, P. Sibanda, Unsteady Casson Nanofluid Flow Over a Stretching Sheet With Thermal Radiation, Convective and Slip Boundary Conditions, Alexandria Engineering Journal, 55(2), (2016), 1025-1035.##I. S. Oyelakin, S. Mondal, P. Sibanda, Unsteady Mixed Convection in Nanofluid Flow Through a Porous Medium with Thermal Radiation Using the Bivariate Spectral Quasilinearization Method, Journal of Nanofluids, 6(2), (2020), 273-281.##I. S. Oyelakin, S. Mondal, P. Sibanda, Nonlinear Radiation in Bioconvective Casson Nanofluid Flow, International Journal of Applied and Computational Mathematics, 5(5), (2019), 1-20.##I. S. Oyelakin, S. Mondal, P. Sibanda, S. S. MOTSA, A Multi-Domain Bivariate Approach for Mixed Convection in a Casson Nanofluid with Heat Generation, Walailak Journal of Science and Technology, 16(9), (2019), 681-699.##R. K. Pandey, N. Kumar, Solution of Lane-Emden Type Equations Using Bernstein Operational Matrix of Differentiation, New Astronomy, 17(3), (2012), 303-308.##K. Parand, A. A. Aghaei, S. Kiani, T. IlkhasZadeh, Z. Khosravi, A Neural Network Approach for Solving Nonlinear Differential Equations of Lane-Emden Type, Engineering with Computers, 40, (2024), 953-969.##K. Parand, A. A. Aghaei, M. Jani, A. Ghodsi, A New Approach to the Numerical Solution of Fredholm Integral Equations Using Least Squares-Support Vector Regression, Math. Comput. Simul., 180, (2021), 114-128.##K. Parand, F. B. Babolghani, Applying the Modified Generalized Laguerre Functions for Solving Steady Flow of a Third Grade Fluid in a Porous Half Space, World Applied Science Journal, 17(4), (2012), 467-472.##K. Parand, F. B. Babolghani, Modified Generalized Laguerre Functions for a Numerical Investigation of Flow and Diffusion of Chemically Reactive Species Over a Nonlinearly Stretching Sheet, World Applied Sciences Journal, 17(12), (2012), 1578-1587.##K. Parand, F. Baharifard, F. B. Babolghani, Comparison Between Rational Gegenbauer and Modified Generalized Laguerre Functions Collocation Methods for Solving the Case of Heat Transfer Equations Arising in Porous Medium, International Journal of Industrial Mathematics, 4(2), (2012), 107-122.##K. Parand, M. Dehghan, A. Pirkhedri, Sinc-Collocation Method for Solving the Blasius Equation, Physics Letters A, 373(44), (2009), 4060-4065.##K. Parand, M. Dehghan, A. Pirkhedri, The Use of Sinc-Collocation Method for Solving Falkner-Skan Boundary-Layer Equation, International journal for numerical methods in fluids,68(1), (2012), 36-47.##K. Parand, M. Dehghan, A. Pirkhedri, The Sinc-Collocation Method for Solving the Thomas-Fermi Equation, Journal of Computational and Applied Mathematics, 237(1), (2013), 244-252.##K. Parand, M. Dehghan, A. Taghavi, Modified Generalized Laguerre Function Tau Method for Solving Laminar Viscous Flow, International Journal of Numerical Methods for Heat & Fluid Flow, 20(7), (2010), 728-743.##K. Parand, Z. Delafkar, N. Pakniat, A. Pirkhedri, M. K. Haji, Collocation Method Using Sinc and Rational Legendre Functions for Solving Volterra's Population Model, Communications in Nonlinear Science and Numerical Simulation, 16(4), (2011), 1811-1819.##K. Parand, Z. Hajimohammadi, Using Modified Generalized Laguerre Functions, QLM and Collocation Method for Solving an Eyring-Powell Problem, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 40(4), (2018), 182.##K. Parand, M. Hemami, Application of Meshfree Method Based on Compactly Supported Radial Basis Function for Solving Unsteady Isothermal Gas Through a Micro-Nano Porous Medium, Iranian Journal of Science and Technology, Transactions A: Science, 41(3), (2017), 677-684.##K. Parand, S. Khaleqi, The Rational Chebyshev of Second Kind Collocation Method for Solving a Class of Astrophysics Problems, The European Physical Journal Plus, 131(2), (2016), 1-24.##K. Parand, Y. Lotfi, J. A. Rad, An Accurate Numerical Analysis of the Laminar Two Dimensional Flow of an Incompressible Eyring-Powell Fluid Over a Linear Stretching Sheet, The European Physical Journal Plus, 132(9), (2017), 397.##K. Parand, M. M. Moayeri, S. Latifi, M. Delkhosh, A Numerical Investigation of the Boundary Layer Flow of an Eyring-Powell Fluid Over a Stretching Sheet via Rational Chebyshev Functions, The European Physical Journal Plus, 132(7), (2017), 325.##K. Parand, M. Nikarya, J. A. Rad, F. Baharifard, A New Reliable Numerical Algorithm Based on the First Kind of Bessel Functions to Solve Prandtl-Blasius Laminar Viscous Flow Over a Semi-Infinite Flat Plate, Zeitschrift für Naturforschung A, 67(12), (2012), 665-673.##K. Parand, M. Razzaghi, Rational Chebyshev Tau Method for Solving Higher-Order Ordinary Differential Equations, International Journal of Computer Mathematics, 81(1), (2004), 73-80.##K. Parand, M. Razzaghi, Rational Chebyshev Tau Method for Solving Volterra's Population Model, Applied Mathematics and Computation, 149(3), (2004), 893-900.##K. Parand, M. Razzaghi, Rational Legendre Approximation for Solving Some Physical Problems on Semi-Infinite Intervals, Physica Scripta,69(5), (2004), 353.##K. Parand, M. Razzaghi, R. Sahleh, M. Jani, Least Squares Support Vector Regression for Solving Volterra Integral Equations, Engineering with Computers, 38, (2022), 789-796.##K. Parand, A. R. Rezaei, A. Taghavi, Lagrangian Method for Solving Lane-Emden Type Equation Arising in Astrophysics on Semi-Infinite Domains, Acta Astronautica, 67(7-8), (2010), 673-680.##K. Parand, M. Shahini, Rational Chebyshev Pseudospectral Approach for Solving Thomas-Fermi Equation, Physics Letters A, 373(2), (2009), 210-213.##K. Parand, M. Shahini, A. Taghavi, Generalized Laguerre Polynomials and Rational Chebyshev Collocation Method for Solving Unsteady Gas Equation, Int. J. Contemp. Math. Sci, 4(21),(2009), 1005-1011.##K. Parand, A. Taghavi, Generalized Laguerre Polynomials Collocation Method for Solving Lane-Emden Equation, Applied Mathematical Sciences, 2(60), (2008), 2955-2961.##K. Parand, A. Taghavi, Rational Scaled Generalized Laguerre Function Collocation Method for Solving the Blasius Equation, Journal of Computational and Applied Mathematics, 233(4), (2009), 980-989.##K. Parand, A. Taghavi, M. Shahini, Comparison Between Rational Chebyshev and Modified Generalized Laguerre Functions Pseudospactral Methods for Solving Lane-Emden and Unsteady Gas Equations, Acta physica polonica B, 40(6), (2009).##J. A. Rad, K. Parand, S. Kazem, A Numerical Investigation to Viscous Flow Over Nonlinearly Stretching Sheet With Chemical Reaction, Heat Transfer and Magnetic Field, International Journal of Applied and Computational Mathematics, 3(2), (2017), 919-935.##B. C. Sakiadis, Boundary-Layer Behavior on Continuous Solid Surfaces: I. BoundaryLayer Equations for Two-Dimensional and Axisymmetric Flow, AIChE Journal, 7(1), (1961), 26-28.##B. C. Sakiadis, Boundary-Layer Behavior on Continuous Solid Surfaces: II. The Boundary Layer on a Continuous Flat Surface", AiChE journal, 7(2), (1961), 221-225.##K. M. Sanni, S. Asghar, M. Jalil, N. F. Okechi, Flow of Viscous Fluid Along a Nonlinearly Stretching Curved Surface, Results in Physics, 7, (2017), 1-4.##I. I. Sharapudinov, Special Series in Laguerre Polynomials and Their Approximation Properties, Siberian Mathematical Journal, 58(2), (2017), 338-362.##J. Shen, Stable and Effcient Spectral Methods in Unbounded Domains Using Laguerre Functions, SIAM Journal on Numerical Analysis, 38(4), (2000), 1113-1133.##Y. Shi, Ch. Tian, Zh. Liu, Ch Liang, A New Laguerre-Based Discontinuous Galerkin Time-Domain Method, Microwave and Optical Technology Letters, 59(7), (2017), 1499-1503.##H. Siyyam, Laguerre Tau Methods for Solving Higher-Order Ordinary Differential Equations", Journal of Computational Analysis and Applications, 3(2), (2001), 173-182.##J. Stoer, R. Bulirsch, Introduction to numerical analysis, Springer Science & Business Media, 12, (2013).##J. A. K. Suykens, J. Vandewalle, Least Squares Support Vector Machine Classifiers, Neural Process. Lett.,9(3), (1999), 293-300.##T. Tajvidi, M. Razzaghi, M. Dehghan, Modified Rational Legendre Approach to Laminar Viscous Flow Over a Semi-Infinite Flat Plate, Chaos, Solitons & Fractals, 35(1), (2008), 59-66.##F. K. Tsou, E. M. Sparrow, R. Jh. Goldstein, Flow and Heat Transfer in the Boundary Layer on a Continuous Moving Surface, International Journal of Heat and Mass Transfer, 10(2), (1967), 219-235.##N. Vukovi, M. Petrovi, Z. Miljkovi, A Comprehensive Experimental Evaluation of Orthogonal Polynomial Expanded Random Vector Functional Link Neural Networks for Regression, Applied Soft Computing, 70, (2018), 1083-1096.##A. W. Wazwaz, A New Algorithm for Solving Differential Equations of Lane-Emden Type, Applied Mathematics and Computation, 118(2-3), (2001), 287-310.##S. Xu, Y. Li, T. Huang, R. Chan, A Sparse Multiwavelet-Based Generalized Laguerre-Volterra Model for Identifying Time-Varying Neural Dynamics from Spiking Activities, Entropy, 19(8), (2017), 425-446.##A. Yıldırım, T. Öziş , Solutions of Singular IVPs of Lane-Emden Type by Homotopy Perturbation Method, Physics Letters A, 369(1-2), (2007), 70-76.##S. A. Yousefi, Legendre Wavelets Method for Solving Differential Equations of Lane-Emden Type, Applied Mathematics and Computation, 181(2), (2006), 1417-1422.##Ş. Yüzbaşı, A Numerical Method for Solving Second-Order Linear Partial Differential Equations Under Dirichlet, Neumann and Robin Boundary Conditions, International Journal of Computational Methods, 14(02), (2017), 1750015.##Ş. Yüzbaşı, M. Sezer, An Improved Bessel Collocation Method With a Residual Error Function to Solve a Class of Lane-Emden Differential Equations, Mathematical and Computer Modelling, 57(5-6), (2013), 1298-1311.##Ch. Zhang, D. Gu, Zh. Wang, H. Li, Effcient Space-Time Spectral Methods for SecondOrder Problems on Unbounded Domains, Journal of Scientific Computing, 72(2), (2017), 679-699.##Di. Zhang, X. Miao, New Unconditionally Stable Scheme for Telegraph Equation Based on Weighted Laguerre Polynomials, Numerical Methods for Partial Differential Equations, 33(5), (2017), 113-130.##R. Zhang, Zh. Wang, B. Guo, Mixed Fourier-Laguerre Spectral and Pseudospectral Methods for Exterior Problems Using Generalized Laguerre Functions, Journal of Scientific Computing, 36(2), (2008), 263-283.##Z. Ziabakhsh, G. Domairry, H. Bararnia, H. Babazadeh, Analytical Solution of Flow and Diffusion of Chemically Reactive Species Over a Nonlinearly Stretching Sheet Immersed in a Porous Medium, Journal of the Taiwan Institute of Chemical Engineers, 41(1), (2010), 22-28.## ##

Mappings Preserving Sum of Products ab + b ◦ a^∗ on Factor Von Neumann Algebras 2 2 Let A and B be two factor von Neumann algebras. In this paper, we proved that a bijective mapping Φ : A → B satisfies Φ(ab+b◦a∗) = Φ(a)Φ(b)+Φ(b)◦Φ(a)∗ (where ◦ is the special Jordan product on A and B), for all elements a, b ∈ A, if and only if Φ is a ∗-ring isomorphism. In particular, if the von Neumann algebras A and B are type I factors, then Φ is a unitary isomorphism or a conjugate unitary isomorphism. 43 51 2020/03/212020/03/272020/06/23 1399/4/3 2022/01/142024/06/132021/04/3 1400/1/14 João Carlos da Motta Ferreira João Carlos da Motta Ferreira Center for Mathematics, Computation and Cognition, Federal University of ABC, Santa Adélia Street 166, 09210-170, Santo André, Brazil Brazil joao.cmferreira@ufabc.edu.br Maria das Graças Bruno Marietto Maria das Graças Bruno Marietto Center for Mathematics, Computation and Cognition, Federal University of ABC, Santa Adélia Street 166, 09210-170, Santo André, Brazil Brazil graca.marietto@ufabc.edu.br ∗-ring isomorphisms Factor von Neumann algebras. P. Ji, Z. Liu, Additivity of Jordan Maps on Standard Jordan Operator Algebras, Linear Algebra Appl., 430, (2009), 335-343.##C. Li, F. Lu, X. Fang, Nonlinear Mappings Preserving Product XY + Y X∗ on Factor Von Neumann Algebras, Linear Algebra Appl., 438, (2013), 2339-2345.##L. Liu, G. Ji, Maps Preserving Product X∗Y + Y X∗ on Factor Von Newmann Algebras, Linear Algebra Appl., 59, (2011), 951-955.##F. Lu, Additivity of Jordan Maps on Standard Operator Algebras, Linear Algebra Appl., 357, (2002), 123-131.##F. Lu, Multiplicative Mappings of Operator Algebras, Linear Algebra Appl., 347, (2002), 283-291.##W. S. Martindale III, When Are Multiplicative Mappings Additive?, Proc. Amer. Math. Soc., 21, (1969), 695-698.## ##

Image Enhancement and Restoration Approach Based on Anisotropic Diffusion 2 2 We propose a new approach for image enhancement, denoising and restoration, using an anisotropic diffusion based on P-M model and L.V and al. equation, replacing the gradient by motion by mean curvature to detect noise direction for each degraded pixel locally, applying the gradient in Gaussian kernel term to restore the degraded pixels and adding a time term supporting the restoration process. For execution progress, the numerical discretization for the terms of PDE modeling (obtained by the approximation by difference finite volumes finite method, Taylor method and Simpsons improved method), concludes an algorithm treats noised image regardless the noise type (salt-pepper or Gaussian or speckle) better than other filters based whether on anisotropic diffusion or total, shown in the experimental results (using MATLAB program), and demonstrated through PSNR and SSIM. 53 67 2020/03/212020/03/272020/06/232019/12/7 1398/9/16 2022/01/142024/06/132021/04/32025/03/10 1403/12/20 Messaouda Gatcha Messaouda Gatcha Faculty of Exact Sciences and Computers, Applied Automation and Industrial Diagnosis Laboratory, Ziane Achour University, Djelfa, Algeria Algeria messaoudamath2017@gmail.com Farid Messelmi Farid Messelmi Faculty of Exact Sciences and Computers, Development in Mechanics and Materials Laboratory, Ziane Achour University, Djelfa, Algeria Algeria foudimath@yahoo.fr Slami Saadi Slami Saadi Faculty of Exact Sciences and Computers, Applied Automation and Industrial Diagnosis Laboratory, Ziane Achour University, Djelfa, Algeria Algeria saadisdz@gmail.com Image restoration Anisotropic diffusion Regularization Noise filters. L. Alvarez, PL. Lions, JM. Morel, Image Selective Smoothing and Edge Detection by Nonlinear Diffusion, SIAM J. NUMER, 29(3), (1992), 845-866.##T. Barbu, A. Miranville, C. Moro, A Qualitative Analysis and Numerical Simulations of a Nonlinear Second-order Anisotropic Diffusion Problem with Non-homogeneous Cauchy-Neumann Boundary Conditions, Applied Mathematics and Computation, 350, (2019), 170-180.##A. Boucher, Image Processing-spatial Convolution-, www.etud.iro, 2016.##V. Choqueuse, Convolution Product-principe and Propieties, Westerner Britann, 2016.##E. Cuervas, H. Becerra, A. Luque, Anisotropic Diffusion Filtering Through Multiobjective Optimization, Mathematics and Computers in Simulation, 181, (2020), 410-429.##M. Gatcha, F. Messelmi, S. Saadi, An Anisotropic Diffusion Adaptive Filter for Image Denoising and Restoration Applied on Satellite Remote Sensing Images: a Case Study, Engineering Technology & Applied Science Research, 12(6), (2022), 9715-9719.##S. Godounov, V. Riabenki, Differences Schemes (Introduction of the Theory), University publication offce, Algeria, 1987.##S. A. Halim, N. N. Wira, Variations of Diffusion Functions on Perona and Malik Model for Noise Removal, Fourth-order Partial Differ, AIP Conference Proceedings, 2266(1), (2020), 1-8.##J. J. Koenderink, The Structure of Images, Biological Cybernetics, 50, (1984), 363-370.##A. Lanza, S. Morigi, IW. Selesnick, F. Sgallari, Sparsity-inducing Nonconvex Non Separable Regularization for Convex Image, Society for Industrial and Applied Mathematics, 12(2), (2019), 1099-1134.##D. Marr, E. Hildreth, Theory of Edge detection, Proceedings of the Royal Society Series B Biological Sciences, 207(1167), (1980), 187-217.##R. R. Nair, E. David, S. Rajagopal, A Robust Anisotropic Diffusion Filter with Low Arithmetic Complexity for Images, EURASIP Journal on Image and Video Processing, 48, (2019).##P. Perona, J. Malik, Scale-space and Edge Detection Using Anisotropic Diffusion, IEEE Transactions On Pattern Analysis And Machine Intelligence, 12(7), (1990), 629-639.##J. C. Russ,The Image Processing Cookbook, 4th edition, USA, 2017.##N. Thakur, N. U. Khan, S. D. Sharma, A Two Phase Ultrasound Image De-speckling Framework by Nonlocal Means on Anisotropic Diffused Image Data, Informatica, 47(2), (2023), 221-234.##A. Theljani, Z. Belhachmi, M. Moakher, High-order Anisotropic Diffusion Operators in Spaces of Variable Exponents and Application to Image Inpainting and Restoration Problems, Nonlinear Analysis Real World Applications, 47, (2019), 251-271.##J. Xu, Y. Hao, M. Li, X. Zhang, A Novel Variational Model for Image Decomposition, Signal and Image and Video Processing, 13, (2019), 967-974.##H. Zeng, X. Xie, J. Ning, Hyperspectral Image Denoising via Global Spatial-spectral Total Variation Regularized Nonconvex Local Low-rank Tensor Approximation, Signal Processing, 178, (2021).## ##

The General (α, β)-metrics with Quadratic Curvatures 2 2 In this paper, we consider the Riemannian curvature of one of the most important metric in Finsler geometry called general (α, β)- metrics, where α is a Riemannian metric and β is 1-form. We give a complete classification of the metrics to be R-quadratic and Ricci-quadratic. 69 78 2020/03/212020/03/272020/06/232019/12/72020/03/10 1398/12/20 2022/01/142024/06/132021/04/32025/03/102022/01/14 1400/10/24 Mehran Gabrani Mehran Gabrani Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran Iran m.gabrani@urmia.ac.ir Bahman Rezaei Bahman Rezaei Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran Iran b.rezaei@urmia.ac.ir Esra Sengelen Sevim Esra Sengelen Sevim Department of Mathematics, Istanbul Bilgi University, 34060, Eski Silahtaraga Elektrik Santrali, Kazim Karabekir Cad. No: 2/13 Eyupsultan, Istanbul, Turkey Turkey esra.sengelen@bilgi.edu.tr General (α β)-metric R-quadratic Ricci-quadratic. S. Bacso, B. Rezaei, On R-quadratic Einstein Finsler space, Publicationes Mathematicae Debrecen, 76(1), (2010), 5.##M. Gabrani, B. Rezaei, On General (α, β)-metric with Isotropic E-curvature, Journal of the Korean Mathematical Society, 55(2), (2018), 415-42.##B. Li, Z. Shen, On Randers Metrics of Quadratic Riemann Curvature, International Journal of Mathematics, 20(03), (2009), 369-76.##B. Rezaei, M. Gabrani, A Class of Finsler Metrics with Quadratic Curvatures, Bull. Iran. Math. Soc, 46, (2020), 53-65.##E. S. Sevim, Z. Shen, L. Zhao, On a Class of Ricci-flat Douglas Metrics, International Journal of Mathematics, 23(06), (2012), p.1250046.##N. Sadeghzadeh, On Finsler Metrics of Quadratic Curvature, Journal of Geometry and Physics, 132, (2018), 75-83.##A. Tayebi, E. Peyghan, On E-curvature of R-quadratic Finsler Metrics, Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, 28(1), (2012), 83-89.##A. Tayebi, M. Razgordani, On Conformally Flat Fourth Root (α, β)-metrics, Differential Geometry and its Applications, 62, (2019), 253-266.##Q. Xia, On a Class of Finsler Metrics of Scalar Flag Curvature, Results Math, 71(1-2), (2017), 483-507.##Q. Xia, Some Results on the Non-Riemannian Quantity H of a Finsler Metric, International Journal of Mathematics, 22(7), (2011), 925-936.##C. Yu, H. Zhu, On a New Class of Finsler Metrics, Differential Geometry and its Applications, 29(2), (2011), 244-254.##C. Yu, H. Zhu, Projectively Flat General (α, β)-metrics with Constant Flag Curvature, Journal of Mathematical Analysis and Applications, 429(2), (2015), 1222-1239.##C. Yu, On Dually Flat General (α, β)-metrics, Differential Geometry and its Applications, 40, (2015), 111-122.##H. Zhu, On a Class of Finsler Metrics with Isotropic Berwald Curvature, Journal of the Korean Mathematical Society, 54(2), (2017), 399-416.##M. Zohrehvand, H. Maleki, On General (α, β)-metrics of Landsberg Type, International Journal of Geometric Methods in Modern Physics, 13(6), (2016), 1650085, 13 pp.##Z. Shen, On R-quadratic Finsler Space, Publ. Math. Debrecen, 58, (2001), 263-274.## ##

Fuzzy Sumudu Transform for System of Fuzzy Differential Equations with Fuzzy Constant Coeffcients 2 2 In this study, we employ fuzzy Sumudu transform to find the solution for system of linear fuzzy differential equations where the system possesses fuzzy constant coeffcients instead of crisp. For this purpose, fuzzy Sumudu transform has been revisited and a brief comparison with fuzzy Laplace transform is provided alongside, particularly on the scale preserving property. For the sake of comparison, we introduce to the literature a time scaling theorem for fuzzy Laplace transform. Next, the system with fuzzy constant coeffcients is interpreted under the strongly generalized differentiability. From here, new procedures for solving the systems are proposed. A numerical example is then carried out for solving a system adapted from fuzzy radioactive decay model. Conclusion is drawn in the last section and some potential research directions are given. 79 100 2020/03/212020/03/272020/06/232019/12/72020/03/102020/04/2 1399/1/14 2022/01/142024/06/132021/04/32025/03/102022/01/142022/08/26 1401/6/4 N. A. Abdul Rahman N. A. Abdul Rahman School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia Malaysia aswad.rahman@usm.my M. Z. Ahmad M. Z. Ahmad Institute of Engineering Mathematics, Universiti Malaysia Perlis, Pauh Putra Main Campus, 02600 Arau, Perlis, Malaysia Malaysia mzaini@unimap.edu.my Fuzzy Sumudu transform Fuzzy differential equations System of fuzzy differential equations Fuzzy Laplace transform Radioactive decay model. N. A. Abdul Rahman, M. Z. Ahmad, Applications of the Fuzzy Sumudu Transform for the Solution of First Order Fuzzy Differential Equations, Entropy, 17(7), (2015), 4582-4601.##N. A. Abdul Rahman, M. Z. Ahmad, Fuzzy Sumudu Transform for Solving Fuzzy Partial Differential Equations, Journal of Nonlinear Science and Applications, 9(5), (2016), 3226-3239.##N. A. Abdul Rahman, M. Z. Ahmad, Solving Fuzzy Fractional Differential Equations Using Fuzzy Sumudu Transform, Journal of Nonlinear Science and Applications, 6, (2017), 19-28.##N. A. Abdul Rahman, M. Z. Ahmad, Solving Fuzzy Volterra Integral Equations via Fuzzy Sumudu Transform, Applied Mathematics and Computational Intelligence, 10(5), (2017), 2620-2632.##M. Z. Ahmad, N. A. Abdul Rahman, Explicit Solution of Fuzzy Differential Equations by Mean of Fuzzy Sumudu Transform, International Journal of Applied Physics and Mathematics, 5(2), (2015), 86-93.##T. Allahviranloo, The Adomian Decomposition Method for Fuzzy System of Linear Equations, Applied Mathematics and Computation, 163(2), (2015), 553-563.##T. Allahviranloo, M. B. Ahmadi, Fuzzy Laplace Transforms, Soft Computing, 14(3), (2010), 235-243.##T. Allahviranloo, N. Ahmady, E. Ahmady, Numerical Solution of Fuzzy Differential Equations by Predictor-corrector Method, Information Sciences, 177(7), (2007), 1633-1647.##B. Bede, S. G. Gal, Generalizations of the Differentiability of Fuzzy-number-valued Functions with Applications to Fuzzy Differential Equations, Fuzzy Sets and Systems, 151(3), (2005), 581-599.##B. Bede, I. J. Rudas, A. L. Bencsik, First Order Linear Fuzzy Differential Equations under Generalized Differentiability, Information Sciences, 177(7), (2007), 1648-1662.##T. Caraballo, D. Cheban, Almost Periodic and Almost Automorphic Solutions of Linear Differential/difference Equations without Favard's Separation Condition. I, Journal of Differential Equations, 246(1), (2009), 108-128.##Y. Chalco-Cano, H. Román-Flores, On New Solutions of Fuzzy Differential Equations, Chaos, Solitons & Fractals, 38(1), (2008), 112-119.##S. S. L. Chang, L. A. Zadeh, On Fuzzy Mapping and Control, IEEE Transactions on Systems, Man, and Cybernetics, (1), (1972), 30-34.##Y. A. Chirkunov, Linear Autonomy Conditions for the Basic Lie Algebra of a System of Linear Differential Equations, Doklady Mathematics, 79(3), (2009), 415-417.##O. S. Fard, N. Ghal-Eh, Numerical Solutions for Linear System of First-order Fuzzy Differential Equations with Fuzzy Constant Coeffcients, Information Sciences, 181(20), (2011), 4765-4779.##M. Friedman, M. Ma, A. Kandel, Numerical Solutions of Fuzzy Differential and Integral Equations, Fuzzy Sets and Systems, 106(1), (1999), 35-48.##Z. Gouyandeh, A. Armand, Numerical Solutions of Fuzzy Linear System Differential Equations and Application of a Radioactivity Decay Model, Communications on Advanced Computational Science with Applications, (2013), 1-11.##Z. Guang-Quan, Fuzzy Continuous Function and its Properties, Fuzzy Sets and Systems, 43(2), (1991), 159-171.##Y. Guo, Y. Wang, Decay of Dissipative Equations and Negative Sobolev Spaces, Communications in Partial Differential Equations, 37(12), (2012), 2165-2208.##A. K. Haydar, Fuzzy Sumudu Transform for Fuzzy nth-order Derivative and Solving Fuzzy Ordinary Differential Equations, International Journal of Science and Research, 4(12), (2015), 1372-1378.##R. Jafari, S. Razvarz, A. Gegov, S. Paul, S. Keshtkar, Fuzzy Sumudu Transform Approach to Solving Fuzzy Differential Equations with z-numbers, In Advanced Fuzzy Logic Approaches in Engineering Science, IGI Global, (2019), 18-48.##O. Kaleva, A Note on Fuzzy Differential Equations, Nonlinear Analysis: Theory, Methods & Applications, 64(5), (2006), 895-900.##A. Kaufmann, M. M. Gupta, Fuzzy Continuous Function and its Properties, Van Nostrand Reinhold, New York, 1985.##A. Khastan, J. J. Nieto, R. Rodríguez-López, Periodic Boundary Value Problems for First-order Linear Differential Equations with Uncertainty Under Generalized Differentiability, Information Sciences, 222, (2013), 544-558.##M. Ma, M. Friedman, A. 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Jesuraj, Solution of Fuzzy Differential Equation of Order 2 by Intuitionistic Fuzzy Numbers (IFS), In International Conference on Intelligent Computing, Information and Control Systems, (2019), 292-298.##M. Wang, Y. Zhang, Two Kinds of Free Boundary Problems for the Diffusive Preypredator Model, Nonlinear Analysis: Real World Applications, 24, (2015), 73-82.##H. C. Wu, The Improper Fuzzy Riemann Integral and its Numerical Integration, Information Sciences, 111(1), (1998), 109-137.##J. Xu, Z. Liao, Z. Hu, Class of Linear Differential Dynamical Systems with Fuzzy Initial Condition, Fuzzy Sets and Systems, 158(21), (2007), 2339-2358.##L. A. Zadeh, Fuzzy sets, Information and Control, 8(3), (1965), 338-353.## ##

On Graded Semi $J_{gr}$-2-absorbing and Graded Weakly Semi $J_{gr}$-2-absorbing Submodules 2 2 Let G be a group with identity e. Let R be a G-graded commutative ring and M a graded R-module. In this paper, we introduce the concepts of graded semi Jgr-2-absorbing and graded weakly semi Jgr-2-absorbing submodules of M and study the behavior of these notions under several constructions. A proper graded submodule N of M is said to be a graded semi Jgr-2-absorbing (resp. graded weakly semi Jgr-2-absorbing) submodule of M if whenever rg ∈ h(R) and mh ∈ h(M) with rg2mh ∈ N (resp. 0 ≠ rg2mh ∈ N), then either rgmh ∈ N + Jgr(M) or rg2∈ (N + Jgr(M) :R M), where Jgr(M) is the graded Jacobson radical. 101 110 2020/03/212020/03/272020/06/232019/12/72020/03/102020/04/22020/04/7 1399/1/19 2022/01/142024/06/132021/04/32025/03/102022/01/142022/08/262022/02/22 1400/12/3 Shatha Alghueiri Shatha Alghueiri Department of Mathematics and Statistics Jordan University of Science and Technology P.O.Box 3030, Irbid 22110, Jordan Jordan ghweiri64@gmail.com Khaldoun Al-Zoubi Khaldoun Al-Zoubi Department of Mathematics and Statistics Jordan University of Science and Technology P.O.Box 3030, Irbid 22110, Jordan Jordan kfzoubi@just.edu.jo Graded semi $J_{gr}$-2-absorbing submodules Graded weakly semi $J_{gr}$-2-absorbing submodules Graded $J_{gr}$-2-absorbing submodules. K. Al-Zoubi, Some Properties of Graded 2-prime Submodules, Asian-Eur. J. Math., 8(2), (2015), 1550016-1- 1550016-5.##K. Al-Zoubi, R. Abu-Dawwas, On Graded 2-absorbing and Weakly Graded 2-absorbing Submodules, J. Math. Sci. Adv. Appl., 28, (2014), 45-60.##K. Al-Zoubi, R. Abu-Dawwas, On Graded Quasi-prime Submodules, Kyungpook Math. J., 55(2), (2015), 259-266.##K. Al-Zoubi, I. Al-Ayyoub, M. Al-Dolat, On Graded 2-absorbing Compactly Packed Modules, Adv. Stud. Contemp. Math. (Kyungshang), 28(3), (2018), 479-486.##K. Al-Zoubi, M. Al-Azaizeh, Some Properties of Graded 2-absorbing and Graded Weakly 2-absorbing Submodules, J. Nonlinear Sci. Appl., 12(8), (2019), 503-508.##K. Al-Zoubi, A. Al-Qderat, Some Properties of Graded Comultiplication Modules, Open Mathematics, 15, (2017), 187-192.##K. Al-Zoubi, S. Alghueiri, On Graded Jgr-2-absorbing and Graded Weakly Jgr-2-absorbing Submodules of Graded Modules Over Graded Commutative Rings, Int. J. Math. Comput. Sci., 16(4), (2021), 1169-1178.##K. Al-Zoubi, S. Alghueiri, On Graded Jgr-semiprime Submodules, Italian Journal Pure and Applied Mathematics, 46, (2021), 361-369.##K. Al-Zoubi, M. Jaradat, R. Abu-Dawwas, On Graded Classical Prime and Graded Prime Submodules, Bull. Iranian Math. Soc., 41(1), (2015), 217-225.##R. Abu-Dawwas, K. Al-Zoubi, On Graded Weakly Classical Prime Submodules, Iran. J. Math. Sci. Inform. 12(1), (2017), 153-161.##R. Abu-Dawwas, M. Bataineh, Graded r-Ideals, Iran. J. Math. Sci. Inform. 14(2), (2019), 1-8.##S. E. Atani, On Graded Prime Submodules, Chiang Mai J. Sci., 33(1), (2006), 3-7.##R. Hazrat, Graded Rings and Graded Grothendieck Groups, CambridgeUniversity Press, Cambridge, 2016.##C. Nastasescu, F. Van Oystaeyen, Graded and Filtered Rings and Modules, Lecture notes in mathematics 758, Berlin-New York: Springer-Verlag, 1982.##C. Nastasescu, F. Van Oystaeyen, Graded Ring Theory, Mathematical Library 28, North Holand, Amsterdam, 1982.##C. Nastasescu, F. Van Oystaeyen, Methods of Graded Rings, LNM 1836. BerlinHeidelberg: Springer-Verlag, 2004.##K. H. Oral, U. Tekir, A. G. Agargun, On Graded Prime and Primary Submodules, Turk. J. Math., 35, (2011), 159-167.## ##

$zeta_varsigma$-$R_0$ and $zeta_varsigma$-$R_1$ Strong Generalized Topological Spaces 2 2 The purpose of the present paper is to introduce the concepts of ζς-R0 and ζς-R1 strong generalized topological spaces are defined by utilizing the notions of (ζ, ς)-open sets and (ζ, ς)-closure operators. Moreover, several characterizations of ζς-R0 and ζς-R1 strong generalized topological spaces are investigated. 111 123 2020/03/212020/03/272020/06/232019/12/72020/03/102020/04/22020/04/72020/04/10 1399/1/22 2022/01/142024/06/132021/04/32025/03/102022/01/142022/08/262022/02/222022/01/10 1400/10/20 Nongluk Viriyapong Nongluk Viriyapong Mathematics and Applied Mathematics Research Unit, Department of Mathematics, Faculty of Science, Mahasarakham University, Maha Sarakham, 44150, Thailand Thailand nongluk.h@msu.ac.th Chawalit Boonpok Chawalit Boonpok Mathematics and Applied Mathematics Research Unit, Department of Mathematics, Faculty of Science, Mahasarakham University, Maha Sarakham, 44150, Thailand Thailand chawalit.b@msu.ac.th Generalized topological space (ζ ς)-open set (ζ ς)-closure $ζ_ς-R_0$ strong generalized topological space $ζ_ς-R_1$ strong generalized topological space. F. G. Arenas, J. Dontchev, M. Ganster, On λ-sets and Daul of Generalized Continuity, Questions and Answers in General Topology, 15, (1997), 3-13.##M. Caldas, D. N. Georgiou, T. Niori, On (Λ, θ)-closed Sets, Questions and Answers in General Topology, 23, (2005), 69-87.##M. Caldas, S. Jafari, T. Noiri, Characterizations of Λθ-R0 and Λθ-R1 Topological Spaces, Acta Mathematica Hungarica, 103, (2004), 85-95.##Á. Császár, Generalized Open Sets, Acta Mathematica Hungarica, 75, (1997), 65-87.##Á. Császár, Generalized Topology, Generalized Continuity, Acta Mathematica Hungarica, 96, (2002), 351-357.##Á. Császár, Extremally Disconnected Generalized Topologies, Annales Universitatis Scientiarium Budapestinensis de Rolando Eötvös Nominatae Sectio Mathematica, 47, (2004), 91-96.##Á. Császár, Generalized Open Sets in Generalized Topologies, Acta Mathematica Hungarica, 106, (2005), 53-66.##Á. Császár, Modification of Generalized Topologies via Hereditary Classes, Acta Mathematica Hungarica, 115, (2007), 29-36.##A. S. Davis, Indexed Systems of Neighborhoods for General Topological Spaces, Journal of the American Mathematical Society, 68, (1961), 886-893.##C. Dorsett, Semi-T2, Semi-R1 and Semi-R0 Topological Spaces, Annales de la Société Scientifique de Bruxelles, 92, (1978), 143-150.##K. K. Dube, A Note on R0 Topological Spaces, Matematički Vesnik, 11, (1974), 203-208.##K. K. Dube, A Note on R1 Topological Spaces, Periodica Mathematica Hungarica, 13, (1982), 267-271.##E. Ekici, B. Roy, New Generalized Topologies on Generalized Topological Spaces Due to Császár, Acta Mathematica Hungarica, 132(1-2), (2011), 117-124.##E. Ekici, Generalized Submaximal Spaces, Acta Mathematica Hungarica, 134(1-2), (2012), 132-138.##S. N. Maheshwari, R. Prasad, On (R0)s-spaces, Portugaliae Mathematica, 34, (1975), 213-217.##H. Maki, Generalized Λ-sets and the Associated Closure Operator, Special Issue in Commemoration of Prof. Kazusada IKEDA's Retirement, (1986), 139-146.##M. G. Murdeshwar, S. A. Naimpally, R1-topological Spaces, Canadian Mathematical Bulletin, 9, (1966), 521-523.##S. A. Naimpally, On R0-topological Spaces, Annales Universitatis Scientiarium Budapestinensis de Rolando Eötvös Nominatae Sectio Mathematica, 10, (1967), 53-54.##B. Roy, E. Ekici, On (∧, µ)-closed Sets in Generalized Topological Spaces, Methods of Functional Analysis and Topology, 17(2), (2011), 174-179.##N. A. Shanin, On Separation in Topological Spaces, Doklady Akademii Nauk SSSR, 38, (1943), 110-113.##N. V. Veličko, H-closed Topological Spaces, American Mathematical Society Translations: Series 2, 78, (1968), 102-118.## ##

On the Independence Graph of Hamming Graph 2 2 The independence graph Ind(G) of a graph G is the graph with vertices as maximum independent sets of G and two vertices are adjacent, if and only if the corresponding maximum independent sets are disjoint. In this work, we find the independence graph of Cartesian product of d copies of complete graphs Kq, which is known as the Hamming graph H(d, q). Greenwell and Lovasz [7] found that the independence number of direct product of d copies of Kq as qd−1. We prove that the independence number of Hamming graph H(d, q), which is cartesian product of d copies of Kq, is also qd−1. As an application of our findings, we find answers for rook problem in higher dimensional square chess board. 125 130 2020/03/212020/03/272020/06/232019/12/72020/03/102020/04/22020/04/72020/04/102020/04/29 1399/2/10 2022/01/142024/06/132021/04/32025/03/102022/01/142022/08/262022/02/222022/01/102022/08/26 1401/6/4 M. Saravanan M. Saravanan Lecturer in Mathematics, Department of Basic Engineering, Government Polytechnic College, Regunathapuram, Papanasam, Thanjavur, India India dr.msaravanan8187@gmail.com KM. Kathiresan KM. Kathiresan Centre for Graph Theory, Ayya Nadar Janaki Ammal College, Sivakasi, India India kathir2esan@yahoo.com Hamming graph Cartesian product Independent set Independence Graph Rook problem. G. Abay-Asmerom, R. Hammack, C. E. Larson, D. T. Taylor, Notes on the Independence Number in the Cartesian Product of Graphs, Discuss. Math. Graph Theory, 31, (2011), 25-35.##F. Alayont, N. Krzywonos, Rook Polynomials in Three and Higher Dimensions, Involve, 6(1), (2013), 35-52.##B. Bresar, B. Zmazek, On the Independence Graph of a Graph, Discrete Math., 272, (2003), 263-268.##A. E. Brouwer, A. M. Cohen, A. Neumair, Distance Regular Graphs, Springer - verlag, NewYork, 1989.##Hon-Chan Chen, Ting-Yem Ho, The Rook Problem on Saw-toothed Chessboards, Appl. Math. Lett., 21, (2008), 1234-1237##T. Derikvand, M. R. Oboudi, On the Number of Maximum Independent Sets of Graphs, Trans. Comb., 3(1), (2014), 29-36##D. Greenwell, L. Lovasz, Applications of Product Colouring, Acta Math. Hungar., 25(3-4), (1974), 335-340.##P. Hell, X. Yu, H. Zhou, Independence Ratios of Graph Powers, Discrete Math., 127, (1994), 213-220.##W. Imrich, S. Klavzar, Product Graphs, Wiley-Interscience, NewYork, 2000.##M. J. Jou, G. J. Chang, The Number of Maximum Independent Sets in Graphs, Taiwan. J. Math., 4(4), (2000), 685-695.##I. Kaplansky, J. Riordan, The Problem of the Rooks and Its Applications, Duke Math. J., 13(2), (1946), 259-268.##S. Klavzar, Some New Bounds and Exact Results on the Independence Number of Cartesian Product Graphs, Ars Combin., 74, (2005), 173-186.##A. Varvak, Rook Numbers and the Normal Ordering Problem, J. Combin. Theory Ser. A, 112, (2005), 292-307.##D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, New Jersey, 2000.##X. Zhu, On the Bounds for the Ultimate Independence Ratio of a Graph, Discrete Math., 156, (1996), 229-236.## ##

Existence Problem for Impulsive Nonlinear Sturm-Liouville Problems on the Whole Line 2 2 The existence problem is studied for the impulsive nonlinear Sturm–Liouville equation on the whole line. Existence and uniqueness results are obtained. 131 149 2020/03/212020/03/272020/06/232019/12/72020/03/102020/04/22020/04/72020/04/102020/04/292020/05/11 1399/2/22 2022/01/142024/06/132021/04/32025/03/102022/01/142022/08/262022/02/222022/01/102022/08/262022/02/14 1400/11/25 Bilender P. Allahverdiev Bilender P. Allahverdiev Department of Mathematics, Khazar University, AZ1096 Baku, Azerbaijan Azerbaijan bilenderpasaoglu@gmail.com Hüseyin Tuna Hüseyin Tuna Department of Mathematics, Burdur Mehmet Akif Ersoy University, 15030 Burdur, Turkey Turkey hustuna@gmail.com Impulsive Sturm-Liouville problem Singular point Weyl limitcircle case Completely continuous operator Fixed point theorems. R. P. Agarwal, D. O'Regan, Boundary Value Problems of Nonsingular Type on the Semiinfinite Interval, Tohoku Math. Journal, 51(3), (1999), 391-397.##R. P. Agarwal, D. O'Regan, Multiple Nonnegative Solutions for Second Order Impulsive Differential Equations, Appl. Math. Comput., 114, (2000), 51-59.##B. P. Allahverdiev, H. Tuna, Existence of Solutions for Nonlinear Singular q-Sturm-Liouville Problems, Ufa Math. J., 12(1), (2020), 91-102.##B. P. Allahverdiev, H. Tuna, Nonlinear Singular Sturm-Liouville Problems with Impulsive Conditions, Facta Univ., Ser. Math. Inf., 34(3), (2019), 439-457.##B. P. Allahverdiev, H. Tuna, On the Solutions for a Nonlinear Singular q-Sturm-Liouville Problems on the Whole Axis, Southeast Asian Bull. Math., 47(2), (2023), 151-164.##D. D. Bainov, P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman: Harlow, 1993.##D. D. Bainov, P. S. Simeonov, Impulsive Differential Equations: Asymptotic Properties of the Solutions, World Scientific: Singapore, 1995.##D. D. Bainov, P. S. Simeonov, Oscillation Theory of Impulsive Differential Equations, International Publications: Orlando, 1998.##M. Benchohra, J. Henderson, S. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation: New York, 2006.##W. Ding, M. Han, Periodic Boundary Value Problem for the Second Order Impulsive Functional Differential Equations, Appl. Math. Comput., 155(3), (2004), 709-726.##N. Dunford, J. T. Schwartz, Linear Operators, Part II, Interscience, New York, 1964.##M. Feng, D. Xie, Multiple Positive Solutions of Multi-point Boundary Value Problem for Second Order Impulsive Differential Equations, J. Comput. Appl. Math., 223(1), (2009), 438-448.##D. Guo, Second Order Impulsive Integro-differential Equations on Unbounded Domains in Banach Spaces, Nonlinear Analysis: Theory, Methods & Applications, 35(4), (1999), 413-423.##G. Sh. Guseinov, I. Yaslan, Boundary Value Problems for Second Order Nonlinear Differential Equations on Infinite Intervals, J. Math. Anal. Appl., 290, (2004), 620-638.##S. G. Hristova, D. D. Bainov, Monotone-iterative Techniques of V. Lakshmikantham for a Boundary Value Problem for Systems of Impulsive Differential-difference Equations, J. Math. Anal. Appl., 197(1), (1996), 1-13.##T. Jankowski, Positive Solutions to Second Order Four-point Boundary Value Problems for Impulsive Differential Equations, Appl. Math. Comput., 202(2), (2008), 550-561.##T. Jankowski, Positive Solutions of Three-point Boundary Value Problems for Second Order Impulsive Differential Equations with Advanced Arguments, Appl. Math. Comput., 197(1), (2008), 179-189.##T. Jankowski, Existence of Solutions for Second Order Impulsive Differential Equations with Deviating Arguments, Nonlinear Analysis: Theory, Methods & Applications, 67(6), (2007), 1764-1774.##A. N. Kolmogorov, S. V. Fomin, Introductory Real Analysis, Translated by R. A. Silverman, Dover Publications, New York, 1970.##M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Gostekhteoretizdat, Moscow, 1956, English transl. Pergamon Press, New York, 1964.##V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific: Singapore, 1989.##I. K. C. Leung, P. N. Shivakumar, On the Eigenvalue Problem −y′′ + f (x) y = λy on a Semi Infinite Interval, Math. Comput. Mod., 46, (2007), 316-330.##B. M. Levitan, I. S.Sargsjan, Sturm-Liouville and Dirac Operators, Mathematics and its Applications (Soviet Series, translated from the Russian), Kluwer Academic Publishers Group, Dordrecht, 1991.##J. Li, J. J. Nieto, Existence of Positive Solutions for Multipoint Boundary Value Problem on the Half-line with Impulses, Boundary Value Problems, 2009, (2009), Article ID 834158, 1-12.##J. Li, J. Shen, Existence of Positive Solution for Second-order Impulsive Boundary Value Problems on Infinity Intervals, Boundary Value Problems, 2006, (2006), Article ID 14594, 1-11.##H. Lian, W. Ge, Existence of Positive Solutions for Sturm-Liouville Boundary Value Problems on the Half-line, J. Math. Anal. Appl., 321, (2006), 781-792.##H. Lian, H. Pang, W. Ge, Triple Positive Solutions for Boundary Value Problems on Infinite Intervals, Nonlinear Anal., 67, (2007), 2199-2207.##X. Lin, D. Jiang, Multiple Positive Solutions of Dirichlet Boundary Value Problems for Second Order Impulsive Differential Equations, J. Math. Anal. Appl., 321(2), (2006), 501-514.##L. Liu, F. Y. Li, Multiple Positive Solution of Nonlinear Two-point Boundary Value Problems, J. Math. Anal. Appl., 203, (1996), 610-625.##Y. Liu, Boundary Value Problems for Second Order Differential Equations on Unbounded Domains in a Banach Space, Appl. Math. Comput., 135(2-3), (2003), 569-583.##X. Liu, D. 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Some Characterizations of Γ−semihypergroups by Soft Generalized Γ-hyperideals 2 2 The aim of this paper is to establish a relationship between soft sets and Γ-semihypergroups. In this aspect, we have introduced soft intersection generalized interior Γ-hyperideals and soft intersection generalized bi-Γ-hyperideals of Γ-semihypergroups with some interesting examples. Moreover, we study some characterizations of regular, intraregular, semisimple and right weakly regular Γ-semihypergroups in terms of soft intersection generalized interior Γ-hyperideals and soft intersection generalized bi-Γ-hyperideals. 151 174 2020/03/212020/03/272020/06/232019/12/72020/03/102020/04/22020/04/72020/04/102020/04/292020/05/112020/05/12 1399/2/23 2022/01/142024/06/132021/04/32025/03/102022/01/142022/08/262022/02/222022/01/102022/08/262022/02/142021/05/6 1400/2/16 Sabahat Ali Khan Sabahat Ali Khan Department of Mathematics, Jamia Millia Islamia, New Delhi-110 025, India India khansabahat361@gmail.com Mohammad Yahya Abbasi Mohammad Yahya Abbasi Department of Mathematics, Jamia Millia Islamia, New Delhi-110 025, India India mabbasi@jmi.ac.in Kostaq Hila Kostaq Hila Department of Mathematical Engineering, Polytechnic University of Tirana, Albania Albania kostaq_hila@yahoo.com Ahmad Raza Ahmad Raza Department of Mathematics, Jamia Millia Islamia, New Delhi-110 025, India India arhanraza@gmail.com Γ-semihypergroups Soft generalized Γ-hyperideals Regular and intra-regular Γ-semihypergroups Interior simple Γ-semihypergroups. 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Symp., New Delhi, 1981, Lecture Notes in Pure and Applied Mathematics 91, Decker, New York, 1984, 301-308.##A. Sezgin, A. O. Atagun, On Operations of Soft Sets, Comput. Math. Appl., 61(5), (2011), 1457-1467.##A. Sezgin, N. Cagman, A. O. Atagun, Soft Intersection Interior Ideals, Quasi-ideals and Generalized bi-ideals; A New Approach to Semigroup Theory II, J. Mult.-Valued Logic Soft Comput., 23(1-2), (2014), 161-207.##A. Sezgin, N. Cagman, A. O. Atagun, M. Ali, Soft Intersection Semigroups, Ideals and Bi-ideals; a New Approach to Semigroup Theory, Filomat, 29(5), (2015), 917-946##J. Tang , B. Davvaz, Y. Luo, A Study on Fuzzy Interior Hyperideals in Ordered Semihypergroups, Ital. J. Pure Appl. Math., 36, (2016), 125-146.##J. Tang, B. Davvaz, X.Y. Xie, N. Yaqoob, On Fuzzy Interior Γ-hyperideals in Ordered Γ-semihypergroups, J. Intell. Fuzzy Systems, 32(3), (2017), 2447-2460.##N. Tipachot, B. Pibaljommee, Fuzzy Interior Hyperideals in Ordered Semihypergroups, Ital. J. Pure Appl. Math., 36, (2016), 859-870.##N. Yaqoob, M. Aslam, B. Davvaz, A. Ghareeb, Structures of Bipolar Fuzzy Γ-hyperideals in Γ-semihypergroups, J. Intell. Fuzzy Systems, 27(6), (2014), 3015-3032.##J. Zhan, N. Cagman, A. S. Sezer, Applications of Soft :union: Sets to Hemirings via SU-h-ideals, J. Intell. Fuzzy Systems, 26(3), (2014), 1363-1370.## ##

Some Hermite-Hadamard Type Inequalities for Convex Functions Defined on Convex Bodies Via Gauss-Ostrogradsky Identity 2 2 In this paper, by the use of Gauss-Ostrogradsky identity, we establish some integral inequalities of Hermite-Hadamard type for functions of three variables defined on closed and bounded convex bodies of the Euclidean space R3. Some examples for 3-dimensional balls are also provided. 175 191 2020/03/212020/03/272020/06/232019/12/72020/03/102020/04/22020/04/72020/04/102020/04/292020/05/112020/05/122020/05/27 1399/3/7 2022/01/142024/06/132021/04/32025/03/102022/01/142022/08/262022/02/222022/01/102022/08/262022/02/142021/05/62022/03/5 1400/12/14 Silvestru Sever Dragomir Silvestru Sever Dragomir Applied Mathematics Research Group, ISILC, Victoria University, PO Box 14428Melbourne City, MC 8001, Australia Australia sever.dragomir@vu.edu.au Hermite-Hadamard inequality Triple integral inequalities GaussOstrogradsky identity. T. M. Apostol, Calculus Volume II, Multi Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability, Second Edition, John Wiley & Sons, New York, London, Sydney, Toronto, 1969.##A. Barani, Hermite-Hadamard and Ostrowski Type Inequalities on Hemispheres, Mediterr. J. Math., 13, (2016), 4253-4263.##M. Bessenyei, The Hermite-Hadamard Inequality on Simplices, Amer. Math. Monthly, 115, (2008), 339-345.##J. de la Cal, J. Cárcamo, Multidimensional Hermite-Hadamard Inequalities and the Convex Order, Journal of Mathematical Analysis and Applications, 324(1), (2006), 248-261.##S. S. Dragomir, On Hadamard's Inequality on a Disk, Journal of Ineq. Pure & Appl. Math., 1(1), (2000), Article 2. [Online https://www.emis.de/journals/JIPAM/article95.html?sid=95].##S. S. Dragomir, On Hadamard's Inequality for the Convex Mappings Defined on a Ball in the Space and Applications, Math. Ineq. & Appl., 3 (2), (2000), 177-187.##S. S. Dragomir, Double Integral Inequalities of Hermite-Hadamard Type for h-convex Functions on Linear Spaces, Analysis (Berlin), 37(1), (2017), 13-22.##S. S. Dragomir, Hermite-Hadamard Type Integral Inequalities for Double Integral on General Domains, Preprint RGMIA Res. Rep. Coll., 22, (2019), Art. 46, 10 pp. [Online http://rgmia.org/papers/v22/v22a46.pdf.]##B. Gavrea, On Hadamard's Inequality for the Convex Mappings Defined a Convex Domain in the Space, Journal of Inequalities in Pure and Applied Mathematics, 1(1), (2000). [https://www.emis.de/journals/JIPAM/article102.html?sid=102].##P. O. Mohammed, M Vivas-Cortez, T Abdeljawad, Y Rangel-Oliveros, Integral Inequalities of Hermite-Hadamard Type for Quasi-convex Functions with Applications, AIMS Math, 5, (2020), 7316-7331.##M. Matłoka, On Hadamard's Inequality for h-convex Function on a Disk, Applied Mathematics and Computation, 235, (2014), 118-123.##F.-C. Mitroi, E. Symeonidis, The Converse of the Hermite-Hadamard Inequality on Simplices, Expo. Math., 30, (2012), 389-396.##E. Neuman, Inequalities Involving Multivariate Convex Functions II, Proc. Amer. Math. Soc., 109, (1990), 965-974.##E. Neuman, J. Pec̆arić, Inequalities Involving Multivariate Convex Functions, J. Math. Anal. Appl., 137, (1989), 541-549.##M. Vivas -Cortez, J. E. H. Hernández, L. A. Azócar, Some New Generalized Jensen and Hermite-Hadamard Inequalities for Operator h-convex Functions, Appl. Math. Inf. Sci, 11(2), (2017), 383-392.##S. Wasowicz, A. Witkowski, On Some Inequality of Hermite-Hadamard Type, Opusc. Math., 32(3), (2012), 591-600.##F.-L. Wang, The Generalizations to Several-dimensions of the Classical Hadamard's Inequality, Mathematics in Practice and Theory, 36(9), 370-373, 2006 (Chinese).##F.-L. Wang, A Family of Mappings Associated with Hadamard's Inequality on a Hypercube, International Scholarly Research Network ISRN Mathematical Analysis, 2011, Article ID 594758, 9 pages doi:10.5402/2011/594758.## ##

A Fuzzy Multivariate Regression Model to Control Outliers and Multicollinearity Based on Exact Predictors and Fuzzy Responses 2 2 Multivariate regression is an approach for modeling the linear relationship between several variables. This paper proposed a ridge methodology with a kernel-based weighted absolute error target with exact predictors and fuzzy responses. Some standard goodness-of-fit criteria were also used to examine the performance of the proposed method. The effectiveness of the proposed method was then illustrated through two numerical examples including a simulation study. The effectiveness and advantages of the proposed fuzzy multiple linear regression model were also examined and compared with some well-established methods through some common goodness-of-fit criteria. The numerical results indicated that our prediction/estimation gives more accurate results in cases where multicollinearity and/or outliers occur in the data set. 193 204 2020/03/212020/03/272020/06/232019/12/72020/03/102020/04/22020/04/72020/04/102020/04/292020/05/112020/05/122020/05/272020/07/16 1399/4/26 2022/01/142024/06/132021/04/32025/03/102022/01/142022/08/262022/02/222022/01/102022/08/262022/02/142021/05/62022/03/52023/10/8 1402/7/16 Gholamreza Hesamian Gholamreza Hesamian Department of Statistics, Payame Noor University, Tehran 19395-3697, Iran Iran gh.hesamian@pnu.ac.ir Mohamad Ghasem Akbari Mohamad Ghasem Akbari Department of Statistics, Ferdowsi University of Mashhad, Mashhad, Iran Iran g_z_akbari@birjand.ac.ir Mehdi Shams Mehdi Shams Department of Statistics, University of Kashan, Kashan, Iran Iran mehdishams@kashanu.ac.ir Goodness-of-fit measure Robust Multicollinearity Kernel function Outlier. M. G. Akbari, G. Hesamian, Linear Model with Exact Inputs and Interval-valued Fuzzy Outputs, IEEE Transactions on Fuzzy Systems, 26, (2018), 518-530.##M. G. Akbari, G. Hesamian, A Partial-robust-ridge Based-regression Model with Fuzzy Predictors-Responses, Journal of Computational and Applied Mathematics, 351, (2019), 290-301.##G. Alfonso, A. F. R. L. de Hierro, C. Roldan, A Fuzzy Regression Model based on Finite Fuzzy Numbers and Its Application to Real-World Financial Data, Journal of Computational and Applied Mathematics, 318, (2017), 47-58.##J. Chachi, M. Roozbeh, A Fuzzy Robust Regression Approach Applied to Bedload Transport Data, Communications in Statistics-Simulation and Computation, 46, (2017), 1703-1714.##S. H. Choi, J. H. Yoon, Fuzzy Regression Based on Non-Parametric Methods, Wseas Transaction on Systems and Control, 13, (2018), 20-25.##S. H. Choi, J. H. Yoon, General Fuzzy Regression Using Least Squares Method, International Journal of Systems Science, 41, (2010), 477-485.##A. F. R. L. de Hierro, J. Martinez-Moreno, C. Aguilar-Pena, C. R. L. de Hierro, Estimation of a Fuzzy Regression Model Using Fuzzy Distances, IEEE Transactions on Fuzzy Systems, 24, (2016), 344-359.##P. D'Urso, R. Massari, A. Santoro, Robust Fuzzy Regression Analysis, Information Sciences, 181, (2011), 4154-4174.##G. Hesamian, M. G. Akbari, M. Asadollahi, Fuzzy Semi-Parametric Partially Linear Model with Fuzzy Inputs and Fuzzy Outputs, Expert Systems with Applications, 71, (2017), 230-239.##G. Hesamian, M. G. Akbari, Fuzzy Quantile Linear Regression Model Adopted with a Semi-Parametric Technique based on Fuzzy Predictors and Fuzzy Responses, Expert Systems with Applications, 118, (2019), 585-597.##G. Hesamian, M. G. Akbari, A Robust Varying Coeffcient Approach to Fuzzy Multiple Regression Model, Journal of Computational and Applied Mathematics, 375, (2020), 1-13.##A. E. Hoerl, R. W. Kennard, Ridge Regression: Biased Estimation for Non-Orthogonal Problems, Technometrics, 12, (1970), 55-67.##G. James, D. Witten, T. Hastie, R. Tibshirani, An Introduction to Statistical Learning: with Applications in R, 8th ed., Springer-Verlag, New York, 2017.##H. Y. Jung, J. H. Yoon, S. H. Choi, Fuzzy Linear Regression Using Rank Transform Method, Fuzzy sets and systems, 274, (2015), 97-108.##U. T. Khan, C. Valeo, A New Fuzzy Linear Rgression Approach for Dissolved Oxygen Prediction, Hydrological Sciences Journal, 60, (2015), 1096-1119.##K. Kula, A. Apaydin, Fuzzy Robust Regression Analysis based on Ranking of Fuzzy Sets, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 16, (2008), 663-681.##K. S. Kula, F. Tank, T. E. Dalkylyc, A Study on Fuzzy Robust Regression and Its Application to Insurance, Mathematical and Computational Applications, 17, (2012), 223-234.##K. H. Lee, First Course on Fuzzy Theory and Applications, Springer-Verlag, Berlin, 2005.##W. J. Lee, H. Y. Jung, J. H. Yoon, S. H. Choi, The Statistical Inferences of Fuzzy Regression Based on Bootstrap Techniques, Soft Computing, 19, (2015), 883-890.##J. Li, W. Zeng, J. Xie, Q. Yin, A New Fuzzy Regression Model based on Least Absolute Deviation, Engineering Applications of Artificial Intelligence, 52, (2016), 54-64.##S. Sheather, A Modern Approach to Regression with R, Springer Science and Business Media, Wiely, New York, 2009.##B. Y. Sohn, Robust Fuzzy Linear Regression Based on M-estimators, Journal of Applied Mathematics and Computing, 18, (2005), 596-597##S. M. Taheri, M. Kelkinnama, Fuzzy Linear Regression Based on Least Absolute Deviations. Iranian Journal of Fuzzy Systems, 9, (2012), 121-140.##H. Tanaka, I. Hayashi, J. Watada, Possibilistic Linear Regression Analysis for Fuzzy Data, European Journal of Operational Research, 40, (1989), 389-396.##G. Wahba, Spline Models for Observational Data, Society for industrial and applied mathematics (Siam), Taiwan, 1990.##L. Wasserman, All of Nonparametric Statistics, Springer, New York, 2007.##W. Zeng, Q. Feng, J. Li, Fuzzy Least Absolute Linear Regression, Applied Soft Computing, 52, (2017), 1009-1019.## ##

New Concepts of Generalized Convexity in Multiobjective Subset Programming 2 2 In this paper, a new class of nonconvex differentiable multiobjective programming problems involving n-set functions with both inequality and equality constraints is considered. Then, under V-r-convexity and/or generalized V-r-convexity hypotheses, several suffcient optimality conditions, saddle point criteria and various mixed duality theorems are proved for such not necessarily convex vector optimization problems involving n-set functions. 205 229 2020/03/212020/03/272020/06/232019/12/72020/03/102020/04/22020/04/72020/04/102020/04/292020/05/112020/05/122020/05/272020/07/162020/07/7 1399/4/17 2022/01/142024/06/132021/04/32025/03/102022/01/142022/08/262022/02/222022/01/102022/08/262022/02/142021/05/62022/03/52023/10/82021/06/28 1400/4/7 Tadeusz Antczak Tadeusz Antczak Faculty of Mathematics and Computer Science, University of Łódź, Banacha 22, 90-238 Łódź, Poland Poland tadeusz.antczak@wmii.uni.lodz.pl Izhar Ahmad Izhar Ahmad Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia Saudi Arabia drizhar@kfupm.edu.sa Vector optimization problem with n-set functions Optimality conditions Saddle point criteria Mixed duality V -r-convex n-set function. I. Ahmad, S. Sharma, Suffciency in Multiobjective Subset Srogramming Involving Generalized Type-I Functions, Journal of Global Optimization, 39, (2007), 473-481.##M. Avriel, r-Convex Functions, Mathematical Programming, 2, (1972), 309-323.##A. Bătătorescu, Optimality Conditions Involving V -Type-I Univexity and Set-Functions, Mathematical Reports, 7, (2005) 1-11.##C. R. Bector, M. Singh, Duality for Multiobjective B-vex Programming Involving n-Set Functions, Journal of Mathematical Analysis and Applications, 202, (1996), 701-726.##D. Begis, R. Glowinski, Applications de la Méthode des éléments Finis à L'approximation d'un Problème de Domaine Optimal, Mèthodes de Rèsolution de Problèmes Approchès, Applied Mathematics and Optimization, 2, (1975), 130-169.##M. Beldiman, A. Paraschiv, O. Cojocaru, On Multiobjective Programming Problems Containing n-Set Functions, Analele Universităţii Bucuresti, Matematică Anul, LVII, (2008), 189-206.##D. Bhatia, A. Mehra, Lagrange Duality in Multiobjective Fractional Programming Problems with n-Set Functions, Journal of Mathematical Analysis and Applications, 236, (1999), 300-311.##J. Cea, A. Gioan, J. Michel, Quelque Résultats sur i'Identification de Domaines, Calcolo, 10, (1973), 133-145.##H. W. Corley, S. D. Roberts, A Partitioning Problem with Applications in Regional Design, Operations Research, 20, (1982), 1010-1019.##H. W. Corley, Optimization Theory for n-set Functions, Journal of Mathematical Analysis and Applications, 127, (1987), 193-205.##G. Dantzing, A. Wald, On the Fundamental Lemma of Neyman and Pearson, The Annals of Mathematical Statistics, 22, (1951), 87-93.##A. Jayswal, I. M. Stancu-Minasian, Multiobjective Subset Programming Problems Involving Generalized D-Type I Univex Functions, Proceedings of the Romanian Academy, Series A, 11, (2010) 19-24.##C. L. Jo, D. S. Kim, G. M. Lee, Duality for Multiobjective Programming Involving n-Set Functions, Optimization, 29, (1994), 45-54.##R. Larsson, Methodology for Topology and Shape Optimization: Application to a RearLower Control Arm, Chalmers University of Technology Göteborg, Sweden, 2016.##L. J. Lin, Optimality of Differentiable Vector-Valued n-set Functions, Journal of Mathematical Analysis and Applications, 149, (1990), 255-270.##L. J. Lin, On the Optimality Conditions of Vector-Valued n-set Functions, Journal of Mathematical Analysis and Applications, 161, (1991), 367-387.##S. K. Mishra, S. Y. Wang, K. K. Lai, J. Shi, New Generalized Invexity for Duality in Multiobjective Programming Problems Involving n-set, in: A. Eberhard, N. Hadjisavvas, D. T. Luc (eds.), Generalized Convexity, Generalized Monotonicity and Applications, Nonconvex Optimization and Applications, 77, Springer, New York, 2005, pp. 321-339.##B. Mond, T. Weir, Generalized Concavity and Duality, in: S. Schaible, W.T. Ziemba (eds.), Generalized Concavity in Optimization and Economics, Academic Press, New York, 1981, pp. 263-279.##R. J. T. Morris, Optimal Constrained Selection of a Measurable Subset, Journal of Mathematical Analysis and Applications, 70, (1979), 546-562.##V. Preda, Some Optimality Conditions for Multiobjective Programming Problems with Set Functions, Revue Roumaine de Mathmatiques Pures et Appliques, 39, (1994), 233-247.##V. Preda, On Duality of Multiobjective Fractional Measurable Subsets Selection Problems, Journal of Mathematical Analysis and Applications, 196, (1995), 514-525.##V. Preda, Duality for Multiobjective Fractional Programming Problems Involving n-Set Functions. In: C. A. Cazacu, C. W. E. Lehto, T. M. Rassias (eds.), Analysis and Topology, World Scientific Publishing Co., River Edge, NJ, 1998, pp. 569-583.##V. Preda, A. Bătătorescu, On Duality for Minmax Generalized B-vex Programming Involving n-Set Functions. Journal of Convex Analysis, 9, ( 2002), 609-623.##V. Preda, I. M. Stancu-Minasian, Optimality and Wolfe Duality for Multiobjective Programming Problems Involving n-Set Functions, In: N. Hadjisavvas, J. E. Martínez-Legaz, J.-P. Penot (eds.), Generalized Convexity and/or Generalized Monotonicity, Lecture Notes in Economics and Mathematical Systems, 502, Springer-Verlag, Berlin, 2001, pp. 349-361.##V. Preda, I. M. Stancu-Minasian, M. Beldiman, A. M. Stancu, Generalized V -Univexity Type-I for Multiobjective Programming with n-Set Functions, Journal of Global Optimization, 44, (2009), 131-148.##V. Preda, I. M. Stancu-Minasian, E. Koller, On Optimality and Duality for Multiobjective Programming Problems Involving Generalized d-Type-I and Related n-Set Functions, Journal of Mathematical Analysis and Applications, 283, (2003), 114-128.##A. M. Stancu, Optimality and Duality for Multiobjective Fractional Programming Problems with n-Set Functions and Generalized V-Type-I Univexity, Revue Roumaine de Mathmatiques Pures et Appliques, 57, (2012), 401-421.##I. M. Stancu-Minasian, V. Preda, Optimality Conditions and Duality for Programming Problems Involving Set and n-Set Functions - a Survey, Journal of Statistics and Management Systems, 5, (2002), 175-207.##A. A. Taflanidis, Robust Stochastic Design of Viscous Dampers for Base Isolation Applications, in: M. Papadrakakis, M. Fragiadakis, N. D. Lagaros (eds.), Computational Methods in Earthquake Engineering, Computational Methods in Applied Sciences, 21, Springer 2011, pp. 305-329.##G. J. Zalmai, Suffciency Criteria and Duality for Nonlinear Programs Involving n-Set Functions, Journal of Mathematical Analysis and Applications, 149, (1990), 322-338.##G. J. Zalmai, Optimality Conditions and Duality for Multiobjective Measurable Subset Selection Problems, Optimization, 22, (1991), 221-238.## ##

On the Terminal Distance Matrix of Graphs 2 2 Let G be a simple connected graph. The terminal distance matrix of G is the distance matrix between all pendant vertices of G. In this paper, we study the terminal distance matrix and compute the characteristic polynomial of this matrix for some rooted trees. Also we obtain lower bounds for the spectral radius of the terminal distance matrix of graphs and characterize those graphs for which the bounds are best possible. 231 241 2020/03/212020/03/272020/06/232019/12/72020/03/102020/04/22020/04/72020/04/102020/04/292020/05/112020/05/122020/05/272020/07/162020/07/72020/07/30 1399/5/9 2022/01/142024/06/132021/04/32025/03/102022/01/142022/08/262022/02/222022/01/102022/08/262022/02/142021/05/62022/03/52023/10/82021/06/282021/01/20 1399/11/1 Abbas Heydari Abbas Heydari Department of Science, Arak University of Technology, Arak, Iran Iran Heydari@arakut.ac.ir Terminal distance matrix Terminal distance spectral radius Terminal Wiener index. F. Buckley, F. Harary, Distance in Graphs, Addison-Wesley, Redwood, 1990.##B. Horvat, T. Pisanski, M. Randić, Terminal Polynomials and Star-like Graphs, MATCH Commun. Math. Comput. Chem., 60, (2008), 493-512.##M. Randić, J. Zupan, D. Vikić-Topić, On Representation of Proteins by Star-like Graphs, J. Mol. Graph. Modell., 26, (2007), 290-305.##E. A. Smolenskii, E. V. Shuvalova, L. K. Maslova, I. V. Chuvaeva , M. S. Molchanova, Reduced Matrix of Topological Distances with a Minimum Number of Independent Parameters: Distance Vectors and Mo lecular Codes, J. Math. Chem., 45, (2009), 1004-1020.##K. Zaretskii, Reconstructing a Tree from the Distances between its Leaves, (In Russian), Uspekhi Mat. Nauk, 20 (1965) 90-92.##Z. Mihalić, D. Veljan, D. Amić, S. Nikolić, D. Plavsić, N. Trinajstić, The Distance Matrix in Chemistry, J. Math. Chem., 11(1) (1992) 223-258.##J. Devillers, A. T. Balaban (Eds.), Topological Indices and Related Descriptors in QSAR and QSPR, Gordon and Breach, Amsterdam, 1999.##G. Indulal, Sharp Bounds on the Distance Spectral Radius and the Distance Energy of Graphs, Linear Algebra and its Applications, 430, (2009), 106-113.##R. Todeschini, V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, Weinheim, 2000.##A. Heydari, On the Spectra of Reduced Distance Matrix of Dendrimers, Trans. Comb., 2(2), (2013), 41-46.##A. Heydari, On Extremal Trees with Respect to Their Terminal Distance Spectral Radius, Australasian J. Combinatorics, 69(1), (2017), 159-168.##A. Heydari, On the Spectra of Reduced Distance Matrix of the Generalized Bethe Trees, Iranian J. Math. Chem., 8(3), (2017), 291-298.## ##

ABSTRACTS IN PERSIAN Vol. 20, No. 1 2 2 Please see the full text contains the Persian abstracts of this volume. 243 258 2020/03/212020/03/272020/06/232019/12/72020/03/102020/04/22020/04/72020/04/102020/04/292020/05/112020/05/122020/05/272020/07/162020/07/72020/07/302025/11/23 1404/9/2 2022/01/142024/06/132021/04/32025/03/102022/01/142022/08/262022/02/222022/01/102022/08/262022/02/142021/05/62022/03/52023/10/82021/06/282021/01/202025/11/23 1404/9/2 The Name of Authors in this Volume The Name of Authors in this Volume ma.hoseinzade@gmail.com ABSTRACTS PERSIAN Vol. 20 No. 1. All references of the papers in Vol20,No1## ##