On Lah-Ribarič Inequality Involving Averages of Convex Functions 2 2 By using the integral arithmetic mean and the Lah-Ribarič inequality we give the extension of Wulbert’s result from [15]. Also, we obtain inequalities with divided differences using the Lah-Ribarič inequality. As a consequence, the convexity of higher order for function defined by divided difference is proved. Further, we construct a new family of exponentially convex functions and Cauchy-type means by exploring at linear functionals with the obtained inequalities. 1 12 2019/01/29 1397/11/9 2024/05/7 1403/2/18 Ana Vukelić Ana Vukelić Faculty of Food Technology and Biotechnology, Mathematics Department, University of Zagreb, Croatia Croatia ana.vukelic@pbf.unizg.hr Lah-Ribarič inequality Divided differences n-convex function (n m)-convex function Exponential convexity. K. E. Atkinson, An Introduction to Numerical Analysis, 2nd ed., Wiley, New York, 1989.##V. Čuljak, I. Franjić, R. Ghulam, J. Pečarić, Schur-convexity of Averages of Convex Function, J. Ineq. Appl., 2011(2011), Article ID 581918.##R. Farwig, D. Zwick, Some Divided Difference Inequalities for n-convex Functions, J. Math. Anal. Appl., 108, (1985), 430-437.##J. Jakšetić, J. Pečarić, G. Roqia, On Jensen's Inequality Involving Averages of Convex Functions, Sarajevo J. Math., 8(20)(1), (2012), 53-68.##S. Karlin, Total Positivity, Stanford Univ. Press, Stanford, 1968.##S. Karlin, W. J. Studden, Tchebycheff Systems: with Applications in Analysis and Statistics, Interscience, New York, 1966.##P. Lah, M. Ribarič, Converse of Jensen's Inequality for Convex Functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., 412-460, (1973), 201-205.##Z. Pavić, J. Pečarić, A. Vukelić, Exponential Convexity and Jensen Inequality for Divided Differences, J. Math. Inequal., 5(2), (2011), 157-168.##Z. Pavić, J. Pečarić, A. Vukelić, Means for Divided Differences and Exponential Convexity, Mediterr. J. Math., 9(1), (2012), 187-198.##J. E. Pečarić, F. Proschan, Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in science and engineering, vol. 187, Academic Press, 1992.##J. Pečarić, J. Perić, Improvements of the Giaccardi and the Petrović Inequality and Related Stolarsky Type Means, An. Univ. Craiova Ser. Mat. Inform., 39, (2012), 65-75.##T. Popoviciu, Sur l'approximation Des Fonctions Convexes D'ordre Superieur, Mathematica, 10, (1934), 49-54.##T. Popoviciu, Les Functions Convexes, Herman and Cie, Éditeurs, Paris (1944).##G. Roqia, J. Pečarić, A. Vukelić, n--exponential Convexity of Divided Differences and Related Stolarsky Type Menas, Math. Ineq. Appl., 16(4), (2013), 1043-1063.##D. E. Wulbert, Favard's Inequality on Average Values of Convex Functions, Math. Comput. Model., 37(2003), 1383-1391.##X. M. Zhang, Y. M. Chu, Convexity of the Integral Arithmetic Mean of a Convex Function, Rocky Mt. J. Math., 40(3), (2010), 1061-1068.## ##

The Zariski Topology on Cl.Spec_g(M) as a Spectral Space 2 2 Let G be a group, R be a G-graded commutative ring with identity, M be a unitary graded R-module, Specg(R) be the set of graded prime ideals of R, and Cl.Specg(M) be the set of all graded classical prime submodules of M. In this paper among other things, the author studied the Zariski topology on both and Cl.Specg(M), and investigate some properties of the Zariski topology on Cl.Specg(M) and some conditions under which the graded classical prime spectrum of M is a spectral for its Zariski topology. 13 39 2019/01/292019/06/6 1398/3/16 2024/05/72024/05/25 1403/3/5 Malik Jaradat Malik Jaradat Department of Mathematics, The International School of Choueifat, UAE UAE mjaradat@mhs.sabis.net Graded classical prime submodule Graded classical prime spectrum Zariski topology. A. Abbasi, D. Hassanzadeh-Lelekaami, Modules and Spertral Space, Comm. Algebra, 40, (2012), 4111-4129.##A. Abbsi, D. Hassanzadeh-Lelekaami, Quasi-Prime Submodules and Developed Zariski Topology, Algebra Colloq., 19, (2012), 1089-1108.##R. Abu-Dawwas, K. Al-Zoubi, On Graded Weakly Classical Prime Submodules, Iran. J. Math. Sci. Inform., 12(1), (2017), 153-161.##K. Al-Zoubi, R. Abu-Dawwas, On Graded Quasi-Prime Submodules, Kyungpook Math. J., 55(2), (2015), 259-266.##K. Al-Zoubi, M. Al-Dolat, On Graded Classical Primary Submodules, Adv. Pure Appl. Math., 7(2), ( 2016), 93-96.##K. Al-Zoubi, M. Jaradat, On the Graded Classical Prime Spectrum of a Graded Module, Proyeccions Journal of Mathematics, 37(3), (2018), 519-533.##K. Al-Zoubi, M. Jaradat, The Zariski Topology on the Graded Classical Prime of a Graded Module Over a Graded Commutative Ring, Matematicki Vesnik, 70(4), (2018), 303-313.##K. Al-ZoubiI, M. Jaradat, R. Abu-Dawwas, On Graded Classical Prime and Graded Prime Submodules, Bull. Iranian Math. Soc., 41(1), (2015), 217-225.##H. Ansari-Toroghy, R. Ovlyaee-Sarmazdh, On the Prime Spectrum of a Module and Zariski Topologies. Comm. Algebra, 38, (2010), 4461-4475.##H. Ansari-Toroghy, S. S. Pourmortazavi, On the Prime Spectrum of Modules, Miskolc Math Notes, 16(2), (2015), 1233-1242.##S. E. Atani, On Graded Prime Submodule, Chiang Mai. J. Sci., 33(1), (2006), 3-7.##S. E. Atani, F. Farzalipour, Notes on the Graded Prime Submodules, International Mathematical Forum, 38(1), (2006), 1871-1880.##S. E. Atani, F. E. K. Saraei, Graded Modules Which Satisfy The Gr-Radical formula, Thai J. Math., 8(1), (2010), 161-170.##M. F. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, Longman Higher Education, New York, 1969.##M. Behboodi, M. R. Haddadi, Classical Zariski Topology of Modules and Spectral Spaces I, International Electrontc Journal Of Algebra, 4(4), (2008), 104-130.##M. Behboodi, M. R. Haddadi, Classical Zariski Topology of Modules and Spectral Spaces II, International Electrontc Journal of Algebra, 4(4), (2008), 131-148.##N. Bourbaki, Algebra Commutative, Hermann, Paris, 1961.##A. Y. Darani, Topology on Specg(M), Bul Acad. Stiinte Repub Mold. Mate., 3(67), (2011), 45-53.##A. Y. Darani, S. Motmaen, Zariski Topology on the Spectrum of Graded Classicl Prime Submodules, Appl. Gen Topol., 14(2), (2013), 159-169.##T. Duralvel, Topology On Spectrume of Module, J. Ramanujan Math. Soc., 9(1), (1994), 25-34.##R. Hazrat, Graded Rings and Graded Grothendieck Groups, Cambridge University Press, Cambridge, 2016.##M. Hochster, Prime Ideal Structure in Commutative Rings, Trans. Amer. Math. Soc, 137, (1969), 43-60.##M. Jesper. Moller, General Topology, Matematisk Institut, Universite tsparken 5, DK-2100 Kobenhavn.##E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry. Boston: Birkh¨auser., 1985.##S. C. Lee, R. Varmazyar, Semiprime Submodules of Graded Multiplication Modules, J. Korean Math. Soc., 49(2), (2012), 435-447.##C. P. Lu, Modules With Noetherian Spertrum, Comm. Algebra, 38(3), (2010), 807-828.##C. P. Lu, The Zariski Topology On the Prime Spectrum of a Module., Houston J. Math, 25(3), (1999), 417-432.##J. R. Munkres, Topology; A first Course, Prentice-Hall, Inc, Eaglewood Cliffs, New Jersey, 1975.##R. L. McCasland, M. E. Moore, t. P Smith, On the Spectrum Over a Commutative Ring, Comm. Algebra, 25, (1997), 79-103.##C. Nastasescu, F. Van Oystaeyen, Graded Ring Theory, Mathematical Library 28, NorthHolland, Amsterdam, 1982.##C. Nastasescu, V. F. Oystaeyen, Methods of Graded Rings, LNM 1836. Berlin-Heidelberg: Springer-Verlag, 2004.##K. H. Oral, U. Tekir, A. G. Agargun, On Graded Prime and Primary Submodules, Turk J. Math, 1(35), (2011), 159-167.##R. Ovlyaee-Sarmazdh, S. Maleki-Roudposhti, On Max-Injective Modules, Journal Of Algebra And Related Topics., 1(1), (2013), 57-66.##N. A. Ozkirisci, K. H. Oral and U. Tekir, Graded Prime Spectrum of a Graded, Iranian Journal Of Science and Technology, 4(4), (2013), 411-420.##M. Refai, On Properties of G-Spec(R), Sci. Math. Jpn., 4, (2001), 491-495.##M. Refai, K. Al-Zoubi, On Graded Primary Ideals, Turk. J. Math., 28, (2004), 217-229.##M. Refai, M. Hailat, S. Obiedat, Graded Radical on Graded Prime Spectra, Far East J. of Math. Sci., part I, (2000), 59-73.##R. N. Uregen, U. Tekir, K. H. Oral, On the :union: of Graded Prime Ideals, Open Phys., 14, (2016), 114-118.##H. A. Tavallaee, M. Zolfaghari, Graded Weakly Semiprime Submodules of Graded Multiplication Modules, Lobachevskii J. Math., 34(1), (2013), 61-67.## ##

On bi–bases of Γ–semihypergroups 2 2 This paper focuses on the Γ–semihypergroups. Our goal seeks to find the conditions of sub–Γ–semihypergroup using bi–bases properties. We provide definitions and explain some properties of bi–bases in Γ– semihypergroups. The findings extend the results from bi–bases of Γ– semigroups. The findings demonstrate that if B is a bi–bases of a Γ– semihypergroup H; then, B is a sub–Γ–semihypergroup of H if and only if for any b, c ∈ B and γ ∈ Γ, b ∈ bγc or c ∈ bγc. 41 50 2019/01/292019/06/62019/11/19 1398/8/28 2024/05/72024/05/252022/07/31 1401/5/9 Apipray Sarakam Apipray Sarakam Department of Mathematics, Faculty of Science, Buriram Rajabhat University, Buriram, Thailand, 31000 Thailand apiplay0805@gmail.com Pongpich Kothep Pongpich Kothep Department of Mathematics, Faculty of Science, Buriram Rajabhat University, Buriram, Thailand, 31000 Thailand Pongpich.iot@bru.ac.th Samkhan Hobanthad Samkhan Hobanthad Department of Mathematics, Faculty of Science, Buriram Rajabhat University, Buriram, Thailand, 31000 Thailand samkan.hb@bru.ac.th Γ–semihypergroup bi–bases sub–Γ–semihypergroup. F. Marty, Sur Uni Generalization de la Notion de Group, Congress Math. Scadenaves, Stockholm, Sweden, (1934), 45-49.##I. Fabrici, One-sided Bases of Semigroup, Matematicky casopis, 22(4), (1972), 286-290.##I. Fabrici, Two-sided Bases of Semigroup, Matematicky casopis, 3, (2009), 181-188.##N. Yaqoob, Rough Approximations in Γ-semihypergroups, Pakistan, Ph.D. Thesis, Department of Mathematics, Quaid-i-Azam University lslamabad, Pakistan, (2016).##N. Yaqoob, M. Aslam, K. Hila, B. Davvaz, Rough Prime Bi-Γ-Hyperideals and Fuzzy Prime Bi-Γ-Hyperideals of Γ-semihypergroupsr, Filomat., 31, (2017), 4167-4183.##P. Kummoon, T. Changphas, On bi-bases of a Semigroups, Quasigroups and Relatedsystems, 25, (2017), 87-94.##P. Kummoon, T. Changphas, Bi-Bases of Γ-semigroups, Thai J. Math., Special Issue, (2017), 75-86.##S. M. Anvariyeh, S. Mirvakili, B. Davvaz, On Γ-Hyperideals in Γ-semihypergroups, Carpathian J. Math., 26, (2010), 11-23.##T. Changphas, P. Summaprab, On two-sided Bases of an Ordered Semigroup, Quasigroups and Related Systems., 22, (2014), 59-66.## ##

G-Injective Envelope of Separable G-C*-algebras 2 2 Argerami and Farenick have found conditions for the injective envelope of a separable C∗-algebra to be a von Neumann algebra. In this paper, we introduce an equivalent version of this result by finding conditions for the G-injective envelope of a separable G-C∗-algebra A to be a von Neumann algebra, when G is a discrete group acting on A. 51 60 2019/01/292019/06/62019/11/192019/11/9 1398/8/18 2024/05/72024/05/252022/07/312021/02/6 1399/11/18 Ali Mahmoodi Ali Mahmoodi Department of Mathematics, Faculty of Science, Science and Research Branch, Islamic Azad University(IAU), Tehran, Iran Iran mahmoodi326@gmail.com Mohammad R. Mardanbeigi Mohammad R. Mardanbeigi Department of Mathematics, Faculty of Science, Science and Research Branch, Islamic Azad University(IAU), Tehran, Iran Iran mrmardanbeigi@srbiau.ac.ir G-W*-algebra G-AW*-algebra G-Injective envelope G-Regular monotone completion Type I C*-algebra G-invariant Essential ideal G-Local multiplier algebra Discrete group. P. Ara, M. Mathieu, Local Multipliers of C∗-algebras, Springer Monographs in Mathematics, London, 2003.##M. Argerami, D. R. Farenick, Local Multipliers Algebras, Injective Envelopes, and Type I W∗-algebras, J. Operator Theory, 59, (2008), 237-245.##M. Argerami, D. R. Farenick, Injective Envelopes of Separable C∗-algebras, Philos. Trans. R. Soc. A, 179, (2005), 43-63.##M.-D. Choi, E. G. Effros, Separable Nuclear C∗-algebras and Injectivity, Duke Math. J, 43, (1976), 309-322.##M. D. choi, E. G. Effros, Injectivity and Operator Spaces, Journal of Functional Analysis, 24, (1977), 156-209.##M. Frank, V. I. Paulsen, Injective Envelopes of C∗-algebras as Operator Modules, Pacific J. Math, 212, (2003), 57-69.##M. Hamana, Injective Envelopes of C∗-algebras, J. Math. Soc. Japen, 31, (1979), 181-197.##M. Hamana, Injective Envelope of C∗-dynamical Systems, Toˆhoku Math. Journ, 37, (1985), 463-487.##M. Hamana, Injective Envelope of Dynamical Systems, Toyama Math. J, 34, (2011), 23-86.##R. V. Kadison, Operator Algebras with a Faithful Weakly-closed Representation, Ann. of Math, 64, (1956), 175-181.##G. K. Pedersen, C∗-algebras and Their Automorphism Groups, London Mathematical Society, Monographs 14, 1979.##K. Saitˆo, J. D. M. Wright, Monotone Complete C∗-algebras and Generic Dynamics, Springer-Varlag, 2015.##M. Takesaki, Theory in Operator Algebras I, Encylopaedia of Mathematical Sciences, Springer-Verlag, New York, 124, 2001.## ##

Coupled Coincidence and Coupled Common Fixed Points of (ψ, ϕ) Contraction Type T-coupling in Metric Spaces 2 2 In this paper, we define (ψ, ϕ)-Contraction Type T-coupling, establish a theorem satisfying such contraction condition, and prove the existence and uniqueness of coupled coincidence and coupled common fixed points in metric space. Here ψ and ϕ are two altering distance functions and T is a SCC-Map for metric spaces. Our results extend and generalize several related results in the existing literature. We also provided two examples to verify our main results. 61 75 2019/01/292019/06/62019/11/192019/11/92018/11/27 1397/9/6 2024/05/72024/05/252022/07/312021/02/62024/05/6 1403/2/17 Fuad Abdulkerim Fuad Abdulkerim Department of Mathematics, Jimma University, Ethiopia Ethiopia Fuadabdulkerim993@gmail.com Kidane Koyas Kidane Koyas Department of Mathematics, Jimma University, Ethiopia Ethiopia kidanekoyas@yahoo.com Solomon Gebregiorgis Solomon Gebregiorgis Department of Mathematics, Jimma University, Ethiopia Ethiopia solomonggty@gmail.com Coupled coincidence point Coupled common fixed Point (ψ ϕ)-Contraction Type T-coupling SCC-Map. H. Aydi, M. Barakat, A. Felhi, H. Isik, On ϕ-contraction Type Couplings in Partial Metric Spaces, Journal of Mathematical Analysis, 8(4), (2017), 78-89.##S. Banach, Sur Les Op'erations Dans Les Ensembles Abstraits et Leur Application Aux Equations Int'egrales, Fund. math, 3(1), (1922), 133-181.##T. G. Bhaskar, V. Lakshmikantham, Fixed Point Theorems in Partially Ordered Metric Spaces and Applications, Nonlinear Analysis: Theory, Methods and Applications, 65(7), (2006), 1379-1393.##N. Bilgili, I. M. Erhan, E. Karapinar, D. Turkoglu, A Note on Coupled Fixed Point Theorems for Mixed g-monotone Mappings in Partially Ordered Metric Spaces, Fixed Point Theory Appl., 2014(120), (2014), 6.##B. S. Choudhury, A. A. Kundu, Coupled Coincidence Point Result in Partially Ordered Metric Spaces for Compatible Mappings, Nonlinear Anal., 73, (2010), 2524-2531.##B. S. Choudhury, A. A. Kundu, Two Coupled Weak Contraction Theorems in Partially Ordered Metric Spaces, Revista de la Real Academia de Ciencias Exactas., Fisicas y Naturales. Serie A. Matematicas, 108, (2014), 335-351.##B. S. Choudhury, P. Maity, Coupled Fixed Point Results in Generalized Metric Spaces, Math. Comput. Modelling., 54, (2011), 73-79.##B. S. Choudhury, P. Maity, Cyclic Coupled Fixed Point Result Using Kannan Type Contractions, Journal of Operators, 108, (2014), 335-351.##B. S. Choudhury, P. Maity, P. Konar, Fixed Point Results for Couplings on Metric Spaces, U.P. Sci. Bull., 79, (2017), 1-12.##F. Gu, W. Shatanawi, Some New Results on Common Coupled Fixed Points of Two Hybrid Pairs of Mappings in Partial Metric Spaces, J. Nonlinear Funct. Anal., 2019 (2019), Article ID 13.##E. Karapinar, B. Alqahtani, A. Fulga, On Ciric Type −ψ− Geraghty Contractions, Thai Journal of Mathematics, 17(1), (2019), 205-216.##M. S. Khan, M. Swaleh, S. Sessa, Fixed Point Theorems by Altering Distances Between the Points, Bull. Aust. Math. Soc., 30, (1984), 1-9.##W. A. Kirk, P. S. Srinivasan, P. Veeramani, Fixed Points for Mappings Satisfying Cyclical Contractive Conditions, Fixed Point Theory, 4, (2003), 78-89.##V. Lakshmikantham, L. Ciric, Coupled Fixed Point Theorems for Nonlinear Contractions in Partially Ordered Metric Spaces, Nonlinear Anal., 70, (2009), 4341-4349.##V. I. Opoitsev, Dynamics of Collective Behavior-Heterogenic Systems, Avtom. Telemeh., 36, (1975a), 124-138.##V. I. Opoitsev, Heterogenic and Combined-concave Operators, Sib. Math. J., 16, (1975b), 781-792.##V. I. Opoitsev, T. A. Khurodze, Nonlinear Operators in Spaces with a Cone, Tbilis. Gos. Univ., (1984), 271.##A. Petrusel, G. Petrusel, Fixed Point Results for Multi-valued Locally Contractive Operators, Appl. Set-Valued Anal. Optim., 2(2), (2020), 175-181.##K. P. R. Rao, Coupled Coincidence Point Theorems for (ψ, α, β)-Weak Contractions in Partially Ordered Partial Metric Spaces, Punjab University Journal of Mathematics, 45, (2013), 51-62.##T. Rashid, Q. H. Khan, Coupled Coincidence Point of ϕ-Contraction Type TCoupling and (ψ, ϕ)-Contraction Type Coupling in Metric Spaces, arXiv preprint arXiv:1710.10054 (2017).##B. Samet, C. Vetro, Coupled Fixed Point Theorems for Multi-valued Nonlinear Contraction Mappings in Partially Ordered Metric Spaces, Nonlinear Analysis: Theory, Methods & Applications, 74(12), (2011), 4260-4268.##M. Simkhah, D. Turkoglu, S. Sedghi, N. Shobe, Suzuki Type Fixed Point Results in p-metric Spaces, Commun. Optim. Theory, 2019(2019), Article ID 13.##N. Van Luong, N. X. Thuan, Coupled Fixed Points in Partially Ordered Metric Spaces and Application, Nonlinear Analysis: Theory, Methods & Applications, 74(3), (2011), 983-992.## ##

A Class of Commutative Semirings with Stable Range 2 II 2 2 The notion and some properties of (strongly) B-rings, in a natural way, are extended to (strongly) B- and (strongly) BJ-semirings which is somewhat similar to the notion of rings having stable range 2. Results are given showing the connection between several types of semirings whose finite sequences satisfy some stability condition, some involving the Jacobson k-radical of the semiring R. Besides some examples and other results, it is shown that R[x], the semiring of polynomials over a semiring R, is not a B-semiring (consequently, not a strongly B-semiring) when R is a zerosumfree semiring. We also study some algebraic properties of the S-relative B- and BJ-semirings with respect to a nonempty subset S of R. 77 93 2019/01/292019/06/62019/11/192019/11/92018/11/272019/11/29 1398/9/8 2024/05/72024/05/252022/07/312021/02/62024/05/62023/01/19 1401/10/29 Elham Mehdi-Nezhad Elham Mehdi-Nezhad Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville 7535, Cape Town, South Africa South Africa emehdinezhad@uwc.ac.za Amir M. Rahimi Amir M. Rahimi School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran Iran amrahimi@ipm.ir (Strongly) B- and (Strongly) B_J-semirings S-relative B- and Srelative B_J-semirings Subtractive ideal (= k-ideal) Simple semiring Gelfand semiring Polynomial semiring Stable range of a commutative semiring. M. S. Abdolyousefi, H. Chen, Elementary Matrix Reduction over J-stable Rings, Comm. Algebra, 45(5), (2017), 1983-1995.##H. Bass, K-theory and Stable Algebra, Publ. IHES, 22, (1964), 5-60.##R. A. Brualdi, H. J. Ryser, Combinatorial Matrix Theory, Cambridge University Press, 1991.##D. Estes, J. Ohm, Stable Range in Commutative Rings, Jour. of Alg., 7, (1967), 343-362.##J. S. Golan, The theory of Semirings (with Applications in Mathematics and Theoretical Computer Science), Pitman Monographs & Surveys in Pure and Appl. Math., 54, Longman Scientific & Tech., Essex, (1992).##W. H. Gustafson, M. E. Moore, I. Reiner, Matrix Completions over Dedekind Rings, Linear Multilinear Algebra, 10, (1981), 141-144.##W. H. Gustafson, R. B. Richter, D. W. Robinson, W. P. Wardlaw, Matrix Completions Over Principal Ideal Rings, Algebra Discrete Math., Hackensack, NJ: World Scientific Publishing, 1, (2008), 151-158.##J. R. Juett, J. L. Williams, Strongly Stable Rank and Applications to Matrix Completion, Comm. Algebra, (2017), 45(9), 3967-3985.##T. Y. Lam, Serre's Problem on Projective Modules, Springer Monogr. Math. Berlin: Springer-Verlag, 2006.##M. Moore, A. Steger, Some Results on Completability in Commutative Rings, Pacific Journal of Mathematics, 37, (1971), 453-460.##A. M. Rahimi, Some Results on Stable Range in Commutative Rings, PhD dissertation, University of Texas at Arlington, 1993.##A. M. Rahimi, A Class of Commutative Semirings with Stable Range 2, Quaestiones Mathematicae, 41(8), (2018), 1061-1071.##A. M. Rahimi, Some Improved Results on B-rings, Missouri J. Math. Sci., 9(3), (1997), 167-169.##A. M. Rahimi, Some Results on n-stable Rings, Missouri J. Math. Sci., 15(2), (2003), 129-139.##M. K. Sen, M. R. Adhikari, On Maximal k-ideals of Semirings, Proc. Amer. Math. Soc, 118, (1993), 699-703.## ##

Nonuniform Semi-orthogonal Wavelet Frames on Non-Archimedean Local Fields of Positive Characteristic 2 2 In this paper, we introduce the notion of nonuniform semiorthogonal wavelet frame associated with nonuniform frame multiresolution analysis on non-Archimedean fields and provide their characterization by means of some basic equations in the frequency domain. 95 109 2019/01/292019/06/62019/11/192019/11/92018/11/272019/11/292019/12/9 1398/9/18 2024/05/72024/05/252022/07/312021/02/62024/05/62023/01/192022/03/5 1400/12/14 Owais Ahmad Owais Ahmad Department of Mathematics, National Institute of Technology, Srinagar-190006, Jammu and Kashmir, India India siawoahmad@gmail.com Neyaz Ahmad Neyaz Ahmad Department of Mathematics, National Institute of Technology, Srinagar-190006, Jammu and Kashmir, India India neyaznit@yahoo.co.in Nonuniform semi-orthogonal wavelet frame Fourier transform Non-Archimedean field. O. Ahmad, N. Ahmad, Construction of Nonuniform Wavelet Frames on NonArchimedean Fields, Math. Phy. Anal. Geom., 23(47), (2020).##O. Ahmad, M. Y. Bhat, N. A. Sheikh, Construction of Parseval Framelets Associated with GMRA on Local Fields of Positive Characteristic, Numer. Funct. Anal. Opt., 42(3), (2021), 344-370.##O. Ahmad, Nonuniform Periodic Wavelet Frames on Non-Archimedean Fields, Annales Universitatis Mariae Curie-Sklodowska, sectio A - Mathematica, (2), (2020), 1-17.##O. Ahmad, N. A. Sheikh, M. A. Ali, Nonuniform Nonhomogeneous Dual Wavelet Frames in Sobolev Spaces in L2(K), Afr. Math., 31, (2020), 1145-1156.##O. Ahmad, N. Ahmad, Explicit Construction of Tight Nonuniform Framelet Packets on Local Fields, Operators and Matrices, 15(1), (2021), 131-149.##O. Ahmad, F. A. Shah, N. A. Sheikh, Gabor Frames on Non-Archimedean Fields,Int. J. Geom. Methods Mod. Phys., 15, (2018), Art. ID 1850079.##O. Ahmad, N. A. Sheikh, On Characterization of Nonuniform Tight Wavelet Frames on Local Fields,Anal. Theory Appl., 34, (2018), 135-146.##J. J. Benedetto, R. L. Benedetto, A Wavelet Theory for Local Fields and Related Groups, J. Geom. Anal., 14, (2004), 423-456.##P. G. Casazza, O. Christensen, Weyl-Heisenberg Frames for Subspaces of L2(R), Proc. Amer. Math. Soc., 129, (2001), 145-154.##O. Christensen, An Introduction to Frames and Riesz Bases, Birkh¨auser, Boston, 2003.##C. K. Chui, X. Shi, Inequalities of Littlewood-Paley Type for Frames and Wavelets, SIAM J. Math. Anal., 24, (1993), 263-277.##I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics, SIAM, Philadelphia, 1992.##I. Daubechies, A. Grossmann, Y. Meyer, Painless Non-orthogonal Expansions, J. Math. Phys., 27(5), (1986), 1271-1283.##R. J. Duffin, A. C. Shaeffer, A Class of Nonharmonic Fourier Series, Trans. Amer. Math. Soc., 72, (1952), 341-366.##J. P. Gabardo, M. Nashed, Nonuniform Multiresolution Analyses and Spectral Pairs, J. Funct. Anal., 158, (1998), 209-241.##H. K. Jiang, D. F. Li, N. Jin, Multiresolution Analysis on Local Fields, J. Math. Anal. Appl., 294, (2004), 523-532.##D. F. Li, H. K. Jiang, The Necessary Condition and Sufficient Conditions for Wavelet Frame on Local Fields, J. Math. Anal. Appl., 345, (2008), 500-510.##S. G. Mallat, Multiresolution Approximations and Wavelet Orthonormal Bases of L2(R), Trans. Amer. Math. Soc., 315, (1989), 69-87.##F. A. Shah, Abdullah, Nonuniform Multiresolution Analysis on Local Fields of Positive Characteristic, Complex Anal. Opert. Theory, 9, (2015), 1589-1608.##F. A. Shah, O. Ahmad, Wave Packet Systems on Local Fields, J. Geom. Phys, 120, (2017), 5-18.##F. A. Shah, M. Y. Bhat, Semi-orthogonal Wavelet Frames on Local Fields, Analysis, 36(3), (2016), 173-181.##X. L. Shi, F. Chen, Necessary Conditions and Sufficient Conditions of Affine Frame, Science in China: Series A, 48, (2005), 1369-1378.##M. H. Taibleson, Fourier Analysis on Local Fields, Princeton University Press, Princeton, NJ, 1975.## ##

Behavior of the Solutions of a Single-species Population Model with Piecewise Constant Argument 2 2 In this paper, we consider a population model with piecewise constant argument and show that every nonoscillatory solution approaches the equilibrium point as t tends to infinity. Moreover, we investigate every positive solution of the model that oscillates about the positive equilibrium point. Also, we give two examples to support the theorems. 111 117 2019/01/292019/06/62019/11/192019/11/92018/11/272019/11/292019/12/92020/01/3 1398/10/13 2024/05/72024/05/252022/07/312021/02/62024/05/62023/01/192022/03/52020/12/16 1399/9/26 Mehtap Lafci Büyükkahraman Mehtap Lafci Büyükkahraman Department of Mathematics, Usak University, Usak, 64200, Turkiye Turkey mehtap.lafci@usak.edu.tr Piecewise constant argument Difference equation Oscillation. A. R. Aftabizadeh, J. Wiener, Oscillatory Properties of First Order Linear Functional Differential Equations, Applicable Analysis, 20(3-4), (1985), 165-187.##A. R. Aftabizadeh, J. Wiener, J. M. Xu, Oscillatory and Periodic Solutions of Delay Differential Equations with Piecewise Constant Argument, Proceedings of the american mathematical society, (1987), 673-679.##M. Akhmet, Nonlinear Hybrid Continuous/discrete-time Models, Springer Science and Business Media, 8, 2011.##H. Bereketoglu, G. Seyhan, F. Karakoc, On a Second Order Differential Equation with Piecewise Constant Mixed Arguments, Carpathian Journal of Mathematics, 27(1), (2011), 1-12.##S. Busenberg, K. L. Cooke, Models of Vertically Transmitted Diseases with Sequentialcontinuous Dynamics, Nonlinear phenomena in mathematical sciences, (1982), 179-187.##K. S. Chiu, T. Li, Oscillatory and Periodic Solutions of Differential Equations with Piecewise Constant Generalized Mixed Arguments, Mathematische Nachrichten, 292(10), (2019), 2153-2164.##K. L. Cooke, J. Wiener, Retarded Differential Equations with Piecewise Constant Delays, Journal of Mathematical Analysis and Applications, 99(1), (1984), 265-297.##K. L. Cooke, J. Wiener, A Survey of Differential Equations with Piecewise Continuous Arguments, Delay differential equations and dynamical systems, (1991), 1-15.##K. Gopalsamy, M. R. S. Kulenovi'c, G. Ladas, On a Logistic Equation with Piecewise Constant Arguments, Differential and Integral equations, 4(1), (1991), 215-223.##K. Gopalsamy, G. Ladas, On the Oscillation and Asymptotic Behavior of N′(t) = N(t)[a + bN(t − τ) − cN2(t − τ), Quarterly of Applied Mathematics, 48, (1990), 433-440.##K. Gopalsamy, P. Liu, Persistence and Global Stability in a Population Model, Journal of Mathematical Analysis and Applications, 224(1), (1998), 59-80.##F. Gurcan, S. Kartal, I. Ozturk, F. Bozkurt, Stability and Bifurcation Analysis of a Mathematical Model for Tumor-immune Interaction with Piecewise Constant Arguments of Delay, Chaos, Solitons and Fractals, 68, (2014), 169-179.##F. Karako¸c, Asymptotic Behaviour of a Population Model with Piecewise Constant Argument, Applied Mathematics Letters, 70, (2017), 7-13.##T. K¨upper, R. Yuan, On Quasi-periodic Solutions of Differential Equations with Piecewise Constant Argument, Journal of mathematical analysis and applications, 267(1), (2002), 173-193.##C. G. Philos, On Oscillations of Some Difference Equations, Funkcial. Ekvac, 34(1), (1991), 157-172.##M. Pinto, Asymptotic Equivalence of Nonlinear and Quasi Linear Differential Equations with Piecewise Constant Arguments, Mathematical and Computer Modelling, 49(9-10), (2009), 1750-1758.##Y. Rong, H. Jialin, The Existence of Almost Periodic Solutions for a Class of Differential Equations with Piecewise Constant Argument, Nonlinear Analysis: Theory, Methods and Applications, 28(8), (1997), 1439-1450.##S. M. Shah, J. Wiener, Advanced Differential Equations with Piecewise Constant Argument Deviations, International Journal of Mathematics and Mathematical Sciences, 6, (1983).## ##

Rotundity of Quotient Spaces in Metric Linear Spaces 2 2 In this paper, we discuss the inheritance of strict convexity, uniform convexity and local uniform convexity by the quotient spaces of metric linear spaces. We also show that, as in the case of normed linear spaces, completeness is a three- space property in metric linear spaces as well. 119 126 2019/01/292019/06/62019/11/192019/11/92018/11/272019/11/292019/12/92020/01/32019/12/13 1398/9/22 2024/05/72024/05/252022/07/312021/02/62024/05/62023/01/192022/03/52020/12/162023/10/29 1402/8/7 Shelly Garg Shelly Garg Department of Mathematics, DAV University, Jalandhar-144 012, India India shellygarg24@gmail.com Harpreet K. Grover Harpreet K. Grover Department of Mathematics, Guru Nanak Dev University, Amritsar-143 005, India India harpreetgr@gmail.com T. D. Narang T. D. Narang Department of Mathematics, Guru Nanak Dev University, Amritsar-143 005, India India tdnarang1948@yahoo.co.in Metric linear space Strict convexity Uniform convexity Local uniform convexity Three-space property. G. C. Ahuja, T. D. Narang, S. Trehan, An Approximation Problem in Metric Linear Spaces, Math. Student, 40, (1972), 447-448.##G. C. Ahuja, T. D. Narang, S. Trehan, Best Approximation on Convex Sets in Metric Linear Spaces, Math. Nachr., 78(1), (1977), 125-130.##Naum I. Akhiezer, Mark G. Krein, Some Questions in The Theory of Moments, Translations of Mathematical Monographs, AMS, Providence, 2, 1962.##G. Albinus, Normaritge Metriken auf Metrisierbaren Lokalkovexen Topologischen Vektor¨ aumen, Math. Nachr., 37, (1968), 177-196.##A. R. Alimov, I. G. Tsar'kov, Approximatively Compact Sets in Asymmetric EfimovSteckin Spaces and Convexity of Almost Suns, Math. Notes, 110(6), (2021), 947-951.##James A. Clarkson, Uniformly Convex Spaces, Trans. Amer. Math. Soc., 40(3), (1936), 396-414.##Shelly Garg, Harpreet K. Grover, T. D. Narang, Different Forms of Convexity in Metric Linear Spaces, Electronic Journal of Mathematical Analysis and Applications, to appear.##Victor L. Klee, Some New Results on Smoothness and Rotundity in Normed Linear Spaces, Math. Ann., 139, (1959), 51-63.##A. R. Lovaglia, Locally Uniformly Convex Banach Spaces, Trans. Amer. Math. Soc., 78(1), (1955), 225-238.##Robert E. Megginson, An Introduction to Banach Space Theory, Springer-Verlag, 1998.##T. D. Narang, Different Types of Convexities in Metric Linear Spaces, Jnanabha, 7, (1977), 17-20.##Stefan Rolewicz, Metric Linear Spaces, Mathematics and Applications, D. Reidel Publishing Company, 1985.##Walter Rudin, Functional Analysis, Tata McGraw- Hill Publishing Company Ltd., 1974.##K. P. R. Sastry, S. V. R. Naidu, Convexity Conditions in Metric Linear Spaces, Mathematics Seminar Notes, 7, (1979), 235-251.##K. P. R. Sastry, S. V. R. Naidu, Uniform Convexity and Strict Convexity in Metric Linear Spaces, Math. Nachr., 104, (1981), 331-347.##I. G. Tsar'kov, Uniform Convexity in Nonsymmetric Spaces, Math. Notes, 110(5), (2021), 773-783.##A. I. Vasil'ev, Chebyshev Sets and Strong Convexity of Metric Linear Spaces, Math.Notes, 25, (1979), 335-340.##A. I. Vasil'ev, Approximative Compactness in Linear Metric Spaces, Math. Nachr., 160, (1993), 329-341.## ##

Numerical Approach of Cattaneo Equation with Time Caputo-Fabrizio Fractional Derivative 2 2 In the paper, we consider a type of Cattaneo equation with time fractional derivative without singular kernel based on fourth-order compact finite difference (CFD) in the space directions. In case of two dimensional, two alternating direction implicit (ADI) methods are proposed to split the equation into two separate one dimensional equations. The time fractional derivation is described in the Caputo-Fabrizio’s sense with scheme of order O(τ2). The solvability, unconditional stability and H1 norm convergence of the scheme are proved. Numerical results confirm the theoretical results and the effectiveness of the proposed scheme. 127 153 2019/01/292019/06/62019/11/192019/11/92018/11/272019/11/292019/12/92020/01/32019/12/132020/01/5 1398/10/15 2024/05/72024/05/252022/07/312021/02/62024/05/62023/01/192022/03/52020/12/162023/10/292021/06/5 1400/3/15 Zoleikha Soori Zoleikha Soori Department of Mathematics, K. N. Toosi University of Technology, P. O. Box: 1676-53381, Tehran, Iran Iran zsoori@mail.kntu.ac.ir Azim Aminataei Azim Aminataei Department of Mathematics, K. N. Toosi University of Technology, P. O. Box: 1676-53381, Tehran, Iran Iran ataei@kntu.ac.ir Caputo-Fabrizio fractional derivative Compact finite difference Cattaneo equation Alternating direction implicit method. T. Akman, B. Yildiz, D. Baleanu, New Discretization of Caputo-Fabrizio Derivative, Comp. Appl. Math., 37, (2018), 3307-3333.##A. Atangana, On the New Fractional Derivative and Application to Nonlinear Fishers Reaction-diffusion Equation, Appl. Math. Comput., 273, (2016), 948-956.##A. Atangana, B. S. T. Alkahtani, Extension of the Resistance, Inductance, Capacitance Electrical Circuit to Fractional Derivative without Singular Kernel, Adv. Mech. Eng., 7, DOI:10.1177/1687814055911937.##A. Atangana, R. T. Alqahtani, Numerical Approximation of the Space-time CaputoFabrizio Fractional Derivative and Application to Groundwater Pollution Equation, Adv. Differ. Equ., 156, DOI:10.1186/s13662-016-0871-x.##A. Atangana, J. J. Nieto, Numerical Solution for the Model of RLC Circuit via the Fractional Derivative without Singular Kernel, Adv. Mech. Eng., 7, (2015), 1-7.##S. M. Aydogan, D. Baleanu, A. Mousalou, S. Rezapour, On Approximate Solutions for Two Higher-order Caputo-Fabrizio Fractional Integro-differential Equations, Adv. Differ. Equ., 221, (2017), 1-11.##M. Caputo, M. Fabrizio, A New Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl., 1, (2015), 73-85.##J. D. Djida, A. Atangana, More Generalized Groundwater Model with Space-time Caputo Fabrizio Fractional Differentiation, Numer. Methods Partial Differ. Equ., 33, (2017), 1616-1627.##B. Fakhr Kazemi, H. Jafari, H, Error Estimate of the MQ-RBF Collocation Method for Fractional Differential Equations with Caputo-Fabrizio Derivative, Math. Sci., 11, (2017), 297-305.##G. Femandz-Anaya, F. J. Valdes-Parada, J. Alvarea-Ramirez, On Generalized Fractional Cattaneo's Equation, Physica A: statical. Mech. Appl., 390, (2011), 4198-4202.##M. A. Firoozjaee, H. Jafari, A. Lia, D. Baleanu, Numerical Approach of Fokker-Planck Equation with Caputo-Fabrizio Fractional Derivative Using Ritz Approximation, J. Compu. Appl. Math., 339, (2018), 367-373.##H. P. Ghazizadeh, M. Maerefat, A. Azim, Explicit and Implicit Finite Difference Schemes for Fractional Cattaneo Equation, J. Comput, Phys, 229, (2010), 7042-7057.##F. El-Ghenbazia, M. Tayeb Meftah, Solution of Nonlocal Schrodinger Equation via the Caputo-Fabrizio Definition for Some Quantum Systems, Reports. Math. Physics, 85, (2020), 57-67.##B. Guo, X. Pu, F. Huang, Fractional Partial Differential Equations and Their Numerical Solutions, Science Press, Beijing, China, 2011.##R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.##Z. Liu, A. Cheng, X. Li, A Second Order Crank-Nicolson Scheme for Fractional Cattaneo Equation Based on New Fractional Derivative, Appl. Math. Comput., 311, (2017), 361-374.##J. R. Loh, Y. T. Toh, On the New Properties of Caputo-Fabrizio Operator and its Application in Deriving Shifted Legendre Operational Matrix, Appl. Numer. Math., 132, (2018), 138-153.##J. Losada, J. J. Nieto, Properties of a New Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl., 1, (2015), 87-92.##K. M. Owolabi, A. Atangana, Analysis and Application of New Fractional AdamsBashforth Scheme with Caputo-Fabrizio Derivative, Chaos. Solitons. Fract, 105, (2017), 111-119.##D. W. Peaceman, H. H. Rachford, The Numerical Solution of Parabolic and Elliptic Differential Equations, J. Soc. Ind. Appl. Math., 3, (1955), 23-41.##I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.##H. T. Qi, X. W. Guo, The Cattaneo-type Time Fractional Heat Conduction Equation for Laser Heating, Fract. Difer. Appl., 66, (2013), 834-831.##H. Qi, X. Jiang, Solution of the Space-time Fractional Cattaneo Diffusion Equation, Physica A: statical. Mech. Appl., 390, (2011), 1876-1883.##J. Ren, G. H. Gao, Efficient and Stable Numerical Methods for the Two-dimensional Fractional Cattaneo Equation, Numer. Algor., 69, (2015), 795-818.##J. Ren, Z. Sun, Efficient Numerical Solution of the Multi-term Time Fractional Diffusionwave Equation, East Asian J. Appl. Math., 5, (2015), 1-28.##A. A. Samarskii, V. B. Andreev, Difference Methods for Elliptic Equation, Nauka, Moscow, 1976.##J. She, M. Chen, A Second-order Accurate Scheme for Two-dimensional Space Fractional Diffusion Equations with Time Caputo-Fabrizio Fractional Derivative, Appl. Numer. Math., 151, (2020), 246-262.##Z. Soori, A. Aminataei, A New Approximation to Caputo-type Fractional Diffusion and Advection Equations on Non-uniform Meshes, Appl. Numer. Math., 144, (2019), 21-41.##Z. Soori, A. Aminataei, Effect of the Nodes Near Boundary Points on the Stability Analysis of Sixth-order Compact Finite Difference ADI Scheme for the Two-dimensional Time Fractional Diffusion-wave Equation, Transactions of A. Razmadze Mathematical Institute, 172, (2018), 582-605.##Z. Soori, A. Aminataei, Numerical Solution of Space Fractional Diffusion Equation by Spline Method Combined with Richardson Extrapolation, Comput. Appl. Mathe., 139, (2020), 1-18.##Z. Soori, A. Aminataei, Sixth-order Non-uniform Combined Compact Difference Scheme for Multi-term Time Fractional Diffusion-wave Equation, Appl. Numer. Math., 131, (2018), 72-94.##J. V. C. Sousa, E. C. Oliveira, Two New Fractional Derivatives of Variable Order with Non-singular Kernel and Fractional Differential Equation, Comp. Appl. Math., 37, (2018), 5375-5394.##Z. Z. Sun, The Method of Order Reduction and Its Application to the Numerical Solutions of Partial Differential Equations, Science Press, Beijing, 2009.##X-J. Yang, H. M. Machado, A New Fractional Operator of Variable Order: Application in the Description of Anomalous Diffusion, Phys A. Stat. Mech. Appl., 481, (2017), 276-283.##X-J. Yang, H. M. Srivastava, T. J. A. Machado, A New Fractional Derivative without Singular Kernel, Therm. Sci., 20, (2016), 753-756.##Y. N. Zhang, Z. Z. Sun, X. Zhao, Compact Alternating Direction Implicit Scheme for the Two-dimensional Fractional Diffusion-wave Equation, Siam J. Numer. Anal., 50, (2012), 1535-1555.## ##

A Hybrid Method to Systems of Fredholm Integral Differential Equations 2 2 The method used in this research consists of a hybrid of the Block-Pulse functions and third-kind Chebyshev polynomials for solving systems of Fredholm integral differential equations. Through the use of an operational matrix representing the derivation, the problem is represented by a system of algebraic equations. Some examples are provided to illustrate the simplicity and effectiveness of the utilized method. In addition, results of the presented method have been compared with those obtained from the Tau method and variational iteration method that reveal the proposed scheme to be more applicable. 155 167 2019/01/292019/06/62019/11/192019/11/92018/11/272019/11/292019/12/92020/01/32019/12/132020/01/52020/01/24 1398/11/4 2024/05/72024/05/252022/07/312021/02/62024/05/62023/01/192022/03/52020/12/162023/10/292021/06/52024/04/28 1403/2/9 Jafar Biazar Jafar Biazar Department of Applied Mathematics, Faculty of Mathematical Science, University of Guilan, Rasht, Iran Iran biazar@guilan.ac.ir yalda Parvari Moghaddam yalda Parvari Moghaddam Department of Applied Mathematics, University Campus 2, University of Guilan, Rasht, Iran Iran yaldaparvari.m@gmail.com Khadijeh Sadri Khadijeh Sadri Mathematics Research Center, Near East University TRNC, Mersin 10, Nicosia 99138, Turkey Turkey khadijeh.sadrikhatouni@neu.edu.tr System of Fredholm integral differential equations Hybrid method Block-pulse functions Third-kind Chebyshev polynomials Operational matrix. J. Biazar, H. Ebrahimi, A Strong Method for Solving System of Integral Differential Equations, Applied Mathematics, 2(9), (2011), 1105-1113.##A. Hosry, R. Nakad, S. Bhalekar, A Hybrid Function Approach to Solving a Class of Fredholm and Volterra Integro-Differential Equations, Math. Comput. Appl., 25(2), (2020), 30.##C. Hsiao, Hybrid Function Method for Solving Fredholm and Volterra Integral Equations of the Second Kind, Journal of Computational and Applied Mathematics, 230(1), (2009), 59-68.##R. Jafari, R. Ezzati, K. Maleknejad, Numerical Solution of Fredholm Integro-Differential Equations By Using Hybrid Function Operational Matrix of Differentiation, Int. J. Industrial Mathematics, 9(4), (2017), 349-358.##Z. H. Jiang, W. Schaufelberger, Block-Pulse Functions and Their Applications in Control Systems, Springer-Verlag, Berlin, Heidelberg, 1992.##K. Maleknejad, Y. Mahmoudi, Numerical Solution of Linear Fredholm Integral Equation by Using Hybrid Taylor and Block-Pulse Functions, Appl. Math. Comput., 149, (2004), 799-806.##K. Maleknejad, M. Tavassoli Kajani, A Hybrid Collocation Method Based on Combining the Third Kind Chebyshev Polynomials and Block-Pulse Functions for Solving HigerOrder Initial Value Problems, Kuwait journal of Science, 43(4), (2016), 1-10.##J. Pour-Mahmoud, M. Y. Rahimi-Ardabili, S. Shahmorad, Numerical Solution of the System of Fredholm Integro-Differential Equations by the Tau Method, Applied Mathematics and Computation, 168(1), (2005), 465-478.##G. P. Rao, L. Sivakumar, Analysis Analysis and Synthesis of Dynamic Systems Containing Time-Delays Via Block-Pulse Functions, Proc. IEE, 125, (1978), 1064-1068.##M. Razzaghi, Fourier Series Approach for the Solution of Linear Two-Point Boundary Value Problems with Time-Varying Coefficients, Int. J. Syst. Sci., 21(9), (1990), 1783-1794.##P. Sannuti, Analysis and Synthesis of Dynamic Systems Via Block-Pulse Functions, Proc. Inst. Elect. Eng., 124(6), (1977), 569-571.##M. Tavassoli Kajani, A. Hadi Vencheh, Solving Second Kind Integral Equations with Hybrid Chebyshev and Block-Pulse Functions, Appl. Math. Comput., 163(1), (2005), 71-77.##X. T. Wang, Y. M. Li, Numerical Solutions of Integro-Differential Systems by Hybrid of General Block-Pulse Functions and the Second Chebyshev Polynomials, Appl. Math. Comput., 209(2), (2009), 266-272.##X. T. Wang, Numerical Solutions of Optimal Control for Time Delay Systems by Hybrid of Block-Pulse Functions and Legendre Polynomials, Appl. Math. Comput., 184(2), (2007), 849-856.##X. T. Wang, Numerical Solution of Time-Varying Systems with a Stretch by General Legendre Wavelets, Appl. Math.comput., 198(2), (2008), 613-620.##A. Wazwaz, Linear and Nonlinear Integral Equations, Methods And Applications, Springer Berlin, Heidelberg, 2011.## ##

On a Group of the Form $2^{4+5}:GL(4,2)$ 2 2 The affine general linear group 25:GL(5, 2) of GL(6, 2) has 6 conjugacy classes of maximal subgroups. The largest two maximal subgroups are of the forms 21+8:GL(4, 2) and 24+5:GL(4, 2). In this article we consider the group 24+5:GL(4, 2), which we denote by ${bar G}$. Firstly we determine its conjugacy classes using the coset analysis technique. The structures of the inertia factor groups are also determined. We then compute all the Fischer matrices and apply the Clifford-Fischer theory to compute the ordinary character table of ${bar G}$. Using information on conjugacy classes, Fischer matrices and both ordinary and projective character tables of the inertia factor groups, we concluded that we need to use the ordinary character tables of all the inertia factor groups to construct the character table of ${bar G}$. The character table of ${bar G}$ is a 75×75 complex valued matrix and we supply it (in the format of Clifford-Fischer theory) at the end of this paper as Table 6. 169 188 2019/01/292019/06/62019/11/192019/11/92018/11/272019/11/292019/12/92020/01/32019/12/132020/01/52020/01/242020/08/10 1399/5/20 2024/05/72024/05/252022/07/312021/02/62024/05/62023/01/192022/03/52020/12/162023/10/292021/06/52024/04/282021/01/5 1399/10/16 A. B. M. Basheer A. B. M. Basheer School of Mathematical and Computer Sciences, University of Limpopo (Turfloop), P Bag X1106, Sovenga 0727, South Africa South Africa ayoubbasheer@gmial.com J. Moori J. Moori School of Mathematical and Statistical Sciences, PAA Focus Area, North-West University (Mahikeng), P. Bag X2046, Mmabatho 2790, South Africa South Africa jamshid.moori@nwu.ac.za A. L. Prins A. L. Prins Department of Pure and Applied Mathematics, University of Fort Hare, Alice, 5700, South Africa South Africa aprins@ufh.ac.za T. T. Seretlo T. T. Seretlo School of Mathematical and Statistical Sciences, PAA Focus Area, North-West University (Mahikeng), P. Bag X2046, Mmabatho 2790, South Africa South Africa thekiso.seretlo@ul.ac.za Group extensions General linear group Character table Inertia factor groups Fischer matrices. A. B. M. Basheer, Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups, PhD Thesis, University of KwaZulu-Natal, Pietermaitzburg, 2012.##A. B. M. Basheer, On a Group Extension Involving the Sporadic Janko Group J2, Ural Mathematical Journal, 10(1), (2024), 28-43.##A. B. M. Basheer, On a Group Involving the Automorphism of the Janko Group J2, Journal of the Indonesian Mathematical Society, 29(2), (2023), 197-216.##A. B. M. Basheer, On a Group Extension Involving the Suzuki Group Sz(8), Afrika Matematika, 34(4), (2023), ##https://doi.org/10.1007/s13370-023-01130-z##A. B. M. Basheer, J. Moori, Fischer Matrices of Dempwolff Group 25·GL(5, 2), International Journal of Group Theory, 1(4), (2012), 43-63.##A. B. M. Basheer, J. Moori, On the Non-split Extension Group 26·Sp(6, 2), Bulletin of the Iranian Mathematical Society, 39(6), (2013), 1189-1212.##A. B. M. Basheer, J. Moori, On the Non-split Extension 22n·Sp(2n, 2), Bulletin of the Iranian Mathematical Society, 41(2), (2015), 499-518.##A. B. M. Basheer, J. Moori, On a Maximal Subgroup of the Thompson Simple Group, Mathematical Communications, 20, (2015), 201-218.##A. B. M. Basheer, J. Moori, A Survey on Clifford-Fischer Theory, London Mathematical Society Lecture Notes Series, 422, published by Cambridge University Press, (2015), 160-172.##A. B. M. Basheer, J. Moori, On a Group of the Form 37:Sp(6, 2), International Journal of Group Theory, 5(2), (2016), 41-59.##A. B. M. Basheer, J. Moori, On a Group of the Form 210:(U5(2):2), Italian Journal of Pure and Applied Mathematics, 37, (2017), 645-658.##A. B. M. Basheer, J. Moori, On Two Groups of the Form 28:A9, Afrika Matematika, 28, (2017), 1011 - 1032.##A. B. M. Basheer, J. Moori, Clifford-Fischer Theory Applied to a Group of the Form 21+6 − :((31+2:8):2), Bulletin of the Iranian Mathematical Society, 43(1), (2017), 41 - 52.##A. B. M. Basheer, F. Ali, M. Alotaibi, On a Maximal Subgroup of the Conway Simple Group Co3, Italian Journal of Pure and Applied Mathematics, 44, (2020), 357-372.##A. B. M. Basheer and J. Moori, On a Maximal Subgroup of the Affine General Linear Group AGL(6, 2), Advances in Group Theory and Applications, 11, (2021), 1-30##W. Bosma, J. J. Cannon, Handbook of Magma Functions, Department of Mathematics, University of Sydeny, November 1994.##J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985.##The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.4.10 ; 2007. (http://www.gap-system.org)##G. Karpilovsky, Projective Representations of Finite Groups, Marcel Dekker, New York and Basel, 1985.##Maxima, a Computer Algebra System. Version 5.18.1; 2009. (http://maxima.sourceforge.net)##J. Moori, On the Groups G+ and G of the Form 210:M22 and 210:M22, PhD Thesis, University of Birmingham, 1975.##J. Moori, On Certain Groups Associated with the Smallest Fischer Group, J. London Math. Soc., 2, (1981), 61 - 67.##A. L. Prins, Fischer-Clifford Applied to a Non-split Extension 25·GL4(2), Palestine Journal of Mathematics, 5 (2016), 71-82.##A. L. Prins, The Projective Character Tables of a Solvable Group 26:(6×2), International Journal of Mathematics and Mathematical Sciences, 2019, (2019), Article ID 8684742, 15 pages.##J. Schmidt, Projective Characters with Corresponding Factor Set, URL (version: 2017-04-13): https://mathoverflow.net/q/165226, (https://mathoverflow.net/users/3710/jack-schmidt).## ##

A Necessary Condition for Zero Divisors in Complex Group Algebra of Torsion-Free Groups 2 2 It is proved that if $sum_{gin G} a_g g$ is a non-zero zero divisor element of the complex group algebra $mathbb{C}G$ of a torsion-free group G then $2sum_{gin G} |a_g|^2 189 194 2019/01/292019/06/62019/11/192019/11/92018/11/272019/11/292019/12/92020/01/32019/12/132020/01/52020/01/242020/08/102021/01/2 1399/10/13 2024/05/72024/05/252022/07/312021/02/62024/05/62023/01/192022/03/52020/12/162023/10/292021/06/52024/04/282021/01/52022/02/16 1400/11/27 Alireza Abdollahi Alireza Abdollahi Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan, Iran 81746-73441 Iran a.abdollahi@math.ui.ac.ir Meisam Soleimani Malekan Meisam Soleimani Malekan Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785-136, Iran Iran msmalekan@gmail.com Hilbert space $ell^2(G)$ Complex group algebras Zero divisor conjecture Torsion-free groups. K. A. Brown, On Zero Divisors in Group Rings, Bull. Lond. Math. Soc., 8, (1976), 251-256.##J. M. Cohen, Zero Divisors in Group Rings, Comm. Algebra, 2, (1974), 1-14.##T. Delzant, Sur lanneau Dun Groupe Hyperbolique, C. R. Acad. Sci. Paris Ser. I Math, 324(4), (1997), 381-384.##G. Elek, On the Analytic Zero Divisor Conjecture of Linnell, Bulletin of the London Mathematical Society, 35(2), (2003), 236-238.##D. R. Farkas, R. L. Snider, K0 and Noetherian Group Rings, J. Algebra, 42, (1976), 192-198.##E. Formanek, The Zero Divisor Question for Supersolvable Groups, Bull. Aust. Math. Soc., 73c, (1973), 67-71.##P. A. Linnell, Zero Divisors and Group von Neumann Algebras, Pacific J. Math., 149, (1991), 349-363.##P. A. Linnell, Division Rings and Group von Neumann Algebras, Forum Math., 5(6), (1993), 561-576.##P. A. Linnell, Analytic Versions of the Zero Divisor Conjecture, In Geometry and cohomology in group theory (Durham 1994), 252. London Math. Soc. Lecture Note, 1998.##P. A. Linnell, P. H. Kropholler, J. A. Moody, Applications of a New K-theoretic Theorem to Soluble Group Rings, Proc. Amer. Math. Soc., 104, (1988), 675-684.##A. I. Malcev, On Embedding of Group Algebras in a Division Algebra (in russian), Dokl. Akad. Nauk., 60, 1948, 1499-1501.##B. H. Neumann, On Ordered Division Rings, Trans. Amer. Math. Soc., 66, (1949), 202-252.## ##

Pointwise Inner and Center Actors of a Lie Crossed Module 2 1 خلاصه فرض کنید L یک لی مدول متقاطع شده و ACT_pi(L) و ACT_z(L) به ترتیب اکتور نقطه ای داخلی و اکتور مرکزی از L باشند. در این مقاله یک شرط لازم و کافی روی L پیدا می کنیم که ACT_pi(L) و ACT_z(L) با هم برابر باشند. 2 Let L be a Lie crossed module and Actpi(L) and Actz(L) be the pointwise inner actor and center actor of L, respectively. We will give a necessary and sufficient condition under which Actpi(L) and Actz(L) are equal. 195 206 2019/01/292019/06/62019/11/192019/11/92018/11/272019/11/292019/12/92020/01/32019/12/132020/01/52020/01/242020/08/102021/01/22020/02/5 1398/11/16 2024/05/72024/05/252022/07/312021/02/62024/05/62023/01/192022/03/52020/12/162023/10/292021/06/52024/04/282021/01/52022/02/162023/02/20 1401/12/1 M. Jamshidi M. Jamshidi Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran Iran mehdijamshidi44@yahoo.com F. Saeedi F. Saeedi Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran Iran saeedi@mashdiau.ac.ir Pointwise Inner Crossed Module Center Actor. A. Allahyari, F. Saeedi, Some Properties of Isoclinism of Lie Algebra Crossed Module, Asian-Eur. J. Math., 13(3), (2020), 2050055 (14 pages).##A. Allahyari, F. Saeedi, On Nilpotency of Outer Pointwise Inner Actor of the Lie Algebra Crossed Modules, Journal of Mathematical Extension, 14(1), (2020), 19-40.##J. M. Casas, Invariants de Modulos Cruzados en Algebras de Lie, Ph.D. Thesis, Universty of Santiago, 1991.##J. M. Casas, M. Ladra, Perfect Crossed Modules in Lie Algebras, Comm. Algebra, 23, (1995), 1625-1644.##J. M. Casas, M. Ladra, The Actor of a Crossed Module in Lie Algebra, Comm. Algebra, 26, (1998), 2065-2089.##J. M.Casas, R. Fernandez-Casado, X. Garcia-Martinez, E. Khmaladze, Actor of a Crossed Module of Leibniz Algebras, Theory Appl. Categ., 33, (2018), 23-42.##B. Edalatzadeh, Capability of Crossed Modules of Lie Algebras, Comm. Algebra, 42, (2014), 3366-3380.##R. Lavendhomme, J. R. Roisin, Cohomologie Non Abelienne de Structures Algebriques, J. Algebra, 67, (1980), 385-414.##K. Norrie, Actions and Automorphisms of Crossed Modules, Bull. Soc. Math. France, 118(2), (1990), 129-146.##S. Sheikh-Mohseni, F. Saeedi, M. Badrkhani-Asl, On Special Subalgebras of Derivations of Lie Algebras, Asian-European J. Math., 8(2), (2015), 1550032 (12 pages).##J. H. C. Whitehead, Combinatorial Homotopy II, Bull. Amer. Math. Soc., 55, (1949), 453-496.## ##

Generalized Kernels of Subsets through Ideals in Topological Spaces 2 2 In this research work, we introduce a generalization of the notion of kernel of a set in topological spaces endowed with an ideal, which is a fundamental tool to obtain new modifications of open sets and closed sets. Using this generalized kernel, we define and characterize new low separation axioms in other contexts obtained from a topological space endowed with an ideal. Also, we study the invariance of these low separation axioms under certain types of continuity defined in this novel theoretical framework. 207 221 2019/01/292019/06/62019/11/192019/11/92018/11/272019/11/292019/12/92020/01/32019/12/132020/01/52020/01/242020/08/102021/01/22020/02/52020/02/19 1398/11/30 2024/05/72024/05/252022/07/312021/02/62024/05/62023/01/192022/03/52020/12/162023/10/292021/06/52024/04/282021/01/52022/02/162023/02/202022/05/24 1401/3/3 José Sanabria José Sanabria Departamento de Matematicas, Facultad de Educacion y Ciencias, Universidad de Sucre, Sincelejo, Colombia Colombia jesanabri@gmail.com Laura Maza Laura Maza Maestrıa en Ciencias Matematicas, Facultad de Ciencias Basicas, Universidad del Atlantico, Barranquilla, Colombia Colombia lamar_91-10@hotmail.com Ennis Rosas Ennis Rosas Departamento de Ciencias Naturales y Exactas, Universidad de la Costa, Barranquilla, Colombia Venezuela ennisrafael@gmail.com Carlos Carpintero Carlos Carpintero Departamento de Ciencias Basicas, Corporacion Universitaria del Caribe-CECAR, Sincelejo, Colombia Colombia carpintero.carlos@gmail.com Topological kernel Ideals Co-local function $(tau^{star} tau^{bullet})$-g-closed set $mathcal{I}$-$lambda$-closed set. A. Al-Omari, T. Noiri, Local Closure Functions in Ideal Topological Spaces, Novi Sad J. Math., 43(2), (2013), 139-149.##F. G. Arenas, J. Dontchev, M. Ganster, On λ-sets and the Dual of Generalized Continuity, Questions Answers Gen. Topology, 15(1), (1997), 3-13.##M. Caldas, D. N. Georgiou, S. Jafari, Study of (Λ, α)-closed Sets and the Related Notions in Topological Spaces, Bull. Malays. Math. Sci. Soc. (2), 30(1), (2007), 23-36.##M. Caldas, D. N. Georgiou, S. Jafari, T. Noiri, On (Λ, θ)-closed Sets, Questions Answers Gen. Topology, 23, (2005), 69-87.##M. Caldas, S. Jafari, T. Noiri, On Λb-sets and the Associated Topology τΛb, Acta Math. Hungar., 110, (2006), 337-345.##J. Dontchev, H. Maki, On sg-closed Sets and Semi-λ-closed Sets, Questions Answers Gen. Topology, 15(2), (1997), 259-266.##M. Ganster, S. Jafari, T. Noiri, On Pre-Λ-sets and Pre-V -sets, Acta Math. Hungar., 95, (2002), 337-343.##D. N. Georgiou, S. Jafari, T. Noiri, Properties of (Λ, δ)-closed Sets in Topological Spaces, Boll. :union:e Mat. Ital., Serie 8, 7-B(3), (2004), 745-756.##D. S. Jankovic, T. R. Hamlett, New Topologies from Old via Ideals, Amer. Math. Monthly, 97, (1990), 295-310.##E. D. Khalimsky, R. Kopperman, P. R. Meyer, Computer Graphics and Connected Topologies on Finite Ordered Sets, Topology Appl., 36, (1990), 1-17.##K. Kuratowski, Topologie I, Monografie Matematyczne tom 3, PWN-Polish Scientific Publishers, Warszawa, 1933.##H. Maki, Generalized Λ-sets and the Associated Closure Operator, The Special Issue in Commemoration of Prof. Kazusada IKEDA's Retirement, (1986), 139-146.##S. Modak, T. Noiri, Some Generalizations of Locally Closed Sets, Iran. J. Math. Sci. Inform., 14(1), (2019), 159-165.##T. Noiri, E. Hatir, Λsp-sets and Some Weak Separation Axioms, Acta. Math. Hungar., 103(3), (2004), 225-232.##T. Noiri, A. Keskin, On ΛI-sets and Some Weak Separation Axioms, Int. J. Math. Anal., 5(11), (2011), 539-548.##J. Sanabria, E. Rosas, C. Carpintero, Una Generalizaci'on de Λs-conjuntos y Vs-conjuntos Mediante Operadores Asociados a una Topolog'ıa y Funciones Asociadas, Rev. Colombiana Mat., 40(2), (2006), 87-103.##J. Sanabria, E. Rosas, C. Carpintero, On Λs I-sets and the Related Notions in Ideal Topological Spaces, Math. Slovaca, 63(6), (2013), 1403-1411.##N. A. Shanin, On Separation in Topological Space, Doklady Akad. Nauk USSR, 38, (1943), 209-216.##N. V. Veliˇcko, H-closed Topological Spaces, Amer. Math. Soc. Transl., 78(2), (1968), 102-118.## ##

ABSTRACTS IN PERSIAN Vol. 19, No. 2 2 2 Please see the full text contains the Persian abstracts of this volume. 223 238 2019/01/292019/06/62019/11/192019/11/92018/11/272019/11/292019/12/92020/01/32019/12/132020/01/52020/01/242020/08/102021/01/22020/02/52020/02/192025/03/11 1403/12/21 2024/05/72024/05/252022/07/312021/02/62024/05/62023/01/192022/03/52020/12/162023/10/292021/06/52024/04/282021/01/52022/02/162023/02/202022/05/242025/03/11 1403/12/21 The Name of Authors in this Volume The Name of Authors in this Volume Academic Center for Education, Culture and Research (ACECR) ma.hoseinzade@gmail.com ABSTRACTS PERSIAN Vol. 19 No. 2. All references of the papers in Vol. 19, No. 2.## ##