2016
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The analysis of a diseasefree equilibrium of Hepatitis B model
2
2
In this paper we study the dynamics of Hepatitis B virus (HBV) infection under administration of a vaccine and treatment, where the disease is transmitted directly from the parents to the offspring and also through contact with infective individuals. Stability of the diseasefree steady state is investigated. The basic reproductive rate, $R_0$, is derived. The results show that the dynamics of the model is completely determined by the basic reproductive number $R_0$. If $R_0<1$, the diseasefree equilibrium is globally stable and the disease always dies out and if $R_0>1$, the diseasefree equilibrium is unstable and the disease is uniformly persistent.
1

1
11


Reza
Akbari
Department of Mathematical Sciences, Payame Noor University, P.O.Box 193953697, Tehran, Iran.
Department of Mathematical Sciences, Payame
Iran
r9reza@yahoo.com


Ali
Vahidian Kamyad
Department of Mathematics Sciences, University of Ferdowsi, Mashhad, Iran.
Department of Mathematics Sciences, University
Iran
avkamyad@yahoo.com


Ali akbar
Heydari
Research Center for Infection Control and Hand Hygiene, Mashhad University Of Medical Sciences, Mashhad, Iran.
Research Center for Infection Control and
Iran
heydariaa@mums.ac.ir


Aghileh
Heydari
Department of Mathematical Sciences, Payame Noor University, P.O.Box 193953697, Tehran, Iran.
Department of Mathematical Sciences, Payame
Iran
a_heidari@pnu.ac.ir
Hepatitis B virus (HBV)
Basic reproduction number ($R_0$)
Compound matrices
DiseaseFree equilibrium state
Global stability
[[1] R.M. Anderson and R.M. May, Infectious Disease of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991. ##[2] S. Bhattacharyyaa and S. Ghosh, Optimal control of vertically transmitted disease, Computational and Mathematical Methods in Medicine. 11(4) (2010) 369387. ##[3] S. Bowong, J.J. Tewa, and J.C. Kamgang, Stability analysis of the transmission dynamics of tuberculosis models, World Journal of Modelling and Simulation. 7(2) (2011) 83100. ##[4] P.V.D. Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math Biosci. 180 (2002) 2948. ##[5] Y. Enatsu, Y. Nakata, and Y. Muroya, Global stability of SIRS epidemic models with a class of nonlinear incidence rates and distributed delays, Published in Acta Mathematica Scientia, 32 (2012) 851865. ##[6] H. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000) 599653. ##[7] J.C. Kamgang and G. Sallet, Computation of threshold conditions for epidemiological models and global stability of the diseasefree equilibrium (DFE), Mathematical Biosciences, 213 (2008) 112. ##[8] A.V. Kamyad, R. Akbari, A.A. Heydari, and A. Heydari, Mathematical Modeling of Transmission Dynamics and Optimal Control of Vaccination and Treatment for Hepatitis B Virus, Computational and Mathematical Methods in Medicine. Volume 2014, Article ID 475451, 15 pages. ##[9] T.K. Kar and S. Jana, A theoretical study on mathematical modelling of an infectious disease with application of optimal control, Bio Systems, 111(1) (2013) 3750. ##[10] T.K. Kar and A. Batabyal, Stability analysis and optimal control of an SIR epidemic model with vaccination, Biosystems. 104 (2011) 127135. ##[11] X. Liu and L. Yang, Stability analysis of an SEIQV epidemic model with saturated incidence rate, Nonlinear Analysis: Real World Applications, 13 (2012) 26712679. ##[12] J. Mann and M. Roberts, Modelling the epidemiology of hepatitis B in New Zealand, Journal of Theoretical Biology. 269 (2011) 266272. ##[13] G.F. Medley and N.A. Lindop, HepatitisB virus endemicity: heterogeneity, catastrophic dynamics and control, Nature Medicine, 7(5) (2001) 619624. ##[14] A.A. Momoh, M.O. Ibrahim, and B.A. Madu, Stability Analysis of an Infectious Disease Free Equilibrium of Hepatitis B Model, Research Journal of Applied Sciences, Engineering and Technology, 3(9) (2011) 905909. ##[15] J.S. Muldowney, Compound matrices and ordinary differential equations, Rocky Mountain J. Math. 20 (1990) 857872. ##[16] J. Pang, J.A. Cui, and X. Zhou, Dynamical behavior of a hepatitis B virus transmission model with vaccination, Journal of Theoretical Biology. 265 (2010) 572578. ##[17] S. Sun, Global Dynamics of a SEIR Model with a Varying Total Population Size and Vaccination, Int. Journal of Math. Analysis, 6(40) (2012) 19851995. ##[18] C. Sun, Y. Lin, and S. Tang, Global stability for an special SEIR epidemic model with nonlinear incidence rates, Chaos, Solitons and Fractals, 33 (2007) 290297. ##[19] S. Thornley, C. Bullen, and M. Roberts, Hepatitis B in a high prevalence New Zealand population: A mathematical model applied to infection control policy, Journal of Theoretical Biology. 254 (2008) 599603. ##[20] P. Van Den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibrium for compartmental models of disease transmission, Mathe matical Biosciences, 180 (2002) 2948. ##[21] K. Wanga, A. Fan, and A. Torres, Global properties of an improved hepatitis B virus model, Nonlinear Analysis: Real World Applications, 11 (2010) 31313138. ##[22] WHO, Hepatitis B Fact Sheet No. 204, The World Health Organisation, Geneva, Switzerland, 2013, http://www.who.int/mediacentre/factsheets/fs204/en/. ##[23] S. Zhang and Y. Zhou, The analysis and application of an HBV model, Applied Mathematical Modelling, 36 (2012) 13021312. ##[24] S.J. Zhao, Z.Y. Xu, and Y. Lu, A mathematical model of hepatitis B virus transmission and its application for vaccination strategy in China, Int.J.Epidemiol. 29 (2000) 744752. ##[25] L. Zou, W. Zhang, and S. Ruan, Modeling the transmission dynamics and control of hepatitis B virus in China, Journal of Theoretical Biology, 262 (2010) 330338. ##]
Growth analysis of entire functions of two complex variables
2
2
In this paper, we introduce the idea of generalized relative order (respectively generalized relative lower order) of entire functions of two complex variables. Hence, we study some growth properties of entire functions of two complex variables on the basis of the definition of generalized relative order and generalized relative lower order of entire functions of two complex variables.
1

13
24


Sanjib
Kumar Datta
Department of Mathematics, University of Kalyani, P.O.Kalyani, DistNadia, PIN741235, West Bengal, India.
Department of Mathematics, University of
Iran
sanjib_kr_datta@yahoo.co.in


Tanmay
Biswas
Rajbari, Rabindrapalli, R. N. Tagore Road, P.O.Krishnagar, DistNadia, PIN741101, West Bengal, India.
Rajbari, Rabindrapalli, R. N. Tagore Road,
Iran
tanmaybiswas_math@rediffmail.com
Entire functions
Generalized relative order
Generalized relative lower order
Two complex variables
Composition
Growth
[[1] A.K. Agarwal, On the properties of entire function of two complex variables, Canadian Journal of Mathematics, 20 (1968) 5157. ##[2] L. Bernal, Crecimiento relativo de funciones enteras. Contribucion al estudio de lasfunciones enteras con ndice exponencial nito, Doctoral Dissertation, University of Seville, Spain, 1984. ##[3] L. Bernal, Orden relativo de crecimiento de funciones enteras, Collect. Math., 39 (1988) 209229. ##[4] D. Banerjee and R. K. Dutta, Relative order of entire functions of two complex variables, International J. of Math. Sci. & Engg. Appls. (IJMSEA), 1(1) (2007) 141154. ##[5] A.B. Fuks, Theory of analytic functions of several complex variables, Moscow, 1963. ##[6] S. Halvarsson, Growth properties of entire functions depending on a parameter, Annales Polonici Mathematici, 14(1) (1996) 7196. ##[7] O.P. Juneja, G.P. Kapoor, and S.K. Bajpai, On the (p,q)order and lower $(p,q)$order of an entire function, J. Reine Angew. Math., 282 (1976) 5367. ##[8] C.O. Kiselman, Order and type as measure of growth for convex or entire functions, Proc. Lond. Math. Soc., 66(3) (1993) 152186. ##[9] C.O. Kiselman, Plurisubharmonic functions and potential theory in several complex variable, a contribution to the book project, Development of Mathematics, 19502000, edited by HeanPaul Pier. ##[10] B.K. Lahiri and D. Banerjee, A note on relative order of entire functions, Bull. Cal. Math. Soc., 97(3) (2005) 201206. ##[11] D. Sato, On the rate of growth of entire functions of fast growth, Bull. Amer. Math. Soc., 69 (1963) 411414. ##[12] E.C. Titchmarsh, The theory of functions, 2nd ed. Oxford University Press, Oxford, 1968.##]
Menger probabilistic normed space is a category topological vector space
2
2
In this paper, we formalize the Menger probabilistic normed space as a category in which its objects are the Menger probabilistic normed spaces and its morphisms are fuzzy continuous operators. Then, we show that the category of probabilistic normed spaces is isomorphicly a subcategory of the category of topological vector spaces. So, we can easily apply the results of topological vector spaces in probabilistic normed spaces.
1

25
32


Ildar
Sadeqi
Department of Mathematics, Faculty of Science, Sahand University of Technology, Tabriz, Iran.
Department of Mathematics, Faculty of Science,
Iran
esadeqi@sut.ac.ir


Farnaz
Yaqub Azari
University of Payame noor, Tabriz, Iran.
University of Payame noor, Tabriz, Iran.
Iran
fyaqubazari@gmail.com
Category of probabilistic normed space
Category of topological vector space
Fuzzy continuous operator
[[1] T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, Fuzzy Math. 11 (2003) 687705. ##[2] G. Constantin and I. Istratfescu, Elements of probabilistic analysis, Kluwer Academic Publishers, 1989. ##[3] P. Freyd, Abelian categories, An Introduction to the theory of functors, Happer & Row, New York, Evanston & London and John Weatherhill, INC, Tokyo, 1964. ##[4] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64 (1994) 395399. ##[5] U. Hohle, A note on the hyporgraph functor, Fuzzy Sets and Systems, 131 (2002) 353356. ##[6] O. Kaleva and S. Seikala, On fuzzy metric spaces, Fuzzy Sets and Systems, 12 (1984) 143154. ##[7] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika, 11 (1975) 326334. ##[8] K. Menger, Statistical metrics. Proc. Nat. Acad. Sci, USA, 28 (1942) 5357. ##[9] S.E. Rodabaugh, Fuzzy addition in the Lfuzzy real line, Fuzzy Sets and Systems, 8 (1982) 3951. ##[10] S.E. Rodabaugh, Complete fuzzy topological hyperelds and fuzzy multiplication in the fuzzy real lines, Fuzzy Sets and Systems, 15 (1985) 285310. ##[11] S.E. Rodabaugh, A theory of fuzzy uniformities with applications to the fuzzy real lines, J. Math. Anal. Appl., 129 (1988) 3770. ##[12] W. Rudin, Functional Analysis, Tata McGrawHill Publishing Company, 1990. ##[13] I. Sadeqi and F. Solaty kia, Fuzzy normed linear space and it's topological structure, Chaos, fractal, solution & Fractals, 40 (2007) 25762589. ##[14] I. Sadeqi and F. Solaty kia, The category of fuzzy normed linear space, The journal of fuzzy mathematics, 3 (2010) 733742. ##[15] I. Sadeqi, F. Solaty kia and F. Yaqub azari, Menger probabilistic normed linear spaces and its topological structure, Fuzzy Inteligent and Systems, (On published data). ##[16] E.S. Santos, Topology versus fuzzy topology, preprint, Youngstown State University, 1977. ##[17] B. Schweizer and A. Sklar, Probablisitic metric spaces, NorthHoland, Amesterdam, 1983.##]
On certain fractional calculus operators involving generalized MittagLeffler function
2
2
The object of this paper is to establish certain generalized fractional integration and differentiation involving generalized MittagLeffler function defined by Salim and Faraj [25]. The considered generalized fractional calculus operators contain the Appell's function $F_3$ [2, p.224] as kernel and are introduced by Saigo and Maeda [23]. The MarichevSaigoMaeda fractional calculus operators are the generalization of the Saigo fractional calculus operators. The established results provide extensions of the results given by Gupta and Parihar [3], Saxena and Saigo [30], Samko et al. [26]. On account of the general nature of the generalized MittagLeffler function and generalized Wright function, a number of known results can be easily found as special cases of our main results.
1

33
45


Dinesh
Kumar
Department of Mathematics & Statistics, Jai Narain Vyas University, Jodhpur  342005, India.
Department of Mathematics & Statistics,
Iran
dinesh_dino03@yahoo.com
MarichevSaigoMaeda fractional calculus operators
Generalized MittagLeffler function
Generalized Wright hypergeometric function
[[1] J. Choi and D. Kumar, Certain unied fractional integrals and derivatives for a product of Aleph function and a general class of multivariable polynomials, Journal of Inequalities and Applications, 2014 (2014), 15 pages. ##[2] A. Erdelyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi, Tables of Integral Transforms, McGrawHill, New York, 1, 1954. ##[3] A. Gupta and C.L. Parihar, Fractional dierintegral operators of the generalized MittagLeer function, Bol. Soc. Paran. Math., 33(1) (2015), 137144. ##[4] H.J. Haubold, A.M. Mathai, and R.K. Saxena, MittagLeer functions and their applications, J. Appl. Math. (Article ID 298628) (2011), 151. ##[5] A.A. Kilbas and M. Saigo, Fractional integrals and derivatives of MittagLeer type function, Doklady Akad. Nauk Belarusi, 39(4) (1995), 2226. ##[6] A.A. Kilbas, M. Saigo and R.K. Saxena, Generalized MittagLeer function and generalized fractional calculus operators, Integral Transform Special Function, 15 (2004), 3149. ##[7] A.A. Kilbas, M. Saigo and J.J. Trujillo, On the generalized Wright function, Fract. Calc. Appl. Anal., 5(4) (2002), 437460. ##[8] Y.C. Kim, K.S. Lee and H.M. Srivastava, Some applications of fractional integral operators and Ruscheweyh derivatives, J. Math. And. Appl., 197(2) (1996), 505517. ##[9] V. Kiryakova, All the special functions are fractional dierintegrals of elementary functions, Journal of Physics A: Mathematical and General, 30(14) (1997), 50855103. ##[10] D. Kumar and J. Daiya, Fractional calculus pertaining to generalized Hfunctions, Global Journal of Science Frontier Research: F Mathematics and Decision Sciences, 14(3) (2014), 2536. ##[11] D. Kumar and S. Kumar, Fractional Calculus of the Generalized MittagLeer Type Function, International Scholarly Research Notices 2014 (2014), Article ID 907432, 6 pages. ##[12] D. Kumar and S.D. Purohit, Fractional dierintegral operators of the generalized MittagLeer type function, Malaya J. Mat., 2(4) (2014), 419425. ##[13] D. Kumar and R.K. Saxena, Generalized fractional calculus of the MSeries involving F3 hypergeometric function, Sohag J. Math., 2(1) (2015), 1722. ##[14] O.I. Marichev, Volterra equation of Mellin convolution type with a Horn function in the kernel, Izvestiya Akademii Nauk BSSR. Seriya FizikoMatematicheskikh Nauk, 1 (1974), 128129, (Russian). ##[15] A.M. Mathai and H.J. Haubold, Special Functions for Applied Scientists, Springer, New York, 2008. ##[16] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Dierential Equations, John Wiley & Sons, New York, NY, USA, 1993. ##[17] G.M. MittagLeer, Sur la nouvelle fonction E (x), C.R. Acad. Sci. Paris 137 (1903), 554558. ##[18] G.M. MittagLeer, Sur la representation analytique d'une branche uniforme d'une function monogene, Acta Math. 29 (1905), 101181. ##[19] J. PanevaKonovska, Inequalities and asymptotic formulae for the three parametric MittagLeer functions, Math. Balkanica, 26 (2012), 203210. ##[20] J. PanevaKonovska, Convergence of series in three parametric MittagLeer functions, Math. Slovaca 62, 2012. ##[21] T.R. Prabhakar, A singular integral equation with a generalized MittagLeer function in the Kernel, Yokohama Math. J. 19 (1971), 715. ##[22] M. Saigo, A remark on integral operators involving the Gauss hypergeometric functions, Math. Rep., College General Ed. Kyushu Univ. 11 (1978), 135143. ##[23] M. Saigo and N. Maeda, More generalization of fractional calculus Transform Methods and Special Functions, Varna, Bulgaria, (1996), 386400. ##[24] T.O. Salim, Some properties relating to the generalized MittagLeer function, Adv. Appl. Math. Anal., 4 (2009), 2130. ##[25] T.O. Salim and A.W. Faraj, A generalization of MittagLeer function and Integral operator associated with fractional calculus, Journal of Fractional Calculus and Application, 3(5) (2012), 113. ##[26] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon et alibi, 1993. ##[27] R.K. Saxena and D. Kumar, Generalized fractional calculus of the Alephfunction involving a general class of polynomials, Acta Mathematica Scientia, 35(5) (2015), 10951110. ##[28] R.K. Saxena, J. Ram and D. Kumar, Generalized fractional dierentiation of the AlephFunction associated with the Appell function F3 , Acta Ciencia Indica, 38M(4) (2012), 781792. ##[29] R.K. Saxena, J. Ram and D. Kumar, On the TwoDimensional SaigoMaeda fractional calculus associated with TwoDimensional Aleph Transform, Le Matematiche, 68 (2013), 267281. ##[30] R.K. Saxena and M. Saigo, Certain properties of the fractional calculus operators associated with generalized MittagLeer function, Fract. Calc. Appl. Anal., 8(2) (2005), 141154. ##[31] H.M. Srivastava and Z. Tomovski, Fractional calculus with an integral operator containing a generalized MittagLeer function in the kernel, Appl. Math. Comput., 211 (2009), 198210. ##[32] A. Wiman, Uber de fundamental satz in der theorie der funktionen E(x), Acta Math. 29 (1905), 191201. ##[33] E.M. Wright, The asymptotic expansion of generalized hypergeometric function, J. London Math. Soc., 10 (1935), 286293.##]
Multistep collocation method for nonlinear delay integral equations
2
2
The main purpose of this paper is to study the numerical solution of nonlinear Volterra integral equations with constant delays, based on the multistep collocation method. These methods for approximating the solution in each subinterval are obtained by fixed number of previous steps and fixed number of collocation points in current and next subintervals. Also, we analyze the convergence of the multistep collocation method when used to approximate smooth solutions of delay integral equations. Finally, numerical results are given showing a marked improvement in comparison with exact solution.
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47
65


Parviz
Darania
Department of Mathematics, Faculty of Science, Urmia University, P.O.Box 5756151818, UrmiaIran.
Department of Mathematics, Faculty of Science,
Iran
p.darania@urmia.ac.ir
Delay integral equations
Collocation method
Multistep collocation method
Convergence
[[1] D. Conte and B. Paternoster, Multistep collocation methods for Volterra integral equations, Appl. Numer. Math., 59 (2009) 17211736. ##[2] V. Horvat, On collocation methods for Volterra integral equations with delay arguments, Mathematical Communications, 4 (1999) 93109. ##[3] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2004. ##[4] H. Brunner, Highorder collocation methods for singular Volterra functional equations of neutral type, Applied Numerical Mathematics, 57 (2007) 533548. ##[5] H. Brunner, Iterated collocation methods for Volterra integral equations with delay arguments, Math. Comp., 62 (1994) 581599. ##[6] H. Brunner, Implicitly linear collocation methods for nonlinear Volterra integral equations, Appl. Numer. Math., 9 (1992) 235247. ##[7] I. Ali, H. Brunner, and T. Tang, Spectral methods for pantographtype dierential and integral equations with multiple delays, Front. Math. China, 4 (2009) 4961. ##[8] H. Brunner, The numerical solution of weakly singular Volterra functional integrodierential equations with variable delays, Comm. Pure Appl. Anal. 5 (2006) 261276. ##[9] Y. Liu, Stability analysis of θmethods for neutral functionaldierential equations, Numer. Math., 70 (1995) 473485. ##[10] T. Tang, X. Xu, and J. Cheng, On the spectral methods for Volterra type integral equations and the convergence analysis, J. Comput. Math., 26 (2008) 825837. ##[11] V. Volterra, Lecons sur les equations integrals et les equations integrodierentielles, GauthierVillars, Paris, 1913. ##[12] J. Cerha, On some linear Volterra delay equations, Casopis Pest Mat., 101 (1976) 111123.##]
On the topological centers of module actions
2
2
In this paper, we study the Arens regularity properties of module actions. We investigate some properties of topological centers of module actions ${Z}^ell_{B^{**}}(A^{**})$ and ${Z}^ell_{A^{**}}(B^{**})$ with some conclusions in group algebras.
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67
74


Kazem
Haghnejad Azar
Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran.
Department of Mathematics, University of
Iran
haghnejad@uma.ac.ir
Arens regularity
Topological centers
Module actions
[[1] R.E. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839848. ##[2] J. Baker, A.T. Lau and J.S. Pym Module homomorphism and topological centers associated with weakly sequentially compact Banach algebras, Journal of Functional Analysis. 158 (1998), 186208. ##[3] F.F. Bonsall and J. Duncan, Complete normed algebras, SpringerVerlag, Berlin 1973. ##[4] H.G. Dales, Banach algebra and automatic continuity, Oxford 2000. ##[5] H.G. Dales, F. Ghahramani and N. Grnbk, Derivation into iterated duals of Banach algebras, Studia Math. 128 1 (1998), 1953. ##[6] M. Eshaghi Gordji and M. Filali, Arens regularity of module actions, Studia Math. 181 3 (2007), 237254. ##[7] E. Hewitt and K.A. Ross, Abstract harmonic analysis, Springer, Berlin, Vol I 1963. ##[8] A.T. Lau and V. Losert, On the second Conjugate Algebra of locally compact groups, J. London Math. Soc. 37 (2)(1988), 464480. ##[9] A.T. Lau and A. Ulger, Topological center of certain dual algebras, Trans. Amer. Math. Soc. 348 (1996), 11911212. ##[10] S. Mohamadzadeh and H.R.E. Vishki, Arens regularity of module actions and the second adjoint of a derivation, Bulletin of the Australian Mathematical Society 77 (2008), 465476. ##[11] J.S. Pym, The convolution of functionals on spaces of bounded functions, Proc. London Math Soc. 15 (1965), 84104. ##[12] A. Ulger, Some stability properties of Arens regular bilinear operators, Proc. Amer. Math. Soc. (1991) 34, 443454. ##[13] A. Ulger, Arens regularity of weakly sequentialy complte Banach algebras, Proc. Amer. Math. Soc. 127 (11) (1999), 32213227.##]
Inverse SturmLiouville problems with a Spectral Parameter in the Boundary and transmission conditions
2
2
In this manuscript, we study the inverse problem for non selfadjoint SturmLiouville operator $D^2+q$ with eigenparameter dependent boundary and discontinuity conditions inside a finite closed interval. By defining a new Hilbert space and using its spectral data of a kind, it is shown that the potential function can be uniquely determined by part of a set of values of eigenfunctions at some interior point and parts of two sets of eigenvalues.
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75
89


Mohammad
Shahriari
Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh, Iran.
Department of Mathematics, Faculty of Science,
Iran
shahriari@maragheh.ac.ir
Inverse SturmLiouville problem
Jump conditions
Non selfadjoint operator
Parameter dependent condition
[[1] Z. Akdogan, M. Demirci, and O. Sh. Mukhtarov, Green function of discontinuous boundary value problem with transmission conditions, Math. Meth. Appl. Sci., 30 (2007) 17191738. ##[2] S. Albeverio, F. Gesztesy, R. HeghKrohn, and H. Holden, Solvable Models in Quantum Mechanics, 2nd ed., AMS Chelsea Publishing, Providence, RI, 2005. ##[3] V.A. Ambartsumyan, Uber eine frage der eigenwerttheorie, Z. Phys. 53 (1929) 690695. ##[4] R.Kh. Amirov, On SturmLiouville operators with discontinuity conditions inside an interval, J. Math. Anal. Appl. 317 (2006) 163176. ##[5] P.A. Binding, P.J. Browne, and B.A. Watson, Inverse spectral problems for left denite SturmLiouville equations with indenite weight, J. Math. Anal. Appl. 271 (2002) 383408. ##[6] G. Borg, Eine umkehrung der SturmLiouvilleschen eigenwertaufgabe, Acta Math. 78 (1945) 196. ##[7] J. Eckhardt and G. Teschl, Uniqueness results for Schrodinger operators on the line with purely discrete spectra, arXiv:1110.2453. ## [8] G. Freiling and V.A. Yurko, Inverse SturmLiouville problems and their applications, NOVA Science Publishers, New York, 2001. ##[9] I.M. Gelfand and B.M. Levitan, On the determination of a dierential equation from its spectral function, Amer. Math. Soc. Transl. Ser. 2 1 (1955) 253304. ##[10] O. Hald, Discontinuous inverse eigenvalue problem, Commun. Pure. Appl. Math. 37 (1984) 539577. ##[11] H. Hochstadt and B. Lieberman, An inverse SturmLiouville problem with mixed given data, SIAM J. Appl. Math., 34(4) (1978), 676680. ##[12] H. Hochstadt, On inverse problems associated with SturmLiouville operators, J. Dierential Equations, 17 (1975) 220235. ##[13] H. Hochstadt, The inverse Sturm{Liouville problem, Comm. pure. Math, 26 (1973) 715729. ##[14] M. Kobayashi, A uniqueness proof for discontinuous inverse SturmLiouville problems with symmetric potentials, Inverse Problems 5 (1989) 767781. ##[15] H. Koyunbakan, A new inverse problem for the diusion operator, Applied Mathematics Letters, 19 (2006) 995999. ##[16] H. Koyunbakan and E.S. Panakhov, Inverse problem for a singular dierential operator, Mathematical and Computer Modeling, 47 (2008) 178185. ##[17] N. Levinson, The inverse SturmLiouville problem, Mat. Tiddskr. 3 (1949) 2530. ##[18] B.M. Levitan, Inverse SturmLiouville Problems, VNU Science Press, 1987. ##[19] V.A. Marchenko, SturmLiouville operators and applications, American Mathematical Society, 2011. ##[20] J.R. McLaughlin, Analytical methods for recovering coecients in dierential equations from spectral data, SIAM Rev. 28 (1986) 5372. ##[21] K. Mochizuki and I. Trooshin, Inverse problem for interior spectral data of the dirac operator, Comm. Korean Math. Soc. 16, No. 3 (2001) 437442. ##[22] K. Mochizuki and I. Trooshin, Inverse problem for interior spectral data of the Dirac operator on a nite interval, Publ. RIMS, Kyoto Univ. 38 (2002), 387395. ##[23] O.Sh. Mukhtarov, Mahir Kadakal, and F.S. Muhtarov, On discontinuous SturmLiouville problems with transmission conditions, J. Math. Kyoto Univ. (JMKYAZ), 444 (2004) 779798. ##[24] Y. Pingwang, An interior inverse problem for SturmLiouville operator with eigenparameter dependent boundary conditions, TAMKANG J. Math., Volume 42, Number 3, 395403, Autumn 2011. ##[25] M. Shahriari and A. Jodayree Akbarfam, Inverse SturmLiouville problem with discontinuity conditions, Sahand Communications in Mathematical Analysis (SCMA) Vol. 1 No. 1(2014), 2940. ##[26] M. Shahriari and A. Jodayree Akbarfam, Inverse problem for SturmLiouville operators with a transmission and parameter dependent boundary conditions, Caspian journal of Mathematical Sciences, preprint. ##[27] M. Shahriari, A. Jodayree Akbarfama, and G. Teschl, Uniqueness for Inverse SturmLiouville Problems with a Finite Number of Transmission Conditions, J. Math. Anal. Appl., 395 (2012) 1929. ##[28] C.T. Shieh and V.A. Yurko, Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl. 347 (2008) 266272. ##[29] G. Teschl, Mathematical Methods in Quantum Mechanics; With Applications to Schrodinger Operators, Graduate Studies in Mathematics 99, Amer. Math. Soc., Providence, RI, 2009. ##[30] C.F. Yang, HochstadtLieberman theorem for Dirac operator with eigenparameter dependent boundary conditions, Nonlinear Analysis, 74(2011)24752484. ##[31] C.F. Yang, Inverse spectral problems for the SturmLiouville operator on a dstar graph, J. Math. Anal. Appl. 365 (2010) 742749. ##[32] C.F. Yang and X.P. Yang, An interior inverse problem for the SturmLiouville operator with discontinuous conditions, Applied Mathematics Letters, 22 (2009) 13151319. ##[33] V. Yurko, Integral transforms connected with discontinuous boundary value problems, Int. Trans. Spec. Functions, 10 (2000) 141164. ##[34] C. Willis, Inverse SturmLiouville problems with two discontinuities, Inverse Problems I (1985) 263289.##]
On multiplicative (strong) linear preservers of majorizations
2
2
In this paper, we study some kinds of majorizations on $textbf{M}_{n}$ and their linear or strong linear preservers. Also, we find the structure of linear or strong linear preservers which are multiplicative, i.e. linear or strong linear preservers like $Phi $ with the property $Phi (AB)=Phi (A)Phi (B)$ for every $A,Bin textbf{M}_{n}$.
1

91
106


Mohammad Ali
Hadian Nadoshan
Department of Mathematics, ValieAsr University of Rafsanjan, Zip Code: 7718897111, Rafsanjan, Iran.
Department of Mathematics, ValieAsr University
Iran
ma.hadiann@gmail.com


Ali
Armandnejad
Department of Mathematics, ValieAsr University of Rafsanjan, Zip Code: 7718897111, Rafsanjan, Iran.
Department of Mathematics, ValieAsr University
Iran
armandnejad@vru.ac.ir
Doubly stochastic matrix
Linear preserver
Multiplicative map
[[1] T. Ando, Majorization, Doubly stochastic matrices, and comparision of eigenvalues, Linear Algebra and its Applications, 118 (1989) 163248. ##[2] A. Armandnejad, F. Akbarzadeh, and Z. Mohammadi, Row and columnmajorization on M_{n;m}, Linear Algebra and its Applications, 437 (2012) 10251032. ##[3] A. Armandnejad and H. Heydari, Linear Preserving gdMajorization Functions from M_{n;m} to M_{n;k}, Bull. Iranian Math. Soc., 37(1) (2011) 215224. ##[4] A. Armandnejad and A. Ilkhanizadeh Manesh, gutMajorization and its Linear Preservers, Electronic Journal of Linear Algebra, 23 (2012) 646654. ##[5] A. Armandnejad, Z. Mohammadi, and F. Akbarzadeh, Linear preservers of Grow and Gcolumn majorization on M_{n;m}, Bull. Iranian Math. Soc., 39(5) (2013) 865880. ##[6] A.M. Hasani and M. Radjabalipour, The structure of linear operators strongly preserving majorizations of matrices, Electronic Journal of Linear Algebra, 15 (2006) 260268. ##[7] C.K. Li and E. Poon, Linear operators preserving directional majorization, Linear Algebra and its Applications, 325 (2001) 1521. ##[8] P. Semrl, Maps on matrix spaces, Linear Algebra and its Applications, 413(23) (2006) 364393.##]
On $n$derivations
2
2
In this article, the notion of $n$derivation is introduced for all integers $ngeq 2$. Although all derivations are $n$derivations, in general these notions are not equivalent. Some properties of ordinary derivations are investigated for $n$derivations. Also, we show that under certain mild condition $n$derivations are derivations.
1

107
115


Mohammad Hossein
Sattari
Department of Mathematics, Faculty of Science, Azarbaijan Shahid Madani University, P.O.Box 5375171379, Tabriz, Iran.
Department of Mathematics, Faculty of Science,
Iran
sattari@azaruniv.ac.ir
$n$derivation
$n$homomorphism
Banach algebra
[[1] F. F. Bonsall and J. Duncan, Complete normed algebras, SpringprVerlag, New York, 1973. ##[2] G. Dales, Banach Algebra and Automatic Continuity, London Mathematical Society Monographs, Volume 24, Clarendon Press, Oxford, 2000. ##[3] H. G. Dales, F. Ghahramani, and N. Gronbaek, Derivations into iterated duals of Banach algebras, Studia Math., 128 (1998), 1954. ##[4] N. Dunford and J. T. Schwartz, Linear operators, Part I, New York, Interscience, 1958. ##[5] F. Ghahramani, Homomorphisms and derivations on weighted convolution algebras, J. London Math. Soc., 21 (1980), 149161. ##[6] M. Hejazian, M. Mirzavaziri, and M. S. Moslehian, nHomomorphism, Bull. Iranian Math. Soc., 31(1) (2005), 1323. ##[7] B. E. Johnson, Local derivations on C* algebras are derivations, Trans. Amer. Math. Soc., 353 (2000), 313325. ##[8] I. Kaplansky, Derivations of Banach algebras, In Seminars on analytic functions, Vol. 2, Princeton Univ. Press, Princeton, 1958. ##[9] E. Samei, Approximately local derivations, J. London Math. Soc., 71(2) (2005), 759778. ##[10] A. M. Sinclair, Continuous derivations on Banach algebras , Proc. Amer. Math. Soc., 20 (1969), 166170. ##[11] I. M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann., 129 (1955) 260264. ##[12] S. Watanabe, A Banach algebra which is an ideal in the second dual space, Sci. Rep. Niigata Univ. Ser., A 11 (1974), 95101.##]