2017
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On the twowavelet localization operators on homogeneous spaces with relatively invariant measures
2
2
In the present paper, we introduce the twowavelet localization operator for the square integrable representation of a homogeneous space with respect to a relatively invariant measure. We show that it is a bounded linear operator. We investigate some properties of the twowavelet localization operator and show that it is a compact operator and is contained in a Schatten $p$class.
1

1
12


Fatemeh
Esmaeelzadeh
Department of Mathematics‎, ‎Bojnourd Branch‎, ‎Islamic Azad University‎, ‎Bojnourd‎, ‎Iran
Department of Mathematics‎, ‎Bojno
Iran
faride.esmaeelzadeh@yahoo.com


Rajab Ali
KamyabiGol
Department of Mathematics‎, ‎Center of Excellency in Analysis on Algebraic Structures(CEAAS)‎, ‎Ferdowsi University Of Mashhad‎, Iran
Department of Mathematics‎, ‎Cente
Iran
kamyabi@ferdowsi.um.ac.ir


Reihaneh
Raisi Tousi
‎Ferdowsi University Of Mashhad
‎Ferdowsi University Of Mashhad
Iran
raisi@ferdowsi.um.ac.ir
homogenous space
square integrable representation
wavelet transform
localization operator
Schatten $p$class operator
[[1] S.T. Ali, JP. Antoine and JP. Gazeau,Coherent States, Wavelets and Their Generalizations, SpringerVerlag, New York, 2000.##[2] V. Catana, Twowavelet localization operators on homogeneous spaces and their traces, Integral Equations Oper. Theory, 62 (2008), 351363.##[3] V. Catana, Schattenvon Neumann norm inequalities for two wavelet localization operators, J. PseudoDiffer. Oper. Appl., 52 (2007), 265277.##[4] F. Esmaeelzadeh, R.A. KamyabiGol and R. Raisi Tousi, On the continuous wavelet transform on homogeneous spases, Int. J. Wavelets Multiresolut Inf. Process., 10(4) (2012), 118.##[5] F. Esmaeelzadeh, R. A. KamyabiGol and R. Raisi Tousi, Localization operators on homogeneous spaces, Bull. Iran. Math. Soc., 39(3) (2013), 455467.##[6] F. Esmaeelzadeh, R. A. KamyabiGol and R. Raisi Tousi, Twowavelet constants for square integrable representations of G/H, Wavel. Linear Algebra, 1 (2014), 6373.##[7] J.M.G. Fell and R.S. Doran, Representation Of *algebras, Locally Compact Groups, and Banach *algebraic Bundles, Vol. 1, Academic Press, 1988.##[8] G.B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, New York, 1995.##[9] H. Reiter and J. Stegeman, Classical Harmonic Analysis and Locally Compact Group, Claredon Press, 2000.##[10] K. Zhu, Operator Theory in Function Spaces, vol. 138, American Mathematical Society, 2007.##[11] M.W. Wong, Wavelet Transform and Localization Operators, Birkhauser Verlag, BaselBostonBerlin, 2002.##]
Characterizing subtopical functions
2
2
In this paper, we first give a characterization of subtopical functions with respect to their lower level sets and epigraph. Next, by using two different classes of elementary functions, we present a characterization of subtopical functions with respect to their polar functions, and investigate the relation between polar functions and support sets of this class of functions. Finally, we obtain more results on the polar of subtopical functions.
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13
23


Hassan
Bakhtiari
Shahid Bahonar University of Kerman
Shahid Bahonar University of Kerman
Iran
hbakhtiari@math.uk.ac.ir


Hossein
Mohebi
Shahid Bahonar University of Kerman
Shahid Bahonar University of Kerman
Iran
hmohebi@uk.ac.ir
subtopical function
elementary function
polar function
pluscoradiant set
abstract convexity
[[1] H. Bakhtiari and H. Mohebi, Characterizing global maximizers of the difference of subtopical functions, J. Appl. Math. Anal. Appl., 450(1) (2017), 6376.##[2] A.R. Doagooei, Subtopical functions and pluscoradiant sets, Optimization, 65(1) (2016), 107119, .##[3] S. Gaubert and J. Gunawardena, A NonLinear Hierarchy for Discrete Event Dynamical Systems, In: Proceedings of the 4th Workshop on Discrete Event Systems Cagliari, Technical Report HPLBRIMS9820, HewlettPackard Labs., Cambridge University Press, Cambridge, 1998.##[4] J. Gunawardena, An Introduction to Idempotency, Cambridge University Press, Cambridge, 1998.##[5] J. Gunawardena, From MaxPlus Algebra to NonExpansive Mappings: A NonLinear Theory for Discrete Event Systems. Theoretical Computer Science, Technical Report HPLBRIMS9907, HewlettPackard Labs., Cambridge University Press, Cambridge, 1999.##[6] J. Gunawardena and M. Keane, On the Existence of Cycle Times for Some Nonexpansive Maps, Technical Report HPLBRIMS95003, HewlettPackard Labs., Cambridge University Press, Cambridge, 1995.##[7] H. Mohebi, Topical functions and their properties in a class of ordered Banach spaces, Appl. Optim., 99 (2005), 343361.##[8] H. Mohebi and M. Samet Abstract convexity of topical functions, J. Glob. Optim., 58(2) (2014), 365375.##[9] A.M. Rubinov, Abstract Convexity and Global Optimization, Kluwer Academic Publishers, DordrechtBostonLondon, 2000.##[10] A.M. Rubinov and I. Singer, Topical and subtopical functions, downward sets and abstract convexity, Optimization, 50(56) (2001), 307351.##[11] I. Singer, On radiant sets, downward sets, topical functions and subtopical functions in lattice ordered groups, Optimization, 53(4) (2004), 393428.##[12] C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific, London, 2002.##]
Linear preservers of MirandaThompson majorization on MM;N
2
2
MirandaThompson majorization is a groupinduced cone ordering on $mathbb{R}^{n}$ induced by the group of generalized permutation with determinants equal to 1. In this paper, we generalize MirandaThompson majorization on the matrices. For $X$, $Yin M_{m,n}$, $X$ is said to be MirandaThompson majorized by $Y$ (denoted by $Xprec_{mt}Y$) if there exists some $Din rm{Conv(G)}$ such that $X=DY$. Also, we characterize linear preservers of this concept on $M_{m,n}$.
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32


Ahmad
Mohammadhasani
Department of Mathematics, Sirjan University of technology, Sirjan, Iran
Department of Mathematics, Sirjan University
Iran
a.mohammadhasani53@gmail.com


Asma
Ilkhanizadeh Manesh
ValieAsr University of Rafsanjan
ValieAsr University of Rafsanjan
Iran
a.ilkhani@vru.ac.ir
Groupinduced cone ordering
Linear preserver
MirandaThompson majorization
[[1] L.B. Beasley, SG. Lee and YH Lee, A characterization of strong preservers of matrix majorization, Linear Algebra Appl., 367 (2003), 341346, ##[2] H. Chiang and C.K. Li, Generalized doubly stochastic matrices and linear preservers, Linear Multilinear Algebra, 53(1) (2005), 111.##[3] A. Giovagnoli and H.P. Wynn, Gmajorization with applications to matrix orderings, Linear Algebra Appl., 67 (1985), 111135.##[4] A.M. Hasani and M. Radjabalipour, On linear preservers of (right) matrix majorization, Linear Algebra Appl., 423 (2007), 255261.##[5] A. Ilkhanizadeh Manesh, Right gutMajorization on M_{n,m}, Electron. J. Linear Algebra, 31(1) (2016), 1326.##[6] F. Khalooei, Linear preservers of twosided matrix majorization, Wavel. Linear Algebra, 1 (2014), 4350.##[7] M. Niezgoda, Cone orderings, group majorizations and similarly separable vectors, Linear Algebra Appl., 436 (2012), 579594.##[8] A.W. Marshall, I. Olkin, and B.C. Arnold, Inequalities: Theory of Majorization and Its Applications, Springer, New York, 2011. ##[9] M. Soleymani and A. Armandnejad, Linear preservers of even majorization on M_{n,m}, Linear Multilinear Algebra, 62(11) (2014), 14371449.##]
Wilson wavelets for solving nonlinear stochastic integral equations
2
2
A new computational method based on Wilson wavelets is proposed for solving a class of nonlinear stochastic It^{o}Volterra integral equations. To do this a new stochastic operational matrix of It^{o} integration for Wilson wavelets is obtained. Block pulse functions (BPFs) and collocation method are used to generate a process to forming this matrix. Using these basis functions and their operational matrices of integration and stochastic integration, the problem under study is transformed to a system of nonlinear algebraic equations which can be simply solved to obtain an approximate solution for the main problem. Moreover, a new technique for computing nonlinear terms in such problems is presented. Furthermore, convergence of Wilson wavelets expansion is investigated. Several examples are presented to show the efficiency and accuracy of the proposed method.
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33
48


Bibi Khadijeh
Mousavi
Department of Pure Mathematica, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman
Department of Pure Mathematica, Faculty of
Iran
khmosavi@gmail.com


Ataollah
Askari Hemmat
Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman
Department of Applied Mathematics, Faculty
Iran
askarihemmat@gmail.com


Mohammad Hossien
Heydari
Shiraz University of Technology, Shiraz,
Shiraz University of Technology, Shiraz,
Iran
heydari@stu.yazd.ac.ir
Wilson wavelets
Nonlinear stochastic It^oVolterra integral equation
Stochastic operational matrix
[[1] A. Abdulle and A. Blumenthal, Stabilized multilevel Monte Carlo method for stiff stochastic differential equations, J. Comput. Phys., 251 (2013), 445460.##[2] E. Babolian and F. Fattahzadeh, Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration, Appl. Math. Comput., 188(1) (2007), 417426.##[3] E. Babolian and A. Shahsavaran, Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelets, J. Comput. Appl. Math., 225(1) (2009), 8795.##[4] M.A. Berger and V.J. Mizel, Volterra equations with Ito integrals I, J. Integral Equations, 2(3) (1980), 187245.##[5] K. Bittner, Wilson bases on the interval, Advances in Gabor Analysis, Birkhäuser Boston, (2003) 197221. ##[6] K. Bittner, Linear approximation and reproduction of polynomials by wilson bases, J. Fourier Anal. Appl., 8(1) (2002), 85108.##[7] K. Bittner, Biorthogonal wilson Bases, Proc. SPIE Wavelet Applications in Signal and Image Processing VII, 3813 (1999), 410421. ##[8] Y. Cao, D. Gillespie and L. Petzod, Adaptive explicitimplicit tauleaping method with automatic tau selection, J. Chem. Phys., 126(22) (2007), 19.##[9] C. Cattani and A. Kudreyko, Harmonic wavelet method towards solution of the Fredholm type integral equations of the second kind, Appl. Math. Comput., 215(12) (2010), 41644171.##[10] J.C. Cortes, L. Jodar and L. Villafuerte, Numerical solution of random differential equations: a mean square approach, Math. Comput. Modelling, 45(78) (2007), 757765.##[11] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.##[12] I. Daubechies, S. Jaffard and J.L. Journe, A simple wilson orthonormal basis with exponential decay, SIAM J. Math. Anal., 22(2) (1991), 554573.##[13] H.G. Feichtinger and T. Strohmer (eds.), Advances in Gabor analysis, Springer Science and Business Media, Davis, U.S.A, 2012.##[14] M.H. Heydari, M.R. Hooshandasl, F.M. Maalek Ghaini and C. Cattani, A computational method for solving stochastic It^{o} Volterra integral equations based on stochastic operational matrix for generalized hat basis functions, J. Comput. Phys., 270(1) (2014), 402415.##[15] M.H. Heydari, M.R. Hooshmandasl, A. Shakiba and C. Cattani, Legendre wavelets Galerkin method for solving nonlinear stochastic integral equations, Nonlinear Dyn., 85(2) (2016), 11851202.##[16] M.H. Heydari, C. Cattani, M.R. Hooshandasl, F.M. Maalek Ghaini, An efficient computational method for solving nonlinear stochastic It^{o} integral equations: Application for stochastic problems in physics, J. Comput. Phys., 283 (2015), 148168.##[17] M.H Heydari, M.R. Hooshmandasl and F. Mohammadi, Legendre wavelets method for solving fractional partial differential equations with Dirichlet boundary conditions, Appl. Math. Comput., 234 (2014), 267276.##[18] M.H. Heydari, M.R. Hooshmandasl, F.M.M. Ghaini and F. Fereidouni, Twodimensional Legendre wavelets for solving fractional poisson equation with Dirichlet boundary conditions, Eng. Anal. Bound. Elem., 37(11) (2013), 13311338.##[19] M.H. Heydari, M.R. Hooshmandasl and F.M. Maleak Ghaini, A good approximate solution for linear equation in a large interval using block pulse functions, J. Math. Ext., 7(1) (2013), 1732.##[20] M.H. Heydari, M.R. Hooshmandasl, F.M. Maalek Ghaini and M. Li, Chebyshev wavelets method for solution of nonlinear fractional integrodifferential equations in a large interval, Adv. Math. Phys., 2013 (2013), DOI. 10.1155/2013/482083.##[21] H. Holden, B. Oksendal, J. Uboe and T. Zhang, Stochastic Partial Differential Equations, second ed., Springer, New york, 1998.##[22] S.K. Kaushik and S. Panwar, An interplay between gabor and wilson frames, J. Funct. Spaces Appl., 2013 (2013), DOI. 10.1155/2013/610917.##[23] M. Khodabin, K. Malekinejad, M. Rostami and M. Nouri, Interpolation solution in generalized stochastic exponential population growth model, Appl. Math. Modelling, 36(3) (2012), 10231033.##[24] M. Khodabin, K. Malekinejad, M. Rostami and M. Nouri, Numerical approach for solving stochastic VolterraFredholm integral equations by stochastic operational matrix, Comput. Math. Appl., 64(6) (2012), 19031913.##[25] M. Khodabin, K. Malekinejad, M. Rostami and M. Nouri, Numerical solution of stochastic differential equations by second order Runge Kutta methods, Appl. Math. Modelling, 53 (2011), 19101920.##[26] P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1999.##[27] J.J. Levin and J.A. Nohel, On a system of integrodifferential equations occurring in reactor dynamics, J. Math. Mech., 9 (1960), 347368.##[28] K. Maleknejad, M. Khodabin and M. Rostami, Numerical solutions of stochastic Volterra integral equations by a stochastic operational matrix based on block pulse functions, Math. Comput. Modelling, 55(34) (2012), 791800.##[29] K. Maleknejad, M. Khodabin and M. Rostami, A numerical method for solving mdimensional stochastic ItoVolterra integral equations by stochastic operational matrix, Comput. Math. Appl., 63(1) (2012), 133143.##[30] J.J. Levin and J.A. Nohel, On a system of integro differential equations occurring in reactor dynamics, J. Math. Mech., 9(3) (1960), 34736.##[31] F. Mohammadi, A Chebyshev wavelet operational method for solving stochastic VolterraFredholm integral equations, Int. J. Appl. Math. Res., 4(2) (2015), 217227.##[32] F. Mohammadi, A waveletbased computational method for solving stochastic It^{o}Volterra integral equations, J. Comput. Phys., 298(1) (2015), 254265.##[33] B.KH. Mousavi, A. Askari hemmat and M. H. Heydari, An application of Wilson system in numerical solution of Fredholm integral equations, PJAA, 2 (2017), 6172. ##[34] B. Oksendal, Stochastic Differential Equations, fifth ed. in: An introduction with Applications, Springer, New York, 1998.##[35] E. Platen and N. BrutiLiberati, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Springer, Berlin, 2010.##[36] S. Yousefi and A. Banifatemi, Numerical solution of Fredholm integral equations by using CAS wavelets, Appl. Math. Comput., 183(1) (2006), 458463.##]
Some results on Haar wavelets matrix through linear algebra
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2
Can we characterize the wavelets through linear transformation? the answer for this question is certainly YES. In this paper we have characterized the Haar wavelet matrix by their linear transformation and proved some theorems on properties of Haar wavelet matrix such as Trace, eigenvalue and eigenvector and diagonalization of a matrix.
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49
59


Siddu
Shiralasetti
Pavate nagar
Pavate nagar
Iran
shiralashettisc@gmail.com


Kumbinarasaiah
S
Pavate nagar
Pavate nagar
Iran
kumbinarasaiah@gmail.com
Linear transformation
Haar wavelets matrix
Eigenvalues and vectors
[[1] J. Astola and L. Yaroslavsky, Advances in Signal Transforms:Theory and Applications, Hindawi Publishing Corporation, 2007.##[2] P. Chang and P. Piau, Haar wavelet matrices designation in numerical solution of ordinary differential equations, Int. J. Appl. Math., 38 (2008), 311.##[3] C.F. Chen and C.H. Hsiao, Haar wavelet method for solving lumped and distributedparameter systems, IEE Proc., Control Theory Appl., 144 (1997), 8794.##[4] C. Capilla, Application of the haar wavelet transform to detect microseismic signal arrivals, Journal of Applied Geophysics, 59 (2006), 3646.##[5] M.D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl., 10 (1975), 285290.##[6] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.##[7] J. Eisenfeld, Block diagonalization and eigenvalues, Linear Algebra Appl., 15 (1976), 205215. ##[8] G. Hariharan and K. Kannan, An overview of haar wavelet method for solving differential and integral equations, World Applied Sciences Journal, 23(12) (2013), 0114.##[9] M.H. Heydaria and F.M. Maalek Ghainia, Legendre wavelets method for numerical solution of timefractional heat equation, Wavel. Linear Algebra, 1 (2014), 1931.##[10] R.D. HILL, Linear transformations which preserve hermitian matrices, Linear Algebra Appl., 6 (1973), 257262. ##[11] C.H. Hsiao, Haar wavelet approach to linear stiff systems, Math. Comput. Simul., 64 (2004), 561567.##[12] I. Aziz and S.U. Islam, New algorithms for the numerical solution of nonlinear fredholm and volterra integral equations using Haar wavelets, J. Comput. Appl. Math., 239 (2013), 333345.##[13] R. Jiwari, A Haar wavelet quasilinearization approach for numerical simulation of burger's equation, Comput. Phys. Commun., 183 (2012), 24132423.##[14] R. Jiwari, A hybrid numerical scheme for the numerical solution of the Burger's Equation, Comput. Phys. Commun., 188 (2015), 5967.##[15] K. Nouri, Application of shannon wavelet for solving boundary value problems of fractional differential equations, Wavel. Linear Algebra, 1 (2014), 3342.##[16] U. Lepik, Numerical solution of evolution equations by the Haar wavelet method, Appl. Math. Comput., 185 (2007), 695704.##[17] U. Lepik, Numerical solution of differential equations using Haar wavelets, Math. Comput. Simul., 68 (2005), 127143.##[18] U. Lepik, H. Hein, Haar Wavelets With Applications, Springer International publishing, 2014.##[19] M. Marcus, Linear transformations on matrices, Journal of research of the notional bureau of standardsB. Mathematical sciences, 75B (1971), 107113. ##[20] J. K. Merikoski, P.H. George and S.H. Wolkowicz, Bounds for ratios of eigenvalues using traces, Linear Algebra Appl., 55 (1983), 105124. ##[21] A. Mohammed, M. Balarabe and A.T. Imam, Rhotrix linear transformation, Advances in Linear Algebra and Matrix Theory, 2 (2012), 4347.##[22] R.K. Mallik, The inverse of a tridiagonal matrix, Linear Algebra Appl., 325 (2001), 109139.##[23] U. Saeed and M. Rehman, Haar wavelet quasilinearization technique for fractional nonlinear differential equations, Appl. Math. Comput., 220 (2013), 630648.##[24] S.C. Shiralashetti and A.B. Deshi, An effective Haar wavelet collocation method for the numerical solution of multiterm fractional differential equation, Nonlinear Dyn., 83 (2016), 293303.##[25] S.C. Shiralashetti, A.B. Deshi and P.B. Mutalik Desai, Haar wavelet collocation method for the numerical solution of singular initial value problems, Ain Shams Engineering Journal, 7 (2016), 663670. ##[26] S.C. Shiralashetti, L.M. Angadi, A.B. Deshi and M.H. Kantli, Haar wavelet method for the numerical solution of KleinGordan equations, AsianEur. J. Math., 9(1), (2016) 114.##[27] S.C. Shiralashetti, M.H. Kantli and A.B. Deshi, Haar wavelet based numerical solution of nonlinear differential equations arising in fluid dynamics, International Journal of Computational Materials Science and Engineering, 5(2) (2016), 113.##[28] S.C. Shiralashetti, L.M. Angadi, A.B. Deshi and M.H. Kantli, Haar wavelet method for the numerical solution of BenjaminBonaMahony equations, Journal of Information and Computing Sciences, 11(2) (2016), 136145.##[29] S.C. Shiralashetti, L.M. Angadi, M.H. Kantli and A.B. Deshi, Numerical solution of parabolic partial differential equations using adaptive gird Haar wavelet collocation method, AsianEur. J. Math., 10(1) (2017), 111.##[30] S.C. Shiralashetti, A.B. Deshi, S.S. Naregal and B. Veeresh, Wavelet series solutions of the nonlinear emdenflower type equations, International Journal of Scientific and Innovative Mathematical Research, 3(2) (2015), 558567.##[31] S.C. Shiralashetti, P.B. Mutalik Desai and A. B. Deshi, A comparative study of finite element method and Haar wavelet collocation method for the numerical solution of nonlinear ordinary differential equations, International Journal of Modern Mathematical Sciences, 13(3) (2015), 228250.##[32] S.U. Islam, I. Aziz, A. Fhaid and A. Shah, A numerical assessment of parabolic partial differential equations using Haar and legendre wavelets, Appl. Math. Modelling, 37 (2013), 94559481.##[33] S.U. Islam, I. Aziz, and A.S. AlFhaid, An improved method based on Haar wavelets for numerical solution of nonlinear integral and Integrodifferential equations of first and higher orders, J. Comput. Appl. Math., 260 (2014), 449469.##[34] G. Strang, Linear Algebra and Its Applications, Cengage Learning, (2005).##]
Projection Inequalities and Their Linear Preservers
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2
This paper introduces an inequality on vectors in $mathbb{R}^n$ which compares vectors in $mathbb{R}^n$ based on the $p$norm of their projections on $mathbb{R}^k$ ($kleq n$). For $p>0$, we say $x$ is $d$projectionally less than or equal to $y$ with respect to $p$norm if $sum_{i=1}^kvert x_ivert^p$ is less than or equal to $ sum_{i=1}^kvert y_ivert^p$, for every $dleq kleq n$. For a relation $sim$ on a set $X$, we say a map $f:X rightarrow X$ is a preserver of that relation, if $xsim y$ implies $f(x)sim f(y)$, for every $x,yin X$. All the linear maps that preserve $d$projectional equality and inequality are characterized in this paper.
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61
67


Mina
Jamshidi
Graduate University of Advanced Technology, Kerman, Iran.
Graduate University of Advanced Technology,
Iran
m.jamshidi@kgut.ac.ir


Farzad
Fatehi
University of Sussex, Brighton, United Kingdom.
University of Sussex, Brighton, United Kingdom.
Iran
f.fatehi@sussex.ac.uk
projectional inequality
Linear preserver
inequality of vectors
[[1] T. Ando, Majorization, doubly stochastic matrices and comparison of eigenvalues, Linear Algebra Appl., 118 (1989), 163248.##[2] A. Armandnejad, S. Mohtashami and M. Jamshidi, On linear preservers of gtridiagonal majorization on R^n, Linear Algebra Appl., 459 (2014), 145153.##[3] A. Armandnejad and M. Jamshidi, Multiplicative isomorphisma at invertible matrices, Miskolc Math. Notes, 15(2) (2014), 287292.##[4] G. Bennet, Majorization versus power majorization, Anal. Math., 12(4) (1986), 283286. ##[5] A. Armandnejad and A. Salemi, The structure of linear preservers of gsmajorization, Bull. Iran. Math. Soc., 32(2) (2006), 3142.##[6] G. Dahl, Matrix majorization, Linear Algebra Appl., 288 (1999), 5373.##[7] A.W. Marshall and Ingram Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, 1979.##]