2012
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Weak amenability of (2N)th dual of a Banach algebra
2
2
In this paper by using some conditions, we show that the weak amenability of
(2n)th dual of a Banach algebra A for some n ⩾ 1 implies the weak amenability of A.
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55
65


Mina
Ettefagh
Department of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran
Department of Mathematics, Tabriz Branch,
Iran
minaettefagh@gmail.com


Sima
Houdfar
Department of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran
Department of Mathematics, Tabriz Branch,
Iran
Banach algebra
Arens porducts
Arens regularity
derivation
weak amenability
[[1] R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839848. ##[2] W.G. Bade, P.C. Curtis and H.G. Dales,, Amenability and weak amenability for Bearling and ##Lipschitz algebra, Proc. London Math. Soc. , 55 (1987), no. 3, 359377. ##[3] A. Bodaghi, M.Ettefagh, M.E. Gordji and A. Medghalchi, Module structures on iterated duals ##of Banach algebras, An.st.Univ.Ovidius Constanta, 18(1) (2010) 6380. ##[4] H.G. Dales, F. Ghahramani, and N. Gronbaek, Derivations into iterated duals of Banach algebras, ##Studia Math, 128 (1998), no.1, 1954. ##[5] H.G. Dales, Banach algebra and Automatic continuity, Oxford university Press, (2000). ##[6] M. Ettefagh, The third dual of a Banach algebra, Studia. Sci. Math. Hung, 45(1) (2008) 111. ##[7] B.E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc, 127 (1972). ##[8] A. Medghalchi and T.Yazdanpanah, Problems concerning nweak amenability of a Banach algebra, ##Czecholovak Math. J, 55(130) (2005) 863876.##]
A note on uniquely (nil) clean ring
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2
A ring R is uniquely (nil) clean in case for any a 2 R there exists a uniquely
idempotent e 2 R such that a e is invertible (nilpotent). Let C =
(
A V
W B
)
be the Morita
Context ring. We determine conditions under which the rings A;B are uniquely (nil) clean.
Moreover we show that the center of a uniquely (nil) clean ring is uniquely (nil) clean.
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67
69


Shervin
Sahebi
Department of Mathematics, Islamic Azad University, Central Tehran Branch, PO.
Code 1416894351, Iran
Department of Mathematics, Islamic Azad University
Iran


Mina
Jahandar
Department of Mathematics, Islamic Azad University, Central Tehran Branch, PO.
Code 1416894351, Iran
Department of Mathematics, Islamic Azad University
Iran
m66.jahandar@gmail.com
Full element
uniquely clean ring
nil clean ring
[[1] M. Y. Ahn, (2003). Weakly clean rings and almost clean rings. Ph.D. Thesis, University of Lowa. ##[2] D. D. Anderson, V. P. Camillo, Commutative rings whose elements are a sum of unit and idempotent. ##Comm. Algebra 30 (2002), pp. 3327{3336. ##[3] B. Li, L. Feng, Fclean rings and rings having many full elements. J. Korean Math. Soc. 2 (2010), pp. ##[4] J. Che, W. K. Nicholson, Y. Zhou, Group rings in which every element is uniquely the sum of a unit ##and idempotent. J. Algebra. 306 (2006), pp. 453{460. ##[5] H. Chen, Morita contexts with many units. Comm. Algebra. 30 (3) (2002), pp. 1499{1512. ##[6] A. J. Diesl, Classes of strongly clean rings. Ph.D. Thesis, University of California, Berkeley, (2006). ##[7] A. Haghany, Hopcity and cohopcity for Morita Contexts. Comm. Algebra. 27(1)(1999), pp. 477{ ##[8] W. K. Nicholson, Y. Zhou, Rings in which elements are uniquely the some of an idempotent and unit. ##Clasy. Math. J. 46(2004), pp. 227{236.##]
A mathematically simple method based on denition for computing eigenvalues, generalized eigenvalues and quadratic eigenvalues of matrices
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2
In this paper, a fundamentally new method, based on the denition, is introduced
for numerical computation of eigenvalues, generalized eigenvalues and quadratic eigenvalues
of matrices. Some examples are provided to show the accuracy and reliability of the proposed
method. It is shown that the proposed method gives other sequences than that of existing
methods but they still are convergent to the desired eigenvalues, generalized eigenvalues and
quadratic eigenvalues of matrices. These examples show an interesting phenomenon in the
procedure: The diagonal matrix that converges to eigenvalues gives them in decreasing order
in the sense of absolute value. Appendices A to C provide Matlab codes that implement the
proposed algorithms. They show that the proposed algorithms are very easy to program.
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71
81


M
Nili Ahmadabadi
Department of Mathematics, Islamic Azad University, Najafabad Branch, Iran.
Department of Mathematics, Islamic Azad University
Iran
nili@phu.iaun.ac.ir
Eigenvalue
Generalized eigenvalue
Quadratic eigenvalue
Numerical computation
Iterative method
[[1] G.H. Golub, H.A. van der Vorst, Eigenvalue computation in the 20th century, J. Comput. Appl. Math., ##123 (2000), pp. 3565. ##[2] N. Papathanasiou, P. Psarrakos, On condition numbers of polynomial eigenvalue problems, Appl. Math. ##Comput., 4 (2010), pp. 1194205. ##[3] J.E. Roman, M. Kammerer, F. Merz and F. Jenko, Fast eigenvalue calculations in a massively parallel ##plasma turbulence code, Parallel Computing, 56 (2010), pp. 33958. ##[4] D.S. Watkins, Understanding the QR Algorithm, SIAM Review, Vol. 24, No. 4. (Oct., 1982), pp. ##427440, Jstor. ##[5] F. Gantmacher, The Theory of Matrices, Vols. I and II, Chelsea, New York, 1959. ##[6] P. Lancaster, LambdaMatrices and Vibrating Systems, Pergamon Press, Oxford, UK, 1969. ##[7] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, London, 1985. ##[8] A. S. Householder, The Theory of Matrices in Numerical Analysis, Blaisdell Publishing Company, New ##York, 1964. ##[9] Chen Gongning, Matrix Theory with Applications ,Higher Education Publishing House ,Beijing, 1990. ##(in Chinese) ##[10] Zhang Xian and Gu Dunhe, A note on A. Brauer's theorem, Linear Algebra Appl., 196 (1994) pp. ##[11] A. Brauer, Limits for the characteristic roots of a matrix IV, Duke Math. J., 19 (1952) pp. 7591. ##[12] Tam Bitshun,Yang Shangjun and Zhang Xiaodong, Invertibility of irreducible matrices, Linear Al ##gebra Appl., 259 (1996) pp. 3970. ##[13] G. Bennet, V. Goodman, and C. M. Newman, Norm of random matrices, Pac. J. Math., 59 (1975) ##pp. 359365. ##[14] B. S. Kashin, On the mean value of certain function connected with the convergence of orthogonal ##series, Anal. Math., 4 (1978) pp. 2735. ##[15] B. S. Kashin, On properties of random matrices associated with unconditional convergence almost ##everywhere, Dokl. Akad. Nauk SSSR, 254 (1980) pp. 13221325. ##[16] R. M. Megrabian, On a characteristic of random matrices connected with unconditional convergence ##almost everywhere, Anal. Math. 14 (1988) pp. 3747. ##[17] Y. Q. Yin, Z. D. Bai and P. R. Krishnaiah, On limit of the largest eigenvalue of the large dimensional ##sample covariance matrix, Center for Multivariate Analysis, Teclm. Report No. 8444, University of ##Pittsburgh, Pittsburgh, PA. (1984). ##[18] Z. D. Bai and Y. Q. Yin, Necessary and sucient conditions for almost sure convergence of the largest ##eigenvalue of Wigner matrix, Center for Multivariate Analysis, Techn. Report No. 8705, University of ##Pittsburgh, Pittsburgh, PA (1987). ##[19] S. Geman, A limit theorem for the norm of random matrices, Ann. Probab., 8, No. 2 (1980) pp. ##[20] K. W. Wachter, The strong limits of random matrix spectra for sample matrices of independent ##elements, Ann. Probab., 6, No. 1 (1978) pp. 118. ##[21] V. L. Girko, Limit theorems for the sums of distribution functions of eigenvalues of random symmetric ##matrices, Ukr. Mat. Zh., 40, No. 1 (1989) pp. 2329. ##M. Nili Ahmadabadi/ JLTA, 01  02 (2013) 7181. 81 ##[22] V. L. Girko,Limit theorems for the distribution of the eigenvalues of random symmetric matrices, ##Teor. Veroyatn. Mat. Stat., 41 (1989) pp. 2329. ##[23] V. L. Girko, The Spectral Theory of Random Matrices [in Russian], Nauka, Moscow (1988). ##[24] V. L. Girko, Limit theorems for the maximal and minimal eigenvalues of random symmetric matrices, ##Teor. Veroyatn. Primen., 35, No. 4 (1990) pp. 677690. ##[25] L. A. Pastur, Spectra of random selfadjoint operators, Usp. Mat. Nauk, 28, No. 1 (1973) pp. 363.##]
Numerical Solution of Heun Equation Via Linear Stochastic Differential Equation
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2
In this paper, we intend to solve special kind of ordinary differential equations which is called
Heun equations, by converting to a corresponding stochastic differential equation(S.D.E.). So, we construct
a stochastic linear equation system from this equation which its solution is based on computing fundamental
matrix of this system and then, this S.D.E. is solved by numerically methods. Moreover, its asymptotic
stability and statistical concepts like expectation and variance of solutions are discussed. Finally, the attained
solutions of these S.D.E.s compared with exact solution of corresponding differential equations.
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83
95


H. R.
Rezazadeh
Department of Mothematics,Karaj Branch,Islamic Azad Univercity,po.code 31485_313 Karaj,Iran
Department of Mothematics,Karaj Branch,Islamic
Iran


M
Maghasedi
Department of Mothematics,Karaj Branch,Islamic Azad Univercity,po.code 31485_313 Karaj,Iran
Department of Mothematics,Karaj Branch,Islamic
Iran
maghasedi@kiau.ac.ir


B
shojaee
Department of Mothematics,Karaj Branch,Islamic Azad Univercity,po.code 31485_313 Karaj,Iran
Department of Mothematics,Karaj Branch,Islamic
Iran
shoujaei@kiau.ac.ir
Heun equation
Wiener process
Stochastic differential equation
Linear equations system
[[1] L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, (1974). ##[2] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differ ential equations, SIAM Review ##43 (2001), 525–546. ##[3] J. Lamperti, A simple construction of certain diffusion processes, J. Math. Kyoto (1964), 161–170. ##[4] H. McKean, Stochastic Integrals, Academic Press, (1969). ##[5] B. K. Oksendal, Stochastic ##H. R. Rezazadeh et al./ JLTA, 01  02 (2013) 8395. 95 ##[10] S. Slavyanov, W. Lay.: Special Functions, A Unified Theory Based on Singularities, Oxford Univ. Press, Oxford.( ##[11] R.S. Borissov, P.P. Fiziev.: Exact Solutions of Teukolsky Master Equation with Continuous Spectrum. Bulg. J. Phys. ##37 (2010) 65–89. ##[12] P.P. Fiziev, Journal of PhysicsMathematical and Theoretical 43, 035203(2010). ##[13] Slavyanov, S.Y., and Lay, W. Special Functions, A Unified Theory Based on Singularities. Oxford Mathematical ##Monographs, (2000). ##[14] Ronveaux, A. ed. heun’s Differential Equations. Oxford University Press,(1995##]
A New Inexact Inverse Subspace Iteration for Generalized Eigenvalue Problems
2
2
In this paper, we represent an inexact inverse subspace iteration method for com
puting a few eigenpairs of the generalized eigenvalue problem Ax = Bx[Q. Ye and P. Zhang,
Inexact inverse subspace iteration for generalized eigenvalue problems, Linear Algebra and
its Application, 434 (2011) 16971715 ]. In particular, the linear convergence property of the
inverse subspace iteration is preserved.
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97
113


M
Amirfakhrian
Department of Mathematics, Islamic Azad University, Central Tehran Branch, PO.
Code 1416894351, Iran.
Department of Mathematics, Islamic Azad University
Iran
majiamir@yahoo.com


F
Mohammad
Department of Mathematics, Islamic Azad University, Central Tehran Branch, PO.
Code 1416894351, Iran.
Department of Mathematics, Islamic Azad University
Iran
f.mohammad456@yahoo.com
Eigenvalue problem
inexact inverse iteration
subspace iteration
innerouter iteration
approximation
Convergence
[[1] J. BernsMuler, I. G. Graham and A. Spence, Inexact inverse iteration for symmtric matrices, Linear ##Algebra Appl, 416 (2006), 389413. ##[2] J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, Templates for the solution of algebraic ##eigenvalue problems: a practical guide, Philadelphia, PA, USA, 2000. ##[3] M. A. Freitag, A. Spence, Convergence rates for inexact inverse iteration with application to precon ##ditioned iterative solves, BIT, 47 (2007), 2744. ##[4] G. H. Golub and Q. Ye, Inexact inverse iterations for the generalized eigenvalue problems, BIT, 40 ##(1999), 672684. ##[5] G. H. Golub and C. F. Van loan, Matrix computation, Baltimore, MD, USA, 1989. ##[6] G. H. Golub, Z. Zhang and H. Zha, Large sparse symmetric eigenvalue prob lems with homogeneous ##linear constraints:the lanczos process with innerouter iteration, Linear Algebra And Its Applications, ##309 (2000), 289306. ##[7] Z. Jia, On convergence of the inexact rayleigh quotient iteration without and with minres, 2009. ##[8] Y. Lai, K. Lin, W. Lin . An inexact inverse iteration for large sparse eigenvalue problems, Numerical ##Linear Algebra With Application, (1997), 425437. ##[9] R. B. Lehoucq and Karl Meerbergen, Using generalized cayley transformations within an inexact ##rational krylov sequence method, SIAM J. Matrix Anal. Appl., 20, 131148. ##[10] R. B. Morgan and D. S. Scott, Preconditioning the lanczos algorithm for sparse symmetric eigenvalue ##problems, SIAM J. Sci. Comput., 14 (1993), no. 3, 585593. ##[11] A. Ruhe, Rational krylov: A practical algorithm for large sparse nonsymmetric matrix pencils, SIAM ##J. Sci. Comput., 19 (1998), no. 5, 15351551. ##[12] A. Ruhe And T. Wiberg, The method of conjugate gradients used in inverse iter ation, BIT, 12 ##(1972), 543554. ##[13] P. Smit and M. H. C. Paardekooper, The eects of inexact solvers in algorithms for sym metric ##eigenvalue problems, Linear Algebra and its Applications, 287 (1999), 337357. ##[14] V. Simoncini and L. Eldn, Inexact rayleigh quotienttype methods for eigenvalue compu tations, ##BIT, 42 (2002), 159182. ##[15] G. Sleijpen and H. Van Der Vorst, A jacobidavidson iteration method for linear eigenvalue problems, ##SIAM J. Matrix Anal. Appl., 17 (2000), 401425. ##[16] D. C. Sorensen and C. Yang, A truncated rq iteration for large scale eigenvalue calculations, SIAM ##J. Matrix Anal. Appl., 19 (1998), no. 4, 10451073. ##[17] A. Stathopoulos, Y. Saad and F. Fischer, Robust preconditioning of large sparse symmetric eigenvalue ##problems, Journal of Computational and Applied Mathematics, 64 (1994), 197215.##]
ModuleAmenability on Module Extension Banach Algebras
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2
Let A be a Banach algebra and E be a Banach Abimodule then S = A E,
the l1direct sum of A and E becomes a module extension Banach algebra when equipped
with the algebras product (a; x):(a′; x′) = (aa′; a:x′ + x:a′). In this paper, we investigate
△amenability for these Banach algebras and we show that for discrete inverse semigroup S
with the set of idempotents ES, the module extension Banach algebra S = l1(ES) l1(S) is
△amenable as a l1(ES)module if and only if l1(ES) is amenable as Banach algebra.
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111
114


D
Ebrahimi baghaa
Department of Mathematics, Faculty of Science, Islamic Azad University, Centeral
Tehran Branch, P. O. Box 13185/768, Tehran, Iran.
Department of Mathematics, Faculty of Science,
Iran
dav.ebrahimibagha@iauctb.ac.ir
Moduleamenability
module extension
Banach algebras
[[1] M.Amini, Module amenability for semigroup algebras, semigroup forum 69 (2004) 243254. ##[2] M.Amini and D.Ebrahimi Bagha, Weak module amenability for semigroup algebras, Semigroup forum ##71 (2005). 1826. ##[3] W.G.Bade, H.G.Dales and Z.A.Lykova, Algebraic and strong splittings of extensions of Banach alge ##bras, Mem. Amer. Math. Soc. 137, no. 656, 1999. ##[4] H.G. DALES, Banach algebras and automatic continuity, London Math. Soc. Monographs, Volume ##24, Clarendon press, Oxford, 2000. ##[5] H.G.Dales, F. Ghahramani and NGronbaek, Drivations into iterated duals of Banach algebras, studia ##Math. 128 (1998) 1954. ##[6] J.Duncan , I.Namioka, Amenability of inverse Semigroup and their Semigroup algebras,Procedings of ##the Royal Society of Edinburgh 80A (1975) 309321. ##[7] D.Ebrahimi Bagha and M.Amini. Module amenability for Banach modules. CUB. A math. Journal. ##Vol13, No.02, (127137). ##[8] B.E.Johnson,Cohomology in Banach algebras,Memoirs of the American Mathematical Sosiety No,127, ##American Mathematical Sosiety,Providence 1972. ##[9] Y.Zhang, Weak Amenability of Module extension of Banach algebras, TrapsAmerMath. Soc ##354(2002) 41314151.##]
EClean Matrices and UnitRegular Matrices
2
2
Let a; b; k 2 K and u ; v 2 U(K). We show for any idempotent e 2 K,
(
a 0
b 0
)
is
eclean i
(
a 0
u(vb + ka) 0
)
is eclean and if
(
a 0
b 0
)
is 0clean,
(
ua 0
u(vb + ka) 0
)
is too.
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115
118


Sh.A
Safari Sabet
Department of Mathematics, Islamic Azad University, Central Tehran Branch,Code
1416894351, Iran;
Department of Mathematics, Islamic Azad University
Iran


S
Razaghi
Department of Mathematics, Islamic Azad University, Central Tehran Branch,Code
1416894351, Iran;
Department of Mathematics, Islamic Azad University
Iran
razaghi somaye@yahoo.com
matrix ring
unimodular column
unitregular
clean
eclean
[[1] V.P. Camillo, D. Khurana, A characterization of unitregular rings, Comm. Algebra 29(2001) 2293 ##[2] V.P. Camillo, H.P.Yu, Exchange rings, units and idempotents, Comm. Algebra 22(1994) 47374749. ##[3] D. Khurana,T.Y. Lam, Clean matrices and unitregular matrices, J. Algebra 280(2004) 683698.##]
Recognition of the group G2(5) by the prime graph
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2
Let G be a nite group. The prime graph of G is a graph (G) with vertex set
(G), the set of all prime divisors of jGj, and two distinct vertices p and q are adjacent by an
edge if G has an element of order pq. In this paper we prove that if (G) = (G2(5)), then G
has a normal subgroup N such that (N) f2; 3; 5g and G=N
=
G2(5).
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115
120


P
Nosratpour
aDepartment of mathematics, ILam Branch, Islamic Azad university, Ilam, Iran;
aDepartment of mathematics, ILam Branch,
Iran
p.nosratpour@ilamiau.ac.ir


M.R
Darafsheh
School of Mathematics, statistics and Computer Science, College of Science, University
of Tehran, Tehran, Iran
School of Mathematics, statistics and Computer
Iran
prime graph
Recognition
linear group