2015
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Classical Wavelet Transforms over Finite Fields
2
2
This article introduces a systematic study for computational aspects of classical wavelet transforms over finite fields using tools from computational harmonic analysis and also theoretical linear algebra. We present a concrete formulation for the Frobenius norm of the classical wavelet transforms over finite fields. It is shown that each vector defined over a finite field can be represented as a finite coherent sum of classical wavelet coefficients.
1

241
257


A.
Ghaani Farashahi
Numerical Harmonic Analysis Group (NuHAG), Faculty of Mathematics,
University of Vienna, Austria
Numerical Harmonic Analysis Group (NuHAG),
Austria
arash.ghaani.farashahi@univie.ac.at
Finite field
classical wavelet group
classical wavelet transforms
dilation operators
[[1] A. Arefijamaal and R.A. KamyabiGol. On the square integrability of quasi regular representation on semidirect product groups, J. Geom. Anal., 19 (3), (2009), 541552. ##[2] A. Arefijamaal and R.A. KamyabiGol. On construction of coherent states associated with semidirect products, Int. J. Wavelets Multiresolut. Inf. Process. 6 (5), (2008), 749759. ##[3] A. Arefijamaal and R.A. KamyabiGol, A Characterization of square integrable representations associated with CWT, J. Sci. Islam. Repub. Iran 18 (2), (2007), 159166. ##[4] G. Caire, R.L. Grossman, and H. Vincent Poor. Wavelet transforms associated with finite cyclic groups. IEEE Transaction On Information Theory 39 (4), (1993), 113119. ##[5] F. Fekri, R.M. Mersereau, and R.W. Schafer. Theory of wavelet transform over finite fields. Proceedings of International Conference on Acoustics, Speech, and Signal Processing., 3, (1999), 12131216. ##[6] K. Flornes, A. Grossmann, M. Holschneider, and B. Torr´esani. Wavelets on discrete fields. Appl. Comput. Harmon. Anal., 1, (1994), 137146. ##[7] A. Ghaani Farashahi, Structure of finite wavelet frames over prime fields, Bull. Iranian Math. Soc., to appear 2016. ##[8] A. Ghaani Farashahi, Wave packet transforms over finite cyclic groups, Linear Algebra Appl., 489 (2016), 7592. ##[9] A. Ghaani Farashahi, Wave packet transform over finite fields, Electronic Journal of Linear Algebra, Volume 30 (2015), 507529. ##[10] A. Ghaani Farashahi. Cyclic wavelet systems in prime dimensional linear vector spaces, Wavelets and Linear Algebra 2 (1), (2015), 1124. ##[11] A. Ghaani Farashahi. Cyclic wave packet transform on finite Abelian groups of prime order. Int. J. Wavelets Multiresolut. Inf. Process., 12 (6), (2014), 1450041 (14 pages). ##[12] A. Ghaani Farashahi and M. MohammadPour. A unified theoretical harmonic analysis approach to the cyclic wavelet transform (CWT) for periodic signals of prime dimensions., Sahand Commun. Math. Anal., 1 (2), (2014), 117. ##[13] A. Ghaani Farashahi and R. KamyabiGol. Gabor transform for a class of nonabelian groups., Bull. Belg. Math. Soc. Simon Stevin., 19 (4), (2012), 683701. ##[14] C. P. Johnston. On the pseudodilation representations of flornes, grossmann, holschneider, and torr´esani. J. Fourier Anal. Appl., 3 (4), (1997), 377385. ##[15] G.L. Mullen and D. Panario. Handbook of Finite Fields., Series: Discrete Mathematics and Its Applications, Chapman and Hall/CRC, 2013. ##[16] R.J. McEliece. Finite Fields for Computer Scientists and Engineers., The Springer International Series in Engineering and Computer Science, 1987. ##[17] O. Pretzel. ErrorCorrecting Codes and Finite Fields., Oxford Applied Mathematics and Computing Science Series, 1996. ##[18] D. Ramakrishnan and R.J. Valenza. Fourier Analysis on Number Fields., SpringerVerlag, New York, 1999. ##[19] R. Reiter and J.D. Stegeman. Classical Harmonic Analysis., 2nd Ed, Oxford University Press, New York, 2000. ##[20] H. Riesel. Prime Numbers and Computer Methods for Factorization., (second edition), Boston: Birkhauser, 1994. ##[21] S. Sarkar and H. Vincent Poor. Cyclic wavelet transforms for arbitrary finite data lengths. Signal Processing 80 (2000), 25412552. ##[22] G. Strang and T. Nguyen. Wavelets and Filter Banks, WellesleyCambridge Press, Wellesley, 1996. ##[23] S. A. Vanstone and P. C. Van Oorschot. An Introduction to Error Correcting Codes with Applications., The Springer International Series in Engineering and Computer Science, 1989. ##[24] A. Vourdas. Harmonic analysis on a Galois field and its subfields. J. Fourier Anal. Appl., 14 (1), (2008), 102123. ##[25] M.W. Wong. Discrete Fourier Analysis. Pseudodifferential Operators Theory and applications Vol. 5, SpringerBirkhauser, 2010.##]
Duals and approximate duals of gframes in Hilbert spaces
2
2
In this paper we get some results and applications for duals and approximate duals of gframes in Hilbert spaces. In particular, we consider the stability of duals and approximate duals under bounded operators and we study duals and approximate duals of gframes in the direct sum of Hilbert spaces. We also obtain some results for perturbations of approximate duals.
1

259
265


M.
Mirzaee Azandaryani
Department of Mathematics, Faculty of Science, University of Qom, Qom, Iran
Department of Mathematics, Faculty of Science,
Iran
morteza_ma62@yahoo.com


A.
Khosravi
Faculty of Mathematical Sciences
and Computer, Kharazmi University, Tehran, Iran
Faculty of Mathematical Sciences
and Computer,
Iran
khosravi_amir@yahoo.com
Frame
gframe
duality
approximate duality
[[1] O. Christensen, R. S. Laugesen, Approximate dual frames in Hilbert spaces and applications to Gabor frames, Sampl. Theory Signal Image Process. 9 (2011), 7790. ##[2] I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), 12711283. ##[3] R. J. Duffin, A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341366. ##4] A. Khosravi, M. Mirzaee Azandaryani, Approximate duality of gframes in Hilbert spaces, Acta Mathematica Scientia. 34B(3) (2014), 639652. ##[5] A. Khosravi, M. Mirzaee Azandaryani, Gframes and direct sums, Bull. Malays. Math. Sci. Soc. 36(2) (2013), 313323. ##[6] A. Khosravi, M. Mirzaee Azandaryani, Fusion frames and gframes in tensor product and direct sum of Hilbert spaces, Appl. Anal. Discrete Math. 6 (2012), 287303. ##[7] A. Khosravi, K. Musazadeh, Fusion frames and gframes, J. Math. Anal. Appl. 342 (2008), 10681083. ##[8] G. J. Murphy, C∗Algebras and Operator Theory. Academic Press, San Diego, 1990. ##[9] W. Sun, Gframes and gRiesz bases, J. Math. Anal. Appl. 322 (2006), 437452.##]
Generalized superconnectedness
2
2
A. Csaszar introduced and extensively studied the notion of generalized open sets. Following Csazar, we introduce a new notion superconnected. The main purpose of this paper is to study generalized superconnected spaces. Various characterizations of generalized superconnected spaces and preservation theorems are discussed.
1

267
273


E.
Bouassida
Department of Mathematics, Faculty of Sciences of Sfax, BP 802, 3038 Sfax, Tunisia
Department of Mathematics, Faculty of Sciences
Tunisia
basdourimed@yahoo.fr


B.
Ghanmi
Department of Mathematics, Faculty of Sciences of Gafsa, Zarroug 2112, Tunisia
Department of Mathematics, Faculty of Sciences
Tunisia
basdourimd@yahoo.fr


R.
Messaoud
Department of Mathematics, Faculty of Sciences of Gafsa, Zarroug 2112, Tunisia
Department of Mathematics, Faculty of Sciences
Tunisia
rimessaoud@yahoo.fr


A.
Missaoui
Department of Mathematics, Faculty of Sciences of Sfax, BP 802, 3038 Sfax, Tunisia
Department of Mathematics, Faculty of Sciences
Tunisia
rimoud@yahoo.fr
Generalized topology
connected
Superconnected
[[1] A. Csaszar, Generalized topology, generalized continuity, Acta math. Hungar. 96 (2002), 351357. ##[2] A. Csaszar, Separation axioms for generalized topologies, Acta math. Hungar. 104 (2004), 6369. ##[3] A. Csaszar, Generalized open sets in generalized topologies, Acta math. Hungar. 106 (2005), 5366. ##[4] A. Csaszar, δand θmodifications of generalized topologies, Acta math. Hungar. 120 (2008), 275279. ##[5] W. K. Min, Some results on generalized topological spaces and generalized systems, Acta math. Hungar. 108 (2005), 171181. ##[6] R. X. Shen, A note on generalized connectedness, Acta math. Hungar. 122 (2009), 231235. ##[7] E. EKICI, Generalized hyperconnectedness, Acta math. Hungar. 133 (12), (2011), 140147.##]
Quotient Arens regularity of $L^1(G)$
2
2
Let $mathcal{A}$ be a Banach algebra with BAI and $E$ be an introverted subspace of $mathcal{A}^prime$. In this paper we study the quotient Arens regularity of $mathcal{A}$ with respect to $E$ and prove that the group algebra $L^1(G)$ for a locally compact group $G$, is quotient Arens regular with respect to certain introverted subspace $E$ of $L^infty(G)$. Some related result are given as well.
1

275
281


A.
ZivariKazempour
Department of Mathematics, University of Ayatollah Borujerdi, Borujerd, Iran
Department of Mathematics, University of
Iran
zivari6526@gmail.com
Arens product
Quotient Arens regular
Introverted subspace
Weakly almost periodic
[[1] R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839848. ##[2] J. F. Berglund, H. D. Junghenn and P. Milnes, Analysis on Semigroups, WileyInterscience, New York, 1989. ##[3] P. Civin and B. Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J. Math. 11 (1961), 847870. ##[4] H. G. Dales, Banach algebras and automatic continuity, London Math. Soc. Monographs 24, Clarenden Press, Oxford, 2000. ##[5] H. G. Dales and A. T. M. Lau, The second duals of Beurling algebras, Mem. Amer. Math. Soc. 177 (2005), 1199. ##[6] H. G. Dales, A. T.M. Lau and D. Strauss, Banach algebras on semigroups and on their compactification, Mem. Amer. Math. Soc. 205 (2010), 1165. ##[7] E. Hewitt and K. Ross, Abstract Harmonic Analysis, Volume II, SpringerVerlag, Berlin, 1970. ##[8] N. Isik, J. Pym and A. Ulger, The second dual of the group algebra of a compact group, J. London Math. Soc. 35 (1987), 135158. ##[9] A. T. M. Lau and A. Ulger, Topological centres of certain dual algebras, Trans. Amer. Math. Soc. 348 (1996), 11911212. ##[10] A. Ulger, Arens regularity of the algebra A⊗Bb , Trans. Amer. Math. Soc, 305 (1988), 623639. ##[11] A. Ulger, Arens regularity sometimes implies RNP, Pacific J. Math. 143 (1990), 377399. ##[12] P. K. Wong, Arens product and the algebra of duble multipliers, Proc. Amem. Math. Soc. 94 (1985), 441444. ##[13] N. J. Young, The irregularity of multiplication in group algebras, Quart. J. Math. 24 (1973), 5962.##]
Some results on higher numerical ranges and radii of quaternion matrices
2
2
Let $n$ and $k$ be two positive integers, $kleq n$ and $A$ be an $n$square quaternion matrix. In this paper, some results on the $k$numerical range of $A$ are investigated. Moreover, the notions of $k$numerical radius, right $k$spectral radius and $k$norm of $A$ are introduced, and some of their algebraic properties are studied.
1

283
288


Gh.
Aghamollaei
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran
Department of Pure Mathematics, Faculty of
Iran
aghamollaei@uk.ac.ir


N.
Haj Aboutalebi
Department of Mathematics, Shahrood Branch, Islamic Azad University, Shahrood, Iran
Department of Mathematics, Shahrood Branch,
Iran
nargesaboutalebi@yahoo.ca
$k$numerical radius
right $k$spectral radius
$k$norm
quaternion matrices
[[1] N. Haj Aboutalebi, Gh. Aghamollaei and H. Momenaee Kermani, Higher numerical ranges of quaternion matrices, Electronic J. Linear Algebra, 30 (2015), 889904. ##[2] R. Kippenhahn, Uber die Wertvorrat einer Matrix, Math. Nachr. 6 (1951), 193228. ##[3] C.K. Li, T.Y. Tam and N.K. Tsing, The generalized spectral radius, numerical radius and spectral norm, Linear Multilinear Algebra, 16 (1984), 215237. ##[4] F. Zhang, Quaternions and matrices of quaternions, Linear Algebra Appl. 251 (1997), 2157.##]
A numerical solution of mixed Volterra Fredholm integral equations of Urysohn type on nonrectangular regions using meshless methods
2
2
In this paper, we propose a new numerical method for solution of Urysohn two dimensional mixed VolterraFredholm integral equations of the second kind on a nonrectangular domain. The method approximates the solution by the discrete collocation method based on inverse multiquadric radial basis functions (RBFs) constructed on a set of disordered data. The method is a meshless method, because it is independent of the geometry of the domain and it does not require any background interpolation or approximation cells. The error analysis of the method is provided. Numerical results are presented, which confirm the theoretical prediction of the convergence behavior of the proposed method.
1

289
304


M.
Nili Ahmadabadi
Department of Mathematics, Najafabad Branch, Islamic Azad University,
Najafabad, Iran
Department of Mathematics, Najafabad Branch,
Iran
mneely59@hotmail.com


H.
Laeli Dastjerdi
Department of Mathematics, Najafabad Branch, Islamic Azad University,
Najafabad, Iran
Department of Mathematics, Najafabad Branch,
Iran
hojatld@gmail.com
Mixed VolterraFredholm integral equations
collocation method
Radial basis functions
Meshless method
Numerical treatment
[[1] H. Wendland, Scattered Data Approximation, Cambridge University Press, 2005. ##[2] W.R. Madych, S.A. Nelson, Multivariate interpolation and conditionally positive definite functions II, Math. Comput. 54 (189) (1990) 211230. ##[3] W.R. Madych, S.A. Nelson, Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation, J. Approx. Theory 70 (1992) 94114. ##[4] P.J. Kauthen, Continuous time collocation methods for VolterraFredholm integral equations, Numer. Math. 56 (1989) 409424. ##[5] H. Brunner, P.J. van der Houwen, The numerical solution of Volterra equations, CWI Monographs, vol. 3, NorthHolland, Amsterdam, 1986. ##[6] H. Brunner, On the numerical solution of nonlinear VolterraFredholm integral equations by collocation methods, SIAM J. Numer. Anal. 27 (4) (1990) 9871000. ##[7] J.P. Kauthen, Continuous time collocation method for VolterraFredholm integral equations, Numer. Math. 56 (1989) 409424. ##[8] K. Maleknejad, M. Hadizadeh, A new computational method for VolterraFredholm integral equations, Comput. Math. Appl. 37 (1999) 18. ##[9] A.M. Wazwaz, A reliable treatment for mixed VolterraFredholm integral equations, Appl. Math. Comput. 127 (2002) 405414. ##[10] E. Banifatemi, M. Razzaghi, S. Yousefi, Twodimensional Legendre Wavelets Method for the mixed VolterraFredholm integral equations, J. Vibr. Control. 13 (2007) 16671675. ##[11] R.L. Hardy, Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res. 76 (1971) 19051915. ##[12] H.R. Thieme, A model for the spatio spread of an epidemic, J. Math. Biol. 4 (1977) 337351. ##[13] G. Han, L. Zhang, Asymptotic expansion for the trapezoidal Nystrom method of linear VolterraFredholm equations, J. Comput. Appl. Math. 51 (1994) 339348. ##[14] M. Hadizadeh, M. Asgari, An effective numerical approximation for the linear class of mixed integral equations, Appl. Math. Comput. 167 (2005) 10901100. ##[15] E. Babolian, A.J. Shaerlar, Two dimensional block pulse functions and application to solve VolterraFredholm integral equations with Galerkin method, Int. J. Contemp. Math. Sci. 6 (2011) 763770. ##[16] H. Laeli Dastjerdi, F. M. Maalek Ghaini, M. Hadizadeh, A meshless approximate solution of mixed VolterraFredholm integral equations, Int. J. Comput. Math. 90 (2013) 527538. ##[17] R.L. Hardy, Theory and applications of the multiquadricbiharmonic method. 20 years of discovery 1968 1988, Comput. Math. Appl. 19 (89) (1990) 163208. ##[18] E.J. Kansa, MultiquadricsA scattered data approximation scheme with applications to computational fluiddynamics. I. Surface approximations and partial derivative estimates, Comput. Math. Appl. 19 (89) (1990) 127145. ##[19] E.J. Kansa, MultiquadricsA scattered data approximation scheme with applications to computational fluiddynamics. II. Solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput. Math. Appl. 19 (89) (1990) 147161. ##[20] A. Cardone, E. Messina, E. Russo, A fast iterative method for discretized VolterraFredholm integral equations, J. Comput. Appl. Math. 189 (2006) 568579. ##[21] Y.C. Hon, X.Z. Mao, An efficient numerical scheme for Burgers equation, Appl. Math. Comput. 95 (1998) 3750. ##[22] Y.C. Hon, K.F. Cheung, X.Z. Mao, E.J. Kansa, Multiquadric solution for shallow water equations, ASCE J. Hydraul. Eng. 125 (1999) 524533. ##[23] M. Zerroukat, H. Power, C.S. Chen, A numerical method for heat transfer problem using collocation and radial basis functions, Int. J. Numer. Meth. Eng. 42 (1992) 12631278. ##[24] K. E. Atkinson, F. A. Potra, Projection and iterated projection methods for nonlinear integral equations, SIAM J. Numer. Anal. 24 (1987) 13521373. ##[25] K. Atkinson, J. Flores, The discrete collocation method for nonlinear integral equations, IMA J. Numer. Anal. 13 (1993) 195213. ##[26] G.E. Fasshauer, Meshfree methods. In: Rieth, M., Schommers, W. (eds.) Handbook of Theoretical and Computational Nanotechnology, American Scientific Publishers, 27 (2006) 3397.##]
A new Approximation to the solution of the linear matrix equation AXB = C
2
2
It is wellknown that the matrix equations play a significant role in several applications in science and engineering. There are various approaches either direct methods or iterative methods to evaluate the solution of these equations. In this research article, the homotopy perturbation method (HPM) will employ to deduce the approximated solution of the linear matrix equation in the form AXB=C. Furthermore, the conditions will be explored to check the convergence of the homotopy series. Numerical examples are also adapted to illustrate the properties of the modified method.
1

305
315


A.
Sadeghi
Department of Mathematics, Robat Karim Branch, Islamic Azad University, Tehran, Iran
Department of Mathematics, Robat Karim Branch,
Iran
drsadeghi.iau@gmail.com
Matrix equation
Homotopy perturbation method
Diagonally dominant matrix
Convergence, Iterative method
[[1] P. Benner, Factorized solution Of Sylvester equations with applications in Control. In: Proceedings of international symposium of mathematics. Theory networks and systems, MTNS 2004. ##[2] P. Benner, Largescale matrix equations of special type, Numer Linear Algebra Appl, 15 (2008), 747754. ##[3] J. Cai, G. Chen, An iterative algorithm for the least squares bisymmetric solutions of the matrix equations A1XB1 = C1 and A2XB2 = C2, Mathematical and Computer Modelling 50 (8), (2009), 12371244. ##[4] B. N. Datta, K. Datta, Theoretical and computational aspects of some linear algebra problems in Control theory. In: Byrnes CI, Lindquist A (eds) Computational and combinatorial methods in systems theory. Elsevier, Amsterdam, pp. 201212, 1986. ##[5] J. Ding, Y.J. Liu, F. Ding, Iterative solutions to matrix equations of form AiXBi = Fi. Comput. Math. Appl. 59 (11), (2010), 35003507. ##[6] A. J. Laub, M. T. Heath, C. Paige, and R. C. Ward, Computation of system balancing transformations and other applications of simultaneous diagonalisation algorithms. IEEE Trans Automat Control 32 (1987), 115122. ##[7] S. J. Liao. The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D. Thesis, Shanghai Jiao Tong University, 1992. ##[8] A. Liao, Y. Lei, Leastsquares solution with the minimumnorm for the matrix equation (AXB, GXH) = (C, D), Computers and Mathematics with Applications 50 (3), (2005), 539549. ##[9] J. H. He. Homotopy perturbation technique. Comput Methods Appl Mech Eng, (1999), 5762. ##[10] J. H. He. A coupling method of a homotopy technique and a perturbation technique for nonlinear problems. Int J Nonlinear Mech, 35 (1), (2000), 3743. ##[11] J. H. He. Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput, 135 (1), (2003), 7379. ##[12] B. Keramati, An approach to the solution of linear system of equations by he’s HPM, Chaos Solitons Fract. doi:10.1016/j.chaos.2007.11.020. ##[13] H. K. Liu, Application of homotopy perturbation methods for solving systems of linear equations, Appl. Math. Comput, 217(12), (2011), 52595264. ##[14] Y.H. Liu, Ranks of least squares solutions of the matrix equation AXB = C, Computers and Mathematics with Applications, 55 (6), (2008), 12701278. ##[15] S.K. Mitra, A pair of simultaneous linear matrix equations A1XB1 = C1 and A2XB2 = C2 and a programming problems, Linear Algebra Appl. 131 (1990), 107123. ##[16] A. Navarra, P.L. Odell, D.M. Young, A representation of the general common solution to the matrix equations A1XB1 = C1 and A2XB2 = C2 with applications, Computers and Mathematics with Applications 41 (8), (2001), 929935. ##[17] M. A. Noor, K. I. Noor, S. Khan, and M. Waseem, Modified homotopy perturbation method for solving system of linear equations, Journal of the Association of Arab Universities for Basic and Applied Sciences, 13 (2013), 3537. ##[18] S. A. Edalatpanah, and M.M. Rashidi, On the application of homotopy perturbation method for solving systems of linear equations, International Scholarly Research Notices, (2014), doi.10.1155/2014/143512. ##[19] A. J. Laub, Matrix analysis for scientists and engineers, SIAM, Philadelphia, PA, 2005. ##[20] Y. Saad, Iterative methods for sparse linear systems, second ed., SIAM, 2003. ##[21] A. Sadeghi, M. I. Ahmad, A. Ahmad, and M. E. Abbasnejad, A note on solving the fuzzy Sylvester matrix equation, journal of computational analysis and applications, 15 (1), (2013), 1022. ##[22] A. Sadeghi, S. Abbasbandy, and M. E. Abbasnejad, The common solution of the pair of fuzzy matrix equations, World Applied Sciences, 15 (2), (2011), 232238. ##[23] A. Sadeghi, M. E. Abbasnejad, and M. I. Ahmad, On solving systems of fuzzy matrix equation, Far East Journal of Applied Mathematics, 59 (1), (2011), 3144. ##[24] J. Saeidian, E. Babolian, and A. Aziz, On a homotopy based method for solving systems of linear equations, TWMS J. Pure Appl. Math., 6(1), (2015), 1526. ##[25] Y. Tian, Ranks of solutions of the matrix equation AXB = C, Linear Multilinear Algebra. 51 (2003), 111125. ##[26] S. Yuan, A. Liao, Y. Lei, Least squares Hermitian solution of the matrix equation (AXB, CXD) = (E, F) with the least norm over the skew field of quaternions, Mathematical and Computer Modelling, 48 (2): (2008), 91100. ##[27] E. Yusufoglu, An improvement to homotopy perturbation method for solving system of linear equations. Comput Math Appl, 58 (2009), 22312235.##]