ORIGINAL_ARTICLE
Classical wavelet systems over finite fields
This article presents an analytic approach to study admissibility conditions related to classical full wavelet systems over finite fields using tools from computational harmonic analysis and theoretical linear algebra. It is shown that for a large class of non-zero window signals (wavelets), the generated classical full wavelet systems constitute a frame whose canonical dual are classical full wavelet frames as well, and hence each vector defined over a finite field can be represented as a finite coherent sum of classical wavelet coefficients as well.
http://wala.vru.ac.ir/article_23236_5150e489e24248c000bc17f050d8b322.pdf
2016-12-01T11:23:20
2018-02-18T11:23:20
1
18
10.22072/wala.2016.23236
Finite field
classical wavelet group
quasi-regular representation
classical wavelet systems
classical dilation operators
Arash
Ghaani Farashahi
arash.ghaani.farashahi@univie.ac.at
true
1
University of Vienna
University of Vienna
University of Vienna
LEAD_AUTHOR
[1] A. Arefijamaal and E. Zekaee, Image processing by alternate dual Gabor frames, Bull. Iran. Math. Soc.,
1
42(6)(2016), 1305 -1314.
2
[2] A. Arefijamaal and E. Zekaee, Signal processing by alternate dual Gabor frames, Appl. Comput. Harmon. Anal., 35(3)(2013), 535-540.
3
[3] A. Arefijamaal and R.A. Kamyabi-Gol, On the square integrability of quasi regular representation on semidirect product groups, J. Geom. Anal., 19(3)(2009), 541-552.
4
[4] A. Arefijamaal and R.A. Kamyabi-Gol, On construction of coherent states associated with semidirect products, Int. J. Wavelets Multiresolut. Inf. Process., 6(5) (2008), 749-759.
5
[5] A. Arefijamaal and R.A. Kamyabi-Gol, A Characterization of square integrable representations associated with CWT, J. Sci. Islam. Repub. Iran 18(2)(2007), 159-166.
6
[6] I. Daubechies, The wavelet transform, time-frequency localization and signal analysis., IEEE Trans. Inform.
7
Theory, 36(5) (1990), 961-1005.
8
[7] K. Flornes, A. Grossmann, M. Holschneider, and B. Torresani, Wavelets on discrete fields, Appl. Comput. Harmon. Anal., 1(2)(1994), 137-146.
9
[8] A. Ghaani Farashahi, Structure of finite wavelet frames over prime fields, Bull. Iranian Math. Soc., to appear.
10
[9] A. Ghaani Farashahi, Wave packet transforms over finite cyclic groups, Linear Algebra Appl., 489(1) (2016), 75-92.
11
[10] A. Ghaani Farashahi, Classical wavelet transforms over finite fields, J. Linear Topol. Algebra, 4 (4) (2015), 241-257.
12
[11] A. Ghaani Farashahi, Wave packet transform over finite fields, Electron. J. Linear Algebra, 30 (2015), 507-529.
13
[12] A. Ghaani Farashahi, Cyclic wavelet systems in prime dimensional linear vector spaces, Wavelets and Linear Algebra, 2 (1) (2015) 11-24.
14
[13] A. Ghaani Farashahi, Cyclic wave packet transform on finite Abelian groups of prime order, Int. J. Wavelets
15
Multiresolut. Inf. Process., 12(6), 1450041 (14 pages), 2014.
16
[14] A. Ghaani Farashahi, M. Mohammad-Pour, A unified theoretical harmonic analysis approach to the cyclic
17
wavelet transform (CWT) for periodic signals of prime dimensions, Sahand Commun. Math. Anal., 1(2)(2014),
18
[15] C. P. Johnston, On the pseudodilation representations of flornes, grossmann, holschneider, and torresani, J. Fourier Anal. Appl., 3(4)(1997), 377-385.
19
[16] G. L. Mullen, D. Panario, Handbook of Finite Fields, Series, Discrete Mathematics and Its Applications, Chapman and Hall/CRC, 2013.
20
[17] R. J. McEliece, Finite Fields for Computer Scientists and Engineers, The Springer International Series in Engineering and Computer Science, 1987.
21
[18] G. Pfander, Gabor Frames in Finite Dimensions, In Finite Frames, Applied and Numerical Harmonic Analysis, 193-239. Birkhauser Boston, 2013.
22
[19] O. Pretzel, Error-Correcting Codes and Finite Fields., Oxford Applied Mathematics and Computing Science
23
Series, 1996.
24
[20] R. Reiter and J.D. Stegeman, Classical Harmonic Analysis, 2nd Ed, Oxford University Press, New York, 2000.
25
[21] H. Riesel, Prime numbers and computer methods for factorization, (second edition), Boston, Birkhauser, 1994.
26
[22] 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.
27
[23] A. Vourdas, Harmonic analysis on a Galois field and its subfields, J. Fourier Anal. Appl., 14(1)(2008), 102-123.
28
ORIGINAL_ARTICLE
Linear combinations of wave packet frames for L^2(R^d)
In this paper we study necessary and sufficient conditions for some types of linear combinations of wave packet frames to be a frame for L2(Rd). Further, we illustrate our results with some examples and applications.
http://wala.vru.ac.ir/article_23237_9748c2d52d39dc6189458e4f3a485660.pdf
2016-12-01T11:23:20
2018-02-18T11:23:20
19
32
10.22072/wala.2016.23237
Frames
Wave Packet Systems
Linear Combinations
Ashok
Sah
ashokmaths2010@gmail.com
true
1
University of Delhi
University of Delhi
University of Delhi
LEAD_AUTHOR
[1] A. Aldroubi, Portraits of frames, Proc. Amer. Math. Soc., 123(6)(1995), 1661–1668.
1
[2] P. G. Casazza, G. Kutyniok. Finite frames: Theory and Applications. Birkhauser, 2012.
2
[3] O. Christensen, Linear combinations of frames and frame packets, Z. Anal. Anwend., 20(4)(2001), 805–815.
3
[4] O. Christensen, An introduction to frames and Riesz bases, Birkhauser, Boston, 2002.
4
[5] O. Christensen, A. Rahimi, Frame properties of wave packet systems in L2(Rd), Adv. Compu. Math., 29(2008), 101–111.
5
[6] A. Cordoba, C. Fefferman, Wave packets and Fourier integral operators, Commun. Partial Differ. Equations,
6
3(11)(1978), 979–1005.
7
[7] W. Czaja, G. Kutyniok, D. Speegle, The geometry of sets of prameters of wave packets, Appl. Comput. Harmon. Anal., 20(1)(2006), 108–125.
8
[8] K. Guo, D. Labate, Some remarks on the unified characterization of reproducing systems, Collect. Math., 57 (3)(2006), 309–318.
9
[9] C. Heil, D. Walnut, Continuous and discrete wavelet transforms, SIAM Rev., 31(4)(1989), 628–666.
10
[10] C. Heil, A Basis Theory Primer, Expanded edition. Applied and Numerical Harmonic Analysis, Birkhauser,
11
Springer, New York, 2011.
12
[11] E. Hernandez, D. Labate, G. Weiss, A unified characterization of reproducing systems generated by a finite family II, J. Geom. Anal., 12(4)(2002), 615–662.
13
[12] E. Hernandez, D. Labate, G. Weiss, E. Wilson, Oversampling, quasi-affine frames and wave packets, Appl. Comput. Harmon. Anal., 16(2004), 111–147.
14
[13] D. Labate, G. Weiss, E. Wilson, An approach to the study of wave packet systems, Contemp. Math., 345(2004), 215–235.
15
[14] M. Lacey, C. Thiele, Lp estimates on the bilinear Hilbert transform for 2<p<1, Ann. Math., 146(1997),
16
693–724.
17
[15] M. Lacey, C. Thiele, On Calderons conjecture, Ann. Math., 149(1999), 475–496.
18
[16] A. K. Sah, L. K. Vashisht, Hilbert transform of irregular wave packet system for L2(R), Poincare J. Anal. Appl.,1(2014), 9–17.
19
[17] A. K. Sah, L. K. Vashisht, Irregular Weyl-Heisenberg wave packet frames in L2(R), Bull. Sci. Math., 139(1)(2015), 61–74.
20
ORIGINAL_ARTICLE
Cartesian decomposition of matrices and some norm inequalities
Let X be an n-square complex matrix with the Cartesian decomposition X = A + i B, where A and B are n times n Hermitian matrices. It is known that $Vert X Vert_p^2 leq 2(Vert A Vert_p^2 + Vert B Vert_p^2)$, where $p geq 2$ and $Vert . Vert_p$ is the Schatten p-norm. In this paper, this inequality and some of its improvements are studied and investigated for the joint C-numerical radius, joint spectral radius, and for the C-spectral norm of matrices.
http://wala.vru.ac.ir/article_23238_64b86b56682082a7a0b0f1d249c90a93.pdf
2016-12-01T11:23:20
2018-02-18T11:23:20
33
42
10.22072/wala.2016.23238
joint C-numerical radius
C-spectral norm
joint spectral radius
Alemeh
Sheikhhosseini
sheikhhosseini@uk.ac.ir
true
1
Department of Pure Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran
Department of Pure Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran
Department of Pure Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran
AUTHOR
Golamreza
Aghamollaei
aghamollaei@uk.ac.ir
true
2
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran
LEAD_AUTHOR
[1] Gh. Aghamollaei, N. Avizeh and Y. Jahanshahi, Generalized numerical ranges of matrix polynomials, Bull. Iran. Math. Soc., 39(5)(2013), 789–803.
1
[2] Gh. Aghamollaei, A. Salemi, Polynomial numerical hulls of matrix polynomials II, Linear Multilinear Algebra, 59(3)(2011), 291–302.
2
[3] R. Bhatia, Matrix Analysis, Springer, Berlin, 1997.
3
[4] R. Bhatia, T. Bhattacharyya, On the joint spectral radius of commuting matrices, Studia Math., 114(1)(1995), 29–37.
4
[5] R. Bhatia, F. Kittaneh, Cartesian decompositions and Schatten norms, Linear Algebra Appl., 318(1-3)(2012), 109–116.
5
[6] M.T. Chien, H. Nakazato, Joint numerical range and its generating hypersurface, Linear Algebra Appl., 432(1)(2010),173–179.
6
[7] R. Drnovsek, V. Muller, On joint numerical radius II, Linear Multilinear Algebra, 62(9)(2014): 1197–1204.
7
[8] R.A. Horn, C.R. Johnson, Matrix Analysis. Second Edition, Cambridge University Press, Cambridge, 2013.
8
[9] F. Kittaneh, M. S. Moslehian and T. Yamazaki, Cartesian decomposition and numerical radius inequalities,
9
Linear Algebra Appl., 471(4)(2015), 46–53.
10
[10] C.K. Li, C-numerical ranges and C-numerical radii, Linear Multilinear Algebra, 37(1-3)(1994), 51–82.
11
[11] C.K. Li and E. Poon, Maps preserving the joint numerical radius distance of operators, Linear Algebra Appl., 437(5)(2012), 1194–1204.
12
[12] C.K. Li, T. Tam and N. Tsing, The generalized spectral radius, numerical radius and spectral norm, Linear
13
Multilinear Algebra, 16(5)(1984), 215–237.
14
[13] A. Salemi and Gh. Aghamollaei, Polynomial numerical hulls of matrix polynomials, Linear Multilinear Algebra, 55(3)(2007), 219–228.
15
[14] B. Simon, Trace Ideals and their Applications, Cambridge University Press, Cambridge, New York, 1979.
16
ORIGINAL_ARTICLE
Pseudoframe multiresolution structure on abelian locally compact groups
Let $G$ be a locally compact abelian group. The concept of a generalized multiresolution structure (GMS) in $L^2(G)$ is discussed which is a generalization of GMS in $L^2(mathbb{R})$. Basically a GMS in $L^2(G)$ consists of an increasing sequence of closed subspaces of $L^2(G)$ and a pseudoframe of translation type at each level. Also, the construction of affine frames for $L^2(G)$ based on a GMS is presented.
http://wala.vru.ac.ir/article_23239_9bab4fd5ee9537b7e6c4e65984d3cc78.pdf
2016-12-01T11:23:20
2018-02-18T11:23:20
43
54
10.22072/wala.2016.23239
Pseudoframe
generalized multiresolution structure
locally compact group, affine pseudoframe
Hamide
Azarmi
azarmi_1347@yahoo.com
true
1
Ph. D. student in Ferdowsi University of Mashhad
Ph. D. student in Ferdowsi University of Mashhad
Ph. D. student in Ferdowsi University of Mashhad
AUTHOR
Radjabali
Kamyabi Gol
kamyabi@um.ac.ir
true
2
Department of pure Mathematics; Ferdowsi University of Mashhad;
Department of pure Mathematics; Ferdowsi University of Mashhad;
Department of pure Mathematics; Ferdowsi University of Mashhad;
AUTHOR
Mohammad
Janfada
janfada@um.ac.ir
true
3
Department of pure Mathematics;Ferdowsi University of Mashhad;
Department of pure Mathematics;Ferdowsi University of Mashhad;
Department of pure Mathematics;Ferdowsi University of Mashhad;
LEAD_AUTHOR
[1] J. J. Benedetto, S. Li, The theory of multiresolution analyses frames and applications to filter banks, Appl. Comp. Harm. Anal., 5 (1998), 389-427.
1
[2] J. J. Benedetto, S. Li, Multiresolution analysis frames with applications, Proceeding ICASSP93 Proceedings of IEEE international conference on Acoustics, speech, and signal processing: digital speech processing Volume III Pages 304-307, 1993.
2
[3] P. G. Casazza O. Christensen, D. Stoeva, Frame expansions in separable Banach space, J. Math. Anal. Appl., 114(1) (2005), 710–723.
3
[4] O. Christensen, On frame multiresolution analysis, In An Introduction to Frames and Riesz Bases, Part of the series Applied and Numerical Harmonic Analysis, 283-311, 2003.
4
[5] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhauser, Boston 2003.
5
[6] S. Dahlke, Multiresolution Analysis and Wavelets on Locally Compact Abelian Groups, Wavelets, Images, and Surface Fitting. P. J. Laurent, A. Le Mehaute, L. L. Schumaker, eds., A. K. Peters, Wellesley, 1994, 141-156.
6
[7] I. Daubechies, Orthonormal bases of compactly supported wavelets, Commun. Pure Appl. Math., 41(9) (1988), 909-996.
7
[8] I. Daubechies, A. Grossmann, Y. Meyer, Painless nonorthogonal expansion, J. Math. Phys. 27 (1986), 1271– 1283.
8
[9] R. Duffin R, S. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), 341–366.
9
[10] Yu. A. Farkov, Orthogonal wavelets on locally compact abelian groups, Funktsional. Anal. i Prilozhen., 31(4) (1997), 86-88, English transl., Funct. Anal. Appl., 31 (1997), 294-296.
10
[11] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, 1995.
11
[12] D. Gabor, Theory of communications, J. Inst. Electr. Eng., 93(26) (1946), 429–457.
12
[13] R. A. Kamyabi Gol, R. Raisi Tousi, The structure of shift invariant spaces on a locally compact abelian group, J. Math. Anal. Appl., 340(1) (2008), 219–225.
13
[14] R. A. Kamyabi Gol, R. Raisi Tousi, Some equivalent multiresolution conditions on locally compact abelian
14
groups, Proc. Math. Sci., 120(3) (2010), 317–331.
15
[15] S. V. Kozyrev, Wavelet theory as p-adic spectral analysis, Izv. Ross. Akad. Nauk Ser. Mat., 66(2) (2002), 149-158, English transl, Izv. Math., 66 (2002), 367–376.
16
[16] W. C. Lang, Wavelet analysis on the Cantor dyadic group, Houston J. Math., 24(3)(1998), 533–544.
17
[17] S. Li, The theory of frame multiresolution analysis and its applications, Ph. D. Thesis, University of Maryland Graduate School, Baltimore, May 1993.
18
[18] S. Li, A theory of generalized multiresolution structure and pseudoframes of translates, J. Fourier Anal. and Appl. 7(1) (2001), 23–40.
19
[19] S. Li, H. Ogawa, A theory of peseudoframes for subspaces with applications, Proc. SPIE 3458, Wavelet Applications in Signal and Imaging Processing VI, 67(1998); doi:10.1117/12.328126.
20
[20] S. Li, H. Ogawa, Pseudoframes for subspaces with applications, J. Fourier Anal. Appl., 10(4)(2004), 409–431.
21
[21] S. G. Mallat, Multiresolution approximations and wavelet orthonormal bases of L2(R), Trans. Amer. Math. Soc., 315(1) (1989), 69–87.
22
[22] Y. Meyer, Wavelets and Operators, Translated by DH Salinger, Cambridge Studies in Advanced Mathematics, 1992.
23
[23] D. P. Petersen, D. Middleton, Sampling reconstruction of wave-number limited functions in N-dimensional Euclidean spaces, Inf. Control, 5(4)(1962), 279–323.
24
ORIGINAL_ARTICLE
Quartic and pantic B-spline operational matrix of fractional integration
In this work, we proposed an effective method based on cubic and pantic B-spline scaling functions to solve partial differential equations of fractional order. Our method is based on dual functions of B-spline scaling functions. We derived the operational matrix of fractional integration of cubic and pantic B-spline scaling functions and used them to transform the mentioned equations to a system of algebraic equations. Some examples are presented to show the applicability and effectivity of the technique.
http://wala.vru.ac.ir/article_23240_9ff0784850a6139d46d8de6393191c71.pdf
2016-12-01T11:23:20
2018-02-18T11:23:20
55
68
10.22072/wala.2016.23240
B-spline
Wavelet
fractional equation
partial differential equation
Operational matrix of integration
Ataollah
Askari Hemmat
askarihemmat@gmail.com
true
1
Depatrment of Mathematics Graduate University of Advanced Technology
Depatrment of Mathematics Graduate University of Advanced Technology
Depatrment of Mathematics Graduate University of Advanced Technology
LEAD_AUTHOR
Tahereh
Ismaeelpour
tismaeelpour@math.uk.ac.ir
true
2
Shahid Bahonar University of Kerman
Shahid Bahonar University of Kerman
Shahid Bahonar University of Kerman
AUTHOR
Habibollah
Saeedi
saeedi@uk.ac.ir
true
3
Shahid Bahonar University of Kerman, Kerman, Iran
Shahid Bahonar University of Kerman, Kerman, Iran
Shahid Bahonar University of Kerman, Kerman, Iran
AUTHOR
[1] M. Dehghan, J. Manafian, A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. Methods Partial Differ. Equations, 26(2)(2010), 448-479.
1
[2] J. Goswami, A. Chan, Fundamentals of wavelets theory, algorithms and applications, John Wiley and Sons, Inc., 1999.
2
[3] T. Ismaeelpour, A. Askari Hemmat and H. Saeedi, B-spline Operational Matrix of Fractional Integration, Optik- International Journal for Light and Electron Optics, 130(2017), 291-305.
3
[4] K. AL-Khaled, Numerical solution of time-fractional partial differential equations Using Sumudu decomposition method, Rom. J. Phys., 60(1-2)(2015),
4
[5] A. A. Kilbas, H. M. Srivastava and J.J. Trujillo, Theory and application of fractional differential equations.
5
North-Holland Mathematics studies, Vol.204, Elsevier, 2006.
6
[6] M. Lakestani, M. Dehghan, S. Irandoust-pakchin. The construction of operational matrix of fractional derivatives using B-spline functions, Commun. Nonlinear Sci. Numer. Simul., 17(3)(2012), 1149 - 1162.
7
[7] Y. Li, Solving a nonlinear fractional differential equations using chebyshev wavelets, Commun. Nonlinear. Sci. Numer. Simul., 15(9)(2010), 2284 - 2292.
8
[8] K.S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley,
9
New York, 1993.
10
[9] D. Sh. Mohammed, Numerical solution of fractional integro-differential equations by least squares method and shifted Chebyshev polynomia, Math. Probl. Eng., 2014.
11
[10] I. Podlubny, Fractional differential equations. Academic Press, New York, 1999.
12
[11] H. Saeedi, Applicaion of Haar wavelets in solving nonlinear fractional Fredholm integro-differential equations, J. Mahani Math. Res. Cent., 2 (1) (2013), 15 - 28.
13
[12] H. Saeedi, M. Mohseni Moghadam, N. Mollahasani, G. N. Chuev, A Cas wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order, Commun. Nonlinear. Sci. Numer. Simul., 16 (2011), 1154-1163.
14
[13] M. Unser, Approximation power of biorthogonal wavelet expansions, IEEE Trans. Signal Process., 44 (39)
15
(1996), 519-527.
16
[14] J.L. Wu, A wavelet operational method for solving fractional partial differential equations numerically, Appl. Math. Comput., 214 (1) (2009), 31 - 40.
17
ORIGINAL_ARTICLE
Triangularization over finite-dimensional division rings using the reduced trace
In this paper we study triangularization of collections of matrices whose entries come from a finite-dimensional division ring. First, we give a generalization of Guralnick's theorem to the case of finite-dimensional division rings and then we show that in this case the reduced trace function is a suitable alternative for trace function by presenting two triangularization results. The first one is a generalization of a result due to Kaplansky and in the second one a triangularizability condition which is dependent on a single element is presented.
http://wala.vru.ac.ir/article_23241_1e8aafe10279931fe51e54733c2efcbc.pdf
2016-12-01T11:23:20
2018-02-18T11:23:20
69
74
10.22072/wala.2016.23241
Triangularizable
Semigroup
Irreducible
Division ring
Reduced trace
Hossein
Momenaee Kermani
momenaee@uk.ac.ir
true
1
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman
LEAD_AUTHOR
[1] D. Z. Docovic, B. H. Smith, Quaternionic matrices: Unitary similarity, simultaneous triangularization and some trace identities, Linear Algebra Appl., 428(4)(2008), 890-910.
1
[2] P. K. Draxl, Skew fields, Cambridge University Press, 1983.
2
[3] R. M. Guralnick, Triangularization of sets of matrices, Linear Multilinear Algebra, 9(2)(1980), 133-140.
3
[4] I. Kaplansky, The Engel-Kolchin theorem revisited. Contributions to Algebra, (Bass, Cassidy and Kovacik, Eds.), Academic Press, New York, 1977.
4
[5] T. Y. Lam, A first course in noncommutative rings. 2nd ed. Springer Verlag, New York, 2001.
5
[6] H. Momenaee Kermani, Triangularizability of algebras over division rings, Bull. Iran. Math. Soc., 34(1)(2008), 73 - 81.
6
[7] H. Momenaee Kermani, Triangularizability over ﬁelds and division rings. Ph. D. thesis, Shahid Bahonar University of Kerman, Kerman, Iran, 2005.
7
[8] M. Radjabalipour, P. Rosenthal and B. R. Yahaghi, Burnside's theorem for matrix rings over division rings,
8
Linear Algebra Appl., 382(2004), 29-44.
9
[9] H. Radjavi, A trace condition equivalent to simultaneous triangularizability, Canada. J. Math., 38(1986), 376 - 386.
10
[10] H. Radjavi and P. Rosenthal, Simultaneous triangularization, Springer-Verlag, New York, 2000.
11
[11] W. S. Sizer, Similarity of sets of matrices over a skew field, Ph.D. thesis, Bedford college, University of London, 1975.
12
[12] B. R. Yahaghi, On F-algebras of algebraic matrices over a subfield F of the center of a division ring, Linear Algebra Appl., 418(2-3)(2006), 599-613.
13
[13] B. R. Yahaghi, Reducibility results on operator semigroups. Ph.D. thesis, Dalhousie University, Halifax, N.S., Canada, 2002.
14