ORIGINAL_ARTICLE
A unified theoretical harmonic analysis approach to the cyclic wavelet transform (CWT) for periodic signals of prime dimensions
The article introduces cyclic dilation groups and finite affine groups for prime integers, and as an application of this theory it presents a unified group theoretical approach for the cyclic wavelet transform (CWT) of prime dimensional periodic signals.
https://scma.maragheh.ac.ir/article_11272_6b84bf3354fa5ed03323941d3c72b4b1.pdf
2014-12-01
1
17
Cyclic wavelet transform (CWT)
Finite affine group
Prime integer
Digital signal processing
Arash
Ghaani Farashahi
ghaanifarashahi@outlook.com
1
Numerical Harmonic Analysis Group (NuHAG), Faculty of Mathematics, University of Vienna, Vienna, Austria.
LEAD_AUTHOR
Mozhgan
Mohammad-Pour
mozhgan.mohammadpour@stu.um.ac.ir
2
Faculty of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran.
AUTHOR
[1] S. T. Ali, J. P. Antoine and J. P. Gazeau, Coherent States, Wavelets and Their Generalizations, Springer-Verlag, New York, 2000.
1
[2] J. P. Antoine, R. Murenzi and P. Vendergheynst, S. T. Ali, Two Dimensional Wavelets and Their Relatives, Cambridge University Press, Cambridge. 2003.
2
[3] J. J. Benedetto and S. Li, The theory of multiresolution analysis frames and applications to filter bands, Appl. Comput. Harmon. Anal., 5 (1998) 389-427.
3
[4] J. J. Benedetto and O. Treiber. Wavelet frames: Multiresolution analysis and extension principles. Debnath, Lokenath, Wavelet transforms and time-frequency signal analysis. Boston, MA: Birkhauser. Applied and Numerical Harmonic Analysis, 3-36 (2001)., 2001.
4
[5] G. Caire, R. L. Grossman and H. Vincent Poor, Wavelet transforms associated with nite cyclic Groups, IEEE Transaction On Information Theory, Vol. 39, No. 4, July 1993.
5
[6] L. Cohen, Time-Frequency Analysis, Prentice-Hall, New York, 1995.
6
[7] I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, 1992.
7
[8] H. G. Feichtinger and G. Zimmermann, A Banach space of test functions for Gabor analysis, Gabor analysis and algorithms, 123-170, Applied and Numerical Harmonic Analysis, Birkhauser, Boston, MA 1998.
8
[9] H. G. Feichtinger and W. Kozek, Quantization of TF lattice-invariant operators on elementary LCA groups, Gabor analysis and algorithms, 233-266, Applied and Numerical Harmonic Analysis, Birkhauser, Boston, MA 1998.
9
[10] H. G. Feichtinger and T. Strohmer, Advances in Gabor Analysis, Series: Applied and Numerical Harmonic Analysis, Birkhuser 2002.
10
[11] H. Feichtinger and F. Luef. Gabor analysis and time-frequency methods, Encyclopedia of Applied and Computational Mathematics, 2012.
11
[12] H. G. Feichtinger, T. Strohmer, and O. Christensen. A group-theoretical approach to Gabor analysis, Opt. Eng., 34 (1697) 1704, 1995.
12
[13] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC press, 1995.
13
[14] D. Gabor, Theory of communication, JIEEE, 93(26), Part III, 1964, 429-457.
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[15] A. Grossmann, J. Morlet and T. Paul, Transforms associated to square integrable group representations I. general results, J. Math. Phys., 26 (10) (1985) 2473-2479
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[16] K. Grochenig, Foundation of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, MA 2001.
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[17] K. Grochenig, Aspects of Gabor analysis on locally compact abelian groups, Gabor analysis and Algorithms, 211-231, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, MA 1998.
17
[18] G. H. Hardy and E.M.Wright, An Introduction to the Theory of Numbers, Oxford Univ. Press (1979).
18
[19] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. 1, 1963.
19
[20] R. Reiter and J. D. Stegeman, Classical Harmonic Analysis, 2nd Ed, Oxford University Press, New York, 2000.
20
[21] S. Mallat, A Wavelet Tour of Signal Processing, 2nd Edition, Academic Press, 1999.
21
[22] S. Mallat, Multiresolution approximations and wavelet orthonormal bases of L2(R), Trans. Amer. Math. Soc., 315 (1989), 69-87.
22
[23] G. L. Mullen and D. Panario, Handbook of Finite Fields, Series: Discrete Mathematics and Its Applications, Chapman and Hall/CRC, 2013.
23
[24] G. Pfander, Gabor Frames in Finite Dimensions. In G. E. Pfander, P. G. Casazza, and G. Kutyniok, editors, Finite Frames, Applied and Numerical Harmonic Analysis, 193-239. Birkhauser Boston, 2013.
24
[25] P. Flandrin, Time-Frequency/Time-Scale Analysis, Wavelet Analysis and its Applications, Vol. 10 Academic Press, San Diego, 1999.
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[26] K. Flornes, A. Grossmann, M. Holschneider and B. Torresani, Wavelets on discrete fields, Appl. Comput. Harmon. Anal., 1 (1994) 137-146.
26
[27] H. Riesel, Prime Numbers and Computer Methods for Factorization (second edition), Boston: Birkhuser, (1994).
27
[28] T. Strohmer., Numerical algorithms for discrete Gabor expansions, Gabor Analysis and Algorithms: Theory and Applications, pages 267{294. Birkhauser Boston, Boston, 1998.
28
[29] T. Strohmer. A Unied Approach to Numerical Algorithms for Discrete Gabor Expansions, In Proc. SampTA - Sampling Theory and Applications, Aveiro/Portugal, pages 297{302, 1997.
29
[30] S. Sarkar, H. Vincent Poor, Cyclic Wavelet Transforms for Arbitrary Finite Data Lengths, Signal Processing, 80 (2000) 2541-2552.
30
[31] G. Strang and T. Nguyen, Wavelets and Filter Banks,Wellesley-Cambridge Press, Wellesley, MA, 1996.
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[32] M. Vetterli and J. Kovacevic, Wavelets and Subband Coding, Prentice Hall, 1995.
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[33] M.V. Wickerhauser, Lectures on Wavelet Packet Algorithms, Technical Report, Washington University, Department of Mathematics, 1992.
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[34] M.W. Wong, Discrete Fourier Analysis, Pseudo-dierential Operators Theory and applications Vol. 5, Birkhauser 2010.
34
[35] Z. Zhang. Periodic wavelet frames. Adv. Comput. Math., 22 (2) (2005) 165-180.
35
ORIGINAL_ARTICLE
On X ̃-frames and conjugate systems in Banach spaces
The generalization of p-frame in Banach spaces is considered in this paper. The concepts of an $\tilde{X}$-frame and a system conjugate to $\tilde{X}$-frame were introduced. Analogues of the results on the existence of conjugate system were obtained. The stability of $\tilde{X}$-frame having a conjugate system is studied.
https://scma.maragheh.ac.ir/article_11273_0356133177a286fc7ca41146dd50c9bf.pdf
2014-12-01
19
29
p-frames
X ̃-frames
Conjugate systems to X ̃
Migdad
Ismailov
miqdadismailov1@rambler.ru
1
Department of Non-harmonic analysis, Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan.
LEAD_AUTHOR
Afet
Jabrailova
afet.cebrayilova@mail.ru
2
Department of Functional analysis, Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan.
AUTHOR
[1] R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952) 341-–366.
1
[2] R. Young, An introduction to nonharmonic Fourier series, New York, 1980.
2
[3] O. Christensen, An Introduction to Frames and Riesz Bases, Appl. Numer. Harmon. Anal., Birkhauser, Boston, MA, 2002.
3
[4] P. G. Casazza and O. Christensen, Perturbation operators and applications to frame theory, J. Fourier Anal. Appl., 3(5) (1997) 543--557.
4
[5] H. G. Feichtinger and K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decomposition, J. Funct. Anal., 86 (1989) 307--340.
5
[6] R. Balan, Stability theorems for Fourier frames and wavelet Riesz bases, J. Fourier Anal. And Appl., 3 (1997) 499--504.
6
[7] P. G. Casazza, The art of frame theory, Taiwanese J. Math., 4(2) (2000) 129--201.
7
[8] K. Gröchenig, Describing functions: atomic decomposition versus frames, Monatsh. Math., 112(1) (1991) 1--41.
8
[9] O. Christensen and C. Heil, Perturbations of Frames and Atomic Decompositions, Math. Nachr., 185 (1997) 33--47.
9
[10] P. G. Casazza, D. Han and D. R. Larson, Frames for Banach space, Contemp. Math., 247 (1999) 149--182.
10
[11] P. K. Jain, S. K. Kaushik and L. K. Vashisht, On perturbations of Banach frames, Int. J. Wavelet multiresolut.Inf. Process, 4(3) (2006) 559--565.
11
[12] A. Aldroubi, Q. Sun and W. Tang, p-frames and shift invariant subspaces of Lp, J. Fourier Anal. Appl., 7 (2001) 1--21.
12
[13] O. Christensen and D. Stoeva, p-frames in separable Banach spaces, Adv. Comp. Math., 18 (2003) 117--126.
13
[14] W. Sun, G-frames and G-Riesz bases, J. Math. Anal. Appl., 322 (2006) 473--452.
14
[15] M.R. Abdollahpour, M.H. Faroughi and A. Rahimi, PG-frames in Banach spaces, Methods of functional Analysis and Topology, 13(3) (2007) 201--210.
15
[16] B.T. Bilalov and F.A.Guliyeva, On The Frame Properties of Degenerate System of Sines, Journal of Function Spaces and Applications, 2012 (2012) Article ID 184186, 12 pages, doi:10.1155/2012/184186.
16
[17] S.R. Sadigova and Z.V. Mamedova, Frames from Cosines with the Degenerate Coefficients, American Journal of Applied Mathematics and Statistics, 1(3) (2013) 36--40.
17
[18] Gar J. Garnett, Bounded Analytic Functions, Moscow, "Mir", 1984, 469 p.
18
ORIGINAL_ARTICLE
On a sequence related to the coprime integers
The asymptotic behaviour of the sequence with general term $P_n=(\varphi(1)+\varphi(2)+\cdots+\varphi(n))/(1+2+\cdots+n)$, is studied which appears in the studying of coprime integers, and an explicit bound for the difference $P_n-6/\pi^2$ is found.
https://scma.maragheh.ac.ir/article_11274_868c353aa19d64f70c97af5928061b30.pdf
2014-12-01
31
37
Euler function
Coprime integers
Asymptotic behaviour of the sequence
Mehdi
Hassani
mehdi.hassani@znu.ac.ir
1
Department of Mathematics, University of Zanjan University Blvd., 45371-38791, Zanjan.
LEAD_AUTHOR
[1] P. Borwein, S. Choi, B. Rooney and A. Weirathmueller (Eds.), The Riemann Hypothesis, Springer, 2008.
1
[2] P. Erdös and H. N. Shapiro, On the changes of sign of a certain error function, Canad. J. Math., 3 (1951) 375--384.
2
[3] E. Landau, Sur les valeurs moyennes de certaines fonctions arithmétiques, Bull. Acad. Royale Belgique, (1911) 443--472.
3
[4] G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge Studies in Advanced Mathematics 46, Cambridge University Press, Cambridge, 1995.
4
[5] A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Mathematische Forschungsberichte, XV. VEB Deutscher Verlag der Wissenschaften, Berlin, 1963.
5
ORIGINAL_ARTICLE
A Class of compact operators on homogeneous spaces
Let $\varpi$ be a representation of the homogeneous space $G/H$, where $G$ be a locally compact group and $H$ be a compact subgroup of $G$. For an admissible wavelet $\zeta$ for $\varpi$ and $\psi \in L^p(G/H),\ \ 1\leq p <\infty$, we determine a class of bounded compact operators which are related to continuous wavelet transforms on homogeneous spaces and they are called localization operators.
https://scma.maragheh.ac.ir/article_11275_2af10d7d0a659c8c7a3a2841a2740d7d.pdf
2014-12-01
39
45
Homogenous space
Square integrable representation
Admissible wavelet
Localization operator
Fatemah
Esmaeelzadeh
esmaeelzadeh@boujnourdiau.ac.ir
1
Department of Mathematics, Bojnourd Branch, Islamic Azad University, Bojnourd, Iran.
LEAD_AUTHOR
Rajab Ali
Kamyabi Gol
kamyabi@ferdowsi.um.ac.ir
2
Department of Mathematics, Center of Excellency in Analysis on Algebraic Structures(CEAAS), Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran.
AUTHOR
Reihaneh
Raisi Tousi
raisi@ferdowsi.um.ac.ir
3
Department of Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran.
AUTHOR
[1] S. T. Ali, J-P. Antoine and J-P. Gazeau, Coherent States, Wavelets and Their Generalizations, Springer-Verlag, New York, 2000.
1
[2] F. Esmaeelzadeh, R. A. Kamyabi Gol and R. Raisi Tousi , On the continuous wavelet transform on homogeneous spases, Int. J. Wavelets. Multiresolut, Vol. 10, No. 4 (2012).
2
[3] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, 1995.
3
[4] K. Zhu, Operator Theory in Function Spaces, Mathematical Surveys and Monographs, Vol. 138, 2007.
4
[5] M. W. Wong, Wavelet Transform and Localization Operators. Birkhauser Verlag, Basel-Boston-Berlin, 2002.
5
ORIGINAL_ARTICLE
Nonstandard finite difference schemes for differential equations
In this paper, the reorganization of the denominator of the discrete derivative and nonlocal approximation of nonlinear terms are used in the design of nonstandard finite difference schemes (NSFDs). Numerical examples confirming then efficiency of schemes, for some differential equations are provided. In order to illustrate the accuracy of the new NSFDs, the numerical results are compared with standard methods.
https://scma.maragheh.ac.ir/article_11277_67e5a322d5f3525fa2ef0d9962143089.pdf
2014-12-01
47
54
Finite difference
Nonstandard
Singular
Predator-prey model
Heat equation
Mohammad
Mehdizadeh Khalsaraei
muhammad.mehdizadeh@gmail.com
1
Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh, Iran.
LEAD_AUTHOR
Fayyaz
Khodadosti
fayyaz64dr@gmail.com
2
Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh, Iran.
AUTHOR
[1] R. E. Mickens, Applications of Nonstandard Finite Difference Schemes, World Scientific, Singapore, 2000.
1
[2] E. Hairer and G. Wanner, Solving ordinary differential equation II: Stiff and Differential-Algebraic problems, Springer, Berlin, 1996.
2
[3] L. C. Evans, Partial Differential Equations, AMS, Providence, 1998.
3
[4] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, New York, 2001.
4
[5] C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Canada., 45 (1965) 1-60.
5
[6] R. Anguelov and J. M. S. Lubuma, Contributions to the mathematics of the nonstandard finite difference method and applications, Numerical Methods for Partial Differential Equations, 17 (2001) 518-543.
6
[7] R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994.
7
and Applied Mathematics, 158 (2003) 19-30.
8
[8] H. Ramos, A non-standard explicit integration scheme for initial-value problems, Appl. Math. Comput., 189 (2007) 710-718.
9
[9] Ramos, Linearization techniques for singularly-perturbed initial-value problems of ordinary differential equations, Appl. Math. Comput., 163 (2005) 1143-1163.
10
ORIGINAL_ARTICLE
Simulation study on copulas
There are several theorical results about order statistics and copulas in the literature that have been mentioned also by Nelsen \cite{p20}. The present study after reviewing some of these results, relies on simulation technique to investigate the mentioned results about order statistics and copulas. The study concentrates on two well known Archimedean Gumbel and Frank families in the case that marginal functions $F(k)$ and $G(k)$ have different distributions.
https://scma.maragheh.ac.ir/article_11278_1b52ef412e1b8b424118ff6f5aaa80c8.pdf
2014-12-01
55
63
Copulas
Order statistics
Generating random variables
Goodness of fit test
Sinem Tuğba Şahin
Tekin
sinemsahin@gazi.edu.tr
1
Gazi University, Faculty of Science, Statistics Department, Ankara, Turkey.
LEAD_AUTHOR
Vadoud
Najjari
fnajjary@yahoo.com
2
Gazi University, Faculty of Science, Statistics Department, Ankara, Turkey.
AUTHOR
H. Hasan
Örkcü
hhorkcu@gazi.edu.tr
3
Gazi University, Faculty of Science, Statistics Department, Ankara, Turkey.
AUTHOR
[1] M. Al-Harthy, S. Begg and R. Bratvold, Copulas: A new technique to model dependence in petroleum decision making, Journal of Petroleum Science and Engineering , 57 (2007) 195-208.
1
[2] S. Çelebioğlu, Archimedean copulas And An Application, Selcuk university journal of science, 22 (2003) 43-52.
2
[3] R. T. Clemen and T. Reilly, Correlations and Copulas for Decision and Risk Analysis, Management Science, 45 (1999) 208-224.
3
[4] N. I. Fisher, Copulas. In: Kotz, S., Read, C. B., Banks, D. L. (Eds.), Encyclopedia of Statistical Sciences, Wiley, New York. 1(1997) 159-163.
4
[5] A. Friend and E. Rogge, Correlation at First Sight, Economic Notes: Review of Banking, Finance and Monetary Economics, 2004.
5
[6] C. Genest and J. MacKay, Copules archimédienneset familles de loisbi dimensionnelles dont les margessontdonnés, Canad. J. Statistics, 14 (1986a) 145-159.
6
[7] C. Genest and J. MacKay, The joy of copula, Bivariate distributions with uniform marginals, Amer. Statistics, 40 (1986b) 280-285.
7
[8] L. Hua, and H. Joe, Tail order and intermediate tail dependence of multivariate copulas, Journal of Multivariate Analysis, 102 (2011) 1454-1471.
8
[9] V. Najjari, T. Bacigàl and H. Bal, An Archimedean copula family with hyperbolic cotangent generator, IJUFKS, Vol. 22 No. 5 (2014) 761-768.
9
[10] V. Najjari and M. G. Ünsal, An Application of Archimedean Copulas for Meteorological Data, GU J Sci, 25(2) (2012) 301-306.
10
[11] R. B. Nelsen, An Introduction to copulas, Springer, New York, Second edition, 2006.
11
[12] J. A. Rodríguez-Lallena, and M. Ubeda-Flores, A new class of bivariate copulas, Statistics and Probability Letters, 66 (2004) 315-325.
12
[13] A. Sklar, Fonctions de répartition à n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris, 8 (1959) 229-231.
13
ORIGINAL_ARTICLE
Characterization of fuzzy complete normed space and fuzzy b-complete set
The present paper introduces the notion of the complete fuzzy norm on a linear space. And, some relations between the fuzzy completeness and ordinary completeness on a linear space is considered, moreover a new form of fuzzy compact spaces, namely b-compact spaces and b-closed spaces are introduced. Some characterizations of their properties are obtained.
https://scma.maragheh.ac.ir/article_11280_7a399dbd2cdc54a494672cf68cd9ec1a.pdf
2014-12-01
65
75
Fuzzy Set
Normed Space
Complete Set
Hamid Reza
Moradi
hamid moradi 68@yahoo.com
1
Department of Mathematics, Qaemshhar Branch, Islamic Azad University, Qaemshahr, Iran.
LEAD_AUTHOR
[1] T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear space, J. Fuzzy Math., 11 (2003) No. 3, 687-705.
1
[2] A. K. Katsaras, Fuzzy topological vector space I, Fuzzy Sets and Systems., 6 (1981) 85-95.
2
[3] A. K. Katsaras, Fuzzy topological vector space II, Fuzzy sets and systems., 12 (1984) 143- 154.
3
[4] A. K. Katsaras, Fuzzy vector spaces and fuzzy topological vector spaces, J. Math. Anal. Appl., 58 (1977) 135-146.
4
[5] S. V. Krishna and K. K. M. Sarma, Fuzzy topological vector spaces – topological generation and normability, Fuzzy sets and systems., 41 (1991) 89-99.
5
[6] S. V. Krishna and K. K. M. Sarma, Fuzzy continuity of linear maps on vector spaces, Fuzzy sets and systems., 45 (1992) 341-354.
6
[7] S. V. Krishna and K. K. M. Sarma, Seperation of fuzzy normed linear spaces, Fuzzy sets and systems., 63 (1994) 207-217.
7
[8] R. Larsen, Functional analysis, Marcel Dekker, Inc. New york, 1973.
8
[9] G. S. Rhie, B. M. Choi and D. S. Kim, On the completeness of fuzzy normed linear spaces, Math. Japonica., 45 (1997) no. 1, 33-37.
9
[10] R. H. Warren, Neighborhoods bases and continuity in fuzzy topological spaces, Rocky Mountain J. Math., 8 (1978) 459-470.
10
ORIGINAL_ARTICLE
Commutative curvature operators over four-dimensional generalized symmetric spaces
Commutative properties of four-dimensional generalized symmetric pseudo-Riemannian manifolds were considered. Specially, in this paper, we studied Skew-Tsankov and Jacobi-Tsankov conditions in 4-dimensional pseudo-Riemannian generalized symmetric manifolds.
https://scma.maragheh.ac.ir/article_11282_d1c0396d2d6f9125c2483b14036ea9bd.pdf
2014-12-01
77
90
Commutative manifold
Pseudo-Riemannian manifold
Cyclic parallel
Locally conformally flat
Curvature operator
Ali
Haji-Badali
haji.badali@bonabu.ac.ir
1
Faculty of Basic Sciences, University of Bonab, , P.O.Box 5551761167, Bonab, Iran.
LEAD_AUTHOR
Masoud
Dehghan
massoud.dehghan@gmail.com
2
Department of Mathematics, Faculty of Science, University of ABCD, P.O.Box xxxx, City, Country.
AUTHOR
Fereshteh
Nourmohammadi
fereshteh−nour87@yahoo.com
3
Faculty of Basic Sciences, University of Bonab, , P.O.Box 5551761167, Bonab, Iran.
AUTHOR
[1] M. Brozos-Vazquez, E. Garcia, P. Gilkey and R. -Vazquez-Lorenzo, Examples of signature (2, 2)-manifolds with commuting curvature operators, J. Phys. A: Math. Theor. 40 (2007) 13149{13159.
1
[2] M. Brozos-Vazquez and P. Gilkey, The global geometry of Riemannian manifolds with commuting curvature operators, J. Fixed Point Theory Appl. 1 (2007) 87-96.
2
[3] M. Brozos-Vazquez and P. Gilkey, Manifolds with commuting Jacobi operators, J. Geom. 86 (2007) 21-30.
3
[4] G. Calvaruso, Harmonicity of vector elds on four-dimensional generalized symmetric spaces, Cent. Eur. J. Math. 10 (2012), 411-425.
4
[5] G. Calvaruso and B. De Leo, Curvature Properties of Four-Dimensional Generalized Symmetric Spaces, J. Geom. 90 (no. 1-2) (2008), 30-46.
5
[6] G. Calvaruso and A. H. Zaeim, Geometric Structures over Four-Dimensional Generalized Symmetric Spaces, Mediterr. J. Math. 10 (2013), 971-987.
6
[7] J. Cerny and O. Kowalski, Classication of generalized symmetric pseudo-Riemannian spaces of dimension n 4, Tensor (N.S.) 38 (1982), 256-267.
7
[8] E. Garcia-Rio, A. Haji-Badali, M. E. Vazquez-Abal and R. Vazqes-Lorenzo, Lorentzian 3-manifold with commuting curvature operators, Int. J. Geom. Meth. Modern Phys. 5 (4) (2008), 557-572.
8
[9] P. Gilkey, Geometric Properties of neutral Operators Dened by the Riemannian Curvature Tensor World Scientic Publishing Co., Inc., River Edge, NJ, 2001.
9
[10] C. Gonzalez and D. Chinea, Estructuras homogeneas sobre espacios simetricos generalizados, Proceedings of the XIIth Portuguese-Spanish Conference on Mathematics, Vol. II, 572{578, Univ. Minho, Braga, 1987.
10
[11] B. Komrakov Jnr., Einstein-Maxwell equation on four-dimensional homogeneous spaces, Lobachevskii J. Math. 8 (2001), 33-165.
11
[12] D. Kotschick and S. Terzic, On formality of generalized symmetric spaces, Math. Proc. Cambridge Philos. Soc. 134 (2003), 491-505.
12
[13] O. Kowalski, Generalized symmetric spaces, Lectures Notes in Math. 805, Springer-Verlag, Berlin-New York, 1980.
13
[14] B. O'Neill, Semi-Riemannian Geometry, Pure and Applied Mathematics 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983.
14
[15] S. Terzic, Real cohomology of generalized symmetric spaces, Fundam. Prikl. Mat. 7(2001), 131-157.
15
[16] S. Terzic, Pontryagin classes of generalized symmetric spaces (Russian), Mat. Zametki 69 (2001), 613{621. Translation in Math. Notes 69 (no. 3{4) (2001), 559-566.
16
[17] Y. Tsankov, A characterization of n-dimensional hypersurface in Euclidean space with commuting curvature operators, Banach Center Publ. 69 (2005) 205-209.
17