2021
2
1
0
115
1

On the rank of the holomorphic solutions of PDE associated to directed graphs
https://ajmc.aut.ac.ir/article_4121.html
10.22060/ajmc.2020.18413.1031
1
Let G be a directed graph with m vertices and n edges, I(B) the binomial ideal associated to the incidence matrix B of the graph G, and IL the lattice ideal associated to the columns of the matrix B. Also let Bi be a submatrix of B after removing the ith column. In this paper it is determined that which minimal prime ideals of I(Bi) are Andean or toral. Then we study the rank of the space of solutions of binomial Dmodule associated to I(Bi) as Agraded ideal, where A is a matrix that, ABi = 0. Afterwards, we define a miniaml cellular cycle and prove that for computing this rank it is enough to consider these components of G. We introduce some bounds for the number of the vertices of the convex hull generated by the columns of the matrix A. Finally an algorthim is introduced by which we can compute the volume of the convex hull corresponded to a cycles with k diagonals, so by Theorem 2.1 the rank of (D / H_A(I(B_i); beta)) can be computed.
0

1
9


Hamid
Damadi
Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran
Iran
hamid.damadi@aut.ac.ir


Farhad
Rahmati
Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran
Iran
frahmati@aut.ac.ir
Directed graph
Binomial Dmodule
Lattice basis ideal
[[1] V. Batyrev, Dual polyhedra and mirror symmetry for CalabiYau hypersurfaces in toric varieties, J. Algebraic Geom., 3(3) (1994) 493535.##[2] B. Braun, An Ehrhart series formula for reflexive polytopes, Electron. J. Combin. 13 (1) (2006), Note 15, 5 pp. (electronic).##[3] A. Dickenstein, L. Matusevich, E. Miller, Combinatorics of binomial primary decomposition, Math. Z. 264 (4) (2010) 745763.##[4] A. Dickenstein, L. Matusevich, E. Miller, Binomial Dmodules, Duke Math. J. 151 (3) (2010) 385429.##[5] D. Eisenbud, B. Sturmfels, Binomial ideals, Duke Math. J. 84 (1) (1996), 145.##[6] E. Ehrhart, Polynˆomes Arithm´etiques et M´ethode des Poly`edres en Combinatoire, Birkhuser, Boston, Basel, Stuttgart, 1977.##[7] K. Fischer, J. Shapiro, Mixed matrices and binomial ideals, J. Pure Appl. Algebra 113 (1) (1996) 3954. [8] I. M. Gelfand, M. I. Graev, A. V. Zelevinski, Holonomic systems of equations and series of hypergeometric type, Dokl. Akad. Nauk SSSR, 295 (1) (1987) 1419.##[9] G. Hegedus, A. M. Kasprzyk, The boundary volume of a lattice polytope, Bull. Aust. Math. Soc. 85 (2012), 84104.##[10] T. Hibi, Dual polytopes of rational convex polytopes, Combinatorica, 12(2) (1992) 237240.##[11] S. Hos¸ten, J. Shapiro, Primary decomposition of lattice basis ideals,J. Symbolic Comput. 29 (2000), no. 45, 625639.##]
1

Recognition by degree primepower graph and order of some characteristically simple groups
https://ajmc.aut.ac.ir/article_4122.html
10.22060/ajmc.2020.18418.1033
1
In this paper, by the order of a group and triviality of Op(G) for some prime p, we give a new characterization for some characteristically simple groups. In fact, we prove that if p ∈ {5, 17, 23, 37, 47, 73} and n 6 p, where n is a natural number, then G ∼= PSL(2, p) n if and only if G = PSL(2, p) n and Op(G) = 1. Recently in [Qin, Yan, Shum and Chen, Comm. Algebra, 2019], the degree primepower graph of a finite group have been introduced and it is proved that the Mathieu groups are uniquely determined by their degree primepower graphs and orders. As a consequence of our results, we show that PSL(2, p) n, where p ∈ {5, 17, 23, 37, 47, 73} and n 6 p are uniquely determined by their degree primepower graphs and orders.
0

11
15


Afsane
Bahri
Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran
Iran
afsanebahri@aut.ac.ir


Behrooz
Khosravi
Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic)
Iran
bkhosravi@aut.ac.ir


Morteza
Baniasad Azad
Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic)
Iran
baniasad84@gmail.com
Degree prime power graph
order
characteristically simple group
Characterization
[[1] M. Baniasad Azad, B. Khosravi, Recognition of some characteristically simple groups by their complex group algebra, Math. Rep (to appear).##[2] I. M. Isaacs, Character theory of finite groups, Academic Press, New York, 1976.##[3] P. Hall, A contribution to the theory of groups of prime power order, Proc. London Math. Soc., 36 (1933) 2995.##[4] M. Khademi, B. Khosravi, Recognition of characteristically simple group A5 × A5 by character degree graph and order, Czechoslovak Math. J., 68 (143) (2018) 11491157.##[5] B. Khosravi, B. Khosravi, B. Khosravi, Z. Momen, Recognition by character degree graph and order of the simple groups of order less than 6000, Miskolc Math. Notes, 15 (2) (2014) 537544.##[6] B. Khosravi, B. Khosravi, B. Khosravi, Z. Momen, Recognition by character degree graph and order of some simple groups, Math. Rep, 18 (68) (2016) 130137.##[7] M. L. Lewis, An overview of graphs associated with character degrees and conjugacy class sizes in finite groups, Rocky Mt. J. Math., 38 (1) (2008) 175211.##[8] O. Manz, R. Staszewski, W. Willems, On the number of components of a graph related to character degrees, Proc. Amer. Math. Soc., 103 (1) (1988) 3137.##[9] M. B. Nathanson, Elementary Methods in Number Theory, Springer edition, NewYork, 2000.##[10] C. Qin, Y. Yan, K. P. Shum, G. Y. Chen, Mathieu groups and its degree primepower graphs, Commun. Algebra, 47 (10) (2019) 41734180.##[11] H. Xu, G. Y. Chen, Y. Yan, A new characterization of simple K3groups by their orders and large degrees of their irreducible characters, Commun. Algebra, 42 (2014) 53745380.##[12] A. V. Zavarnitsine, Finite simple groups with narrow prime spectrum, Sib. Elektron. Mat. Izv 6 (2009) 112.##]
1

A class of operator related weighted composition operators between Zygmund space
https://ajmc.aut.ac.ir/article_4132.html
10.22060/ajmc.2020.18833.1041
1
Let D be the open unit disk in the complex plane C and H(D) be the set of all analytic functions on D. Let u, v ∈ H(D) and ϕ be an analytic selfmap of D. A class of operator related weighted composition operators is defined as follow
Tu,v,ϕf(z) = u(z)f(ϕ(z)) + v(z)f 0 (ϕ(z)), f ∈ H(D), z ∈ D.
In this work, we obtain some new characterizations for boundedness and essential norm of operator Tu,v,ϕ between Zygmund space.
0

17
25


Ebrahim
Abbasi
Department of Mathematics&lrm;, &lrm;Mahabad Branch&lrm;, &lrm;Islamic Azad University&lrm;, &lrm;Mahabad&lrm;, &lrm;Iran
Iran
ebrahimabbasi81@gmail.com
Boundedness
Compactness
essential norm
Zygmund space
[[1] C. C. Cowen, B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, FL, 1995.##[2] K. Esmaeili, M. Lindstr¨om, Weighted composition operators between Zygmund type spaces and their essential norms, Integr. Equ. Oper. Theory, 75 (2013) 473490.##[3] Z. Jiang, Producttype operators from Zygmund spaces to BlochOrlicz spaces, Complex Var. Elliptic Equ., 62 (11) (2017) 16451664.##[4] Y. Liu, Y. Yu, On a Stevi´cSharma operator from Hardy spaces to the logarithmic Bloch spaces, J. Inequal. Appl., 2015 (2015) 2015:22, DOI 10.1186/s1366001505471.##[5] Y. Liu, Y. Yu, Products of composition, multiplication and radial derivative operators from logarithmic Bloch spaces to weightedtype spaces on the unit ball, J. Math. Anal. Appl., 423 (1) (2015) 7693.##[6] Y. Liu, Y. Yu, On an extension of Stevi´cSharma operator from the general space to weightedtype spaces on the unit ball, Complex Anal. Oper. Theory, 11 (2) (2017) 261288.##[7] Y. Liu, Y. Yu, On Stevi´cSharma type operators from the Besov spaces into the weightedtype space H∞ µ , Math. Inequal. Appl., 22 (3) (2019) 10371053.##[8] B. D. Maccluer, R. Zhao, Essential norms of weighted composition operators between Blochtype spaces, Rocky Mountain J. Math., 33 (2003) 14371458.##[9] S. Stevi´c, A. Sharma, A. Bhat, Products of multiplication composition and differentiation operators on weighted Bergman spaces, Appl. Math. Comput., 217 (2011) 81158125.##[10] S. Stevi´c, A. Sharma, A. Bhat, Essential norm of products of multiplications composition and differentiation operators on weighted Bergman spaces, Appl. Math. Comput., 218 (2011) 23862397.##[11] S. Stevi´c, A. Sharma, On a producttype operator between Hardy and αBloch spaces of the upper halfplane, J. Inequal. Appl., 2018:273, 2018.##[12] M. Tjani, Compact composition operators on some M¨obius invariant Banach space, PhD dissertation, Michigan State university, 1996.##[13] H. Wulan, D. Zheng, K. Zhu, Compact composition operators on BMOA and the Bloch space, Proc. Amer. Math. Soc., 137 (2009) 38613868.##[14] S. Y, Q. Hu, Weighted Composition Operators on the Zygmund Space, Abstract and Applied Analysis, 2012 (2012), Article ID 462482, 18 pages doi:10.1155/2012/462482.##[15] Y. Yu, Y. Liu, On Stevi´c type operator from H∞ space to the logarithmic Bloch spaces, Complex Anal. Oper. Theory, 9 (8) (2015) 17591780.##[16] F. Zhang, Y. Liu, On the compactness of the Stevi´cSharma operator on the logarithmic Bloch spaces, Math. Inequal. Appl., 19(2) (2016) 625642.##[17] F. Zhang, Y. Liu, On a Stevi´cSharma operator from Hardy spaces to Zygmundtype spaces on the unit disk, Complex Anal. Oper. Theory, 12 (1) (2018) 81100.##[18] K. Zhu, Bloch type spaces of analytic functions, Rocky Mountain J. Math., 23 (3) (1993) 11431177.##]
1

On GDWRanders metrics on tangent Lie groups
https://ajmc.aut.ac.ir/article_4160.html
10.22060/ajmc.2020.18572.1038
1
Let G be a Lie group equipped with a leftinvariant Randers metric F. Suppose that F v and F c denote the vertical and complete lift of F on T G, respectively. We give the necessary and sufficient conditions under which F v and F c are generalized DouglasWeyl metrics. Then, we characterize all 2step nilpotent Lie groups G such that their tangent Lie groups (T G, Fc ) are generalized DouglasWeyl Randers metrics.
0

27
36


Mona
Atashafrouz
Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran
Iran
m.atashafrooz@aut.ac.ir


Behzad
Najafi
Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran
Iran
behzad.najafi@aut.ac.ir


Akbar
Tayebi
Department of Mathematics, Faculty of Science, University of Qom, Qom, Iran
Iran
akbar.tayebi@gmail.com
Leftinvariant metric
Douglas metric
Generalized DouglasWeyl Metrics
Randers Metric
[[1] F. Asgari and H. R. Salimi Moghaddam, Left invariant Randers metrics of Berwald type on tangent Lie groups, Int. J. Geom. Meth. Modern. Phys. 15(01) (2018), 1850015.##[2] F. Asgari and H. R. Salimi Moghaddam, Riemannian geometry of two families of tangent Lie groups, Bull. Iran. Math. Soc. 44 (2018), 193203.##[3] M. Atshafrouz and B. Najafi, On ChengShen conjecture in Finsler geometry, Int. J. Math (2020). https://doi.org/10.1142/S0129167X20500305.##[4] S. B´acs´o and M. Matsumoto, On Finsler spaces of Douglas type, A generalization of notion of Berwald space, Publ. Math. Debrecen. 51 (1997), 385406.##[5] S. B´acs´o and I. Papp, A note on generalized Douglas space, Periodica. Math. Hung. 48 (2004), 181184.##[6] J. Hilgert and K. H. Neeb, Structure and geometry of Lie groups, Springer Monographs in Mathematics, 2012.##[7] S. Homolya and O. Kowalski, Simply connected twostep homogeneous nilmanifolds of dimension 5, Note. Math. 26 (2006), 6977.##[8] R. S. Ingarden, On the geometrically absolute optical representation in the electron microscope, Trav. Soc. Sci. Lett. Wrochlaw. Ser. B. 3 (1957), 60 pp.##[9] O. Kowalski and M. Sekizawa, Natural transformations of Riemannian metrics on manifolds to metrics on tangent bundles, Bull. Tokyo. Gakugei. Univ. Sect. IV. 40 (1988), 129.##[10] B. Najafi, Z. Shen and A. Tayebi, On a projective class of Finsler metrics, Publ. Math. Debrecen. 70 (2007), 211219.##[11] B. Najafi and A. Tayebi, A new quantity in Finsler geometry, C. R. Acad. Sci. Paris. Ser. I. 349 (2011), 8183.##[12] M. Nasehi, On 5dimensional 2step homogeneous Randers nilmanifolds of Douglas type, Bull. Iranian Math. Soc. 43 (2017), 695706.##[13] DN. Pham, On the tangent Lie group of a symplectic Lie group, Riccerche di Mathematica. 68(2) (2019), 669704.##[14] G. Randers, On an asymmetric metric in the fourspace of general relativity, Phys. Rve. 59 (1941), 195199.##[15] H.R. Salimi Moghaddam, On the Randers metrics on twostep homogeneous nilmanifolds of dimension five, Int. J. Geom. Meth. Mod. Phys. 8 (2011), 501510.##[16] S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. 10 (1958), 338354. [17] A. Tayebi, H. Sadeghi and E. Peyghan, On generalized DouglasWeyl spaces, Bull. Malays. Math. Sci. Soc. (2) 36(3) (2013), 587594.##[18] A. Tayebi and H. Sadeghi, On generalized DouglasWeyl (α, β)metrics, Acta Mathematica Sinica, English Series. 31(10) (2015), 16111620.##[19] K. Yano and S. Kobayashi, Prolongations of tensor fields and connections to tangent bundles I, J. Math. Soc. Japan. 18(2) (1966), 194210.##[20] K. Yano and S. Ishihara, Tangent and cotangent bundles, volume 16 of Pure and Applied Mathematics. Marcel Dekker, Inc., 1973.##]
1

Conservation law and Lie symmetry analysis of Foam Drainage equation
https://ajmc.aut.ac.ir/article_4161.html
10.22060/ajmc.2020.18460.1036
1
In this paper, using the Lie group analysis method, we study the group invariant of the Foam Drainage equation. It shows that this equation can be reduced to ODE. Also we apply the Liegroup classical, and the nonclassical method due to Bluman and Cole to deduce symmetries of the Foam Drainage equation. and we prove that the nonclassical method applied to the equation leads to new reductions, which cannot be obtained by Lie classical symmetries. Also this paper shows how to construct directly the local conservation laws for this equation.
0

37
44


Mehdi
Nadjafikhah
School of Mathematics, Iran University of Science and Technology
Iran
m_nadjafikhah@iust.ac.ir


Omid
Chekini
School of Mathematics, Iran University of Science and Technology
Iran
omid_chgini@mathdep.iust.ac.ir
Lie group analysis
Conservation law
Optimal system
Foam Drainage equation
Nonclassical symmetry
[[1] M. F. Ashby, A. G. Evans, N. A. Fleck, L. J. Gibson, J. W. Hutchinson, H. N. G. Wasley, Metal Foams: A Design Guide, Society of Automotive Engineers, Boston, Mass, USA, 2000.##[2] J. Banhart, Metallschaume, MIT, Bermen, Germany, 1997.##[3] A. Bhakta, E. Ruckenstein, Decay of standing foams: drainage, coalescence and collapse, Advances in Colloid and Interface Science, vol. 70, no. 13 (1997)1123.##[4] M. Durand, D. Langevin, Physicochemical approach to the theory of foam drainage, European Physical Journal E, vol. 7, no. 1 (2002) 3544.##[5] L. J. Gibson, M. F. Ashby, Cellular Solids: Structure & Properties, Cambridge University Press, Cambridge, UK, 1997.##[6] L. J. Gibson, M. F. Ashby, Cellular Solids: Structure and Properties, Cambridge University Press, Cambridge, UK, 1999.##[7] S. Hilgenfeldt, S. A. Koehler, H. A. Stone, Dynamics of coarsening foams: accelerated and selflimiting drainage, Physical Review Letters, vol. 86, no. 20 (2001) 47044707.##[8] S. A. Koehler, H. A. Stone, M. P. Brenner, J. Eggers, Dynamics of foam drainage, Physical Review E, vol. 58, no. 2 (1998) 20972106.##[9] R. A. Leonard, R. Lemlich, A study of interstitial liquid flow in foam, AIChE Journal, vol. 11 (1965) 1829.##[10] S. Lie, F. Engel, Theorie der transformationsgruppen; Teubner: Leipzig, Germany, 1888.##[11] S. Lie, Vorlesungen ¨uber differentialgleichungen mit bekannten infinitesimalen transformationen; Teubner: Leipzig, Germany, 1891.##[12] A. H. Nayfeh, Introduction to Perturbation Techniques, John Wiley & Sons, New York, NY, USA, 1981.##[13] E. Noether, Invariante Variationsprobleme. Nachr. v. d. Ges. d. Wiss. zu G¨ottingen, Math. Phys. Kl. 1918, 235–257; English translation, Transp. Th. Stat. Phys. 1 (1971) 186207.##[14] R. K. Prud’homme, S. A. Khan, Foams: Theory, Measurements and Applications, Marcel Dekker, New York, NY, USA, 1996.##[15] H. A. Stone, S. A. Koehler, S. Hilgenfeldt, M. Durand, Perspectives on foam drainage and the influence of interfacial rheology, Journal of Physics Condensed Matter, vol. 15, no. 1 (2003) S283S290.##[16] S. D. Stoyanov, V. N. Paunov, E. S. Basheva, I. B. Ivanov, A. Mehreteab, G. Broze, Motion of the front between thick and thin film: hydrodynamic theory and experiment with vertical foam films, Langmuir, vol. 13, no. 6 (1997) 14001407.##[17] G. Verbist, D. Weaire, Soluble model for foam drainage, Europhysics Letters, vol. 26 (1994) 631634.##[18] G. Verbist, D. Weaire, A soluble model for foam drainage, Europhysics Letters, vol. 26, no. 8 (1994) 631.##[19] G. Verbist, D. Weaire, A. M. Kraynik, The foam drainage equation, Journal of Physics Condensed Matter, vol. 8, no. 21 (1996) 37153731.##[20] D. Weaire, S. Hutzler, N. Pittet, D. Pardal, Steadystate drainage of an aqueous foam, Physical Review Letters, vol. 71, no. 16 (1993) 26702673.##[21] D. Weaire, S. Findlay, G. Verbist, Measurement of foam drainage using AC conductivity, Journal of Physics: Condensed Matter, vol. 7, no. 16 (1995) L217L222.##[22] D. Weaire, S. Hutzler, G. Verbist, E. A. J. Peters, A review of foam drainage, Advances in Chemical Physics, vol. 102 (1997) 315374.##[23] D. L. Weaire, S. Hutzler, The Physics of Foams, Oxford University Press, Oxford, UK, 2000.##[24] D. Weaire, S. Hutzler, S. Cox, N. Kern, M. D. Alonso, D. D. Drenckhan, The fluid dynamics of foams, Journal of Physics Condensed Matter, vol. 15, no. 1 (2003) S65S73.##[25] J. I. B. Wilson, Essay review, scholarly froth and engineering skeletons, Contemporary Physics, vol. 44 (2003) 153155.##]
1

An extension of the Cardioid distributions on circle
https://ajmc.aut.ac.ir/article_4209.html
10.22060/ajmc.2020.18285.1029
1
A new family of distributions on the circle is introduced which is a generalization of the Cardioid distributions. The elementary properties such as mean, variance, and the characteristic function are computed. The distribution is shown to be either unimodal or bimodal. The modes are computed. The symmetry of the distribution is characterized. The parameters are shown to be canonic (i.e. uniquely determined by the distribution). This implies that the estimation problem is welldefined. We also show that this new family is a subset of distributions whose Fourier series has degree at most 2 and study the implications of this property. Finally, we study the maximum likelihood estimation for this family.
0

45
52


Erfan
Salavati
Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran
Iran
erfan.salavati@aut.ac.ir
Circular Distributions
Cardioid Distribution
Von Mises Distribution
[1] E. A. Yfantis, L. E. Borgman, An extension of the von Mises distribution, Communications in StatisticsTheory and Methods 11 (1982) 16951706.##[2] K. V. Mardia, P. E. Jupp, Directional Statistics, Wiley series in probability and statistics. Wiley, Chichester 2000.##[3] K. V. Mardia, Probability and mathematical statistics: statistics of directional data, Academic Press, London 1972.##[4] S. Kato, M. C. Jones, A family of distributions on the circle with links to, and applications arising from, M¨obius transformation, Journal of the American Statistical Association 105 (2010) 249262##]
1

Estimation of the parameter of L´evy distribution using ranked set sampling
https://ajmc.aut.ac.ir/article_4216.html
10.22060/ajmc.2020.18499.1037
1
Ranked set sampling is a statistical technique for data collection that generally leads to more efficient estimators than competitors based on simple random sampling. In this paper, we consider estimation of scale parameter of L´evy distribution using a ranked set sample. We derive the best linear unbiased estimator and its variance, based on a ranked set sample. Also we compare numerically, variance of this estimator with mean square error of the maximum likelihood, a median based estimator and an estimator based on Laplace transform. It turns out that the best linear unbiased estimator based on ranked set sampling is more efficient than other mentioned estimators.
0

53
60


Sahar
Dorniani
Department of Statistics, Roudehen
Branch, Islamic Azad University, Tehran, Iran
Iran
sdorniani@yahoo.com


Adel
Mohammadpour
Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran
Iran
adelmohammadpour1@gmail.com


Nader
Nematollahi
Department of Statistics, Allameh Tabataba'i
University, Tehran, Iran.
Iran
na_nemat@yahoo.com
Best linear unbiased estimator
Levy distribution
Ranked set sampling
Scale parameter
[[1] P. Besbeas, B. J. T. Morgan, Efficient and robust estimation for the onesided stable distribution of index 1 2 . Statist. Probab. Lett., 66 (2004) 251257.##[2] Z. Chen, Z. Bai, B. K. Sinha, Ranked set sampling: Theory and applications, Springer, New York, 2004.##[3] T. R. Dell, J. L. Clutter, Ranked set sampling theory with order statistics background, Biometrics 28 (1972) 545555.##[4] H. Fei, B. K. Sinha, Z. Wu, Estimation of parameters in twoparameter Weibull and extremevalue distributions using ranked set sample, J. Statist. Res. 28 (1994), 149161.##[5] A. Hatefi, M. Jafari Jozani, D. Ziou, Estimation and classification for finite Mixture models under ranked set sampling, Statistica Sinica 24 (2014) 675698.##[6] K. Lam, B. K. Sinha, Z. Wu, Estimation of parameters in a twoparameter exponential distribution using ranked set sample, Ann. Inst. Statist. Math., 46 (1994) 723736.##[7] K. Lam, B. K. Sinha, Z. Wu, Estimation of location and scale parameters of a Logistic distribution using ranked set sample, In: Nagaraja, Sen, Morrison, (Eds), Papers in Honor of Herbert A. David. (1995) 187197.##[8] G. A. McIntyre, A method for unbiased selective sampling using rankedset sampling, Austral. J. Agric. Res. 3 (1952) 385390. [9] M. Mohammadi, M. Mohammadpour, Existence of order statistics moments of αstable distributions, Preprint, 2010.##[10] N. Ni Chuiv,B. K. Sinha, Z. Wu, Estimation of the location parameter of a Cauchy distribution using a ranked set sample, Technical Report, University of Maryland Baltimore County, 1994.##[11] N. Ni Chuiv, B. K. Sinha, On some aspects of ranked set sampling in parametric estimation, In: Balakrishnan, Rao (Eds), Handbook of Statistics, 17 (1998) 337377.##[12] A. Piryatinska, Inference for L´evy models and their application in medicine and statistical physics, Ph.D. Thesis, Department of statistics, Case Western Reserve University, Cleveland, OH, 2005.##[13] G. Samorodnitsky, Extrema of skewed stable process, Stochastic Process. Appl. 30 (1988) 1739.##[14] E. Scalas, K. Kim, The art of fitting financial time series with L´evy stable distributions, 2006, http://mpra.ub.unimuenchen.de/336/.##[15] B. K. Sinha, S. Purkayastha, On some aspects of ranked set sampling for estimation of Normal and Exponential parameters, Statist. Decisions 14 (1996) 223240.##[16] K. Takahasi, K. Wakimato, On unbiased estimates of the population mean based on the sample stratified by means of ordering, Ann. Inst. Statist. Math. 20 (1968) 131.##[17] V. V. Uchaikin, V. M. Zolotarev, Chance and Stability: stable distributions and their applications, Utrecht: VSP Press, 1999. [18] G. Zheng, R. Modarres, A robust estimate of the correlation coefficient for bivariate normal distribution using ranked set, J. Statist. Plann. Inference 136 (2006) 298309.##]
1

Some conformal vector fields and conformal Ricci solitons on N(k)contact metric manifolds
https://ajmc.aut.ac.ir/article_4246.html
10.22060/ajmc.2021.19220.1043
1
The target of this paper is to study N(k)contact metric manifolds with some types of conformal vector fields like φholomorphic planar conformal vector fields and Ricci biconformal vector fields. We also characterize N(k)contact metric manifolds allowing conformal Ricci almost soliton. Obtained results are supported by examples.
0

61
71


Uday
De
Department of Pure Mathematics, Faculty of Science, University of Calcutta, Kolkata, India
India
uc_de@yahoo.com


Arpan
Sardar
Department of Mathematics, Faculty of Science, University of Kalyani, West Bengal, India
India
arpansardar51@gmail.com


Avijit
Sarkar
Department of Mathematics, Faculty of Science, University of Kalyani, West Bengal
India
avjaj@yahoo.co.in
N(k)contact metric manifolds
φholomorphic planar conformal vector fields
conformal vector fields
conformal Ricci solitons
[[1] G. P. Alfonso, M. M. S. Jose, Biconformal vector fields and their applications, arXiv:mathph/0311014v2.##[2] C. Baikoussis, D. E. Blair, T. Koufogiorgos, A decomposition of the curvature tensor of a contact manifold satisfying R(X, Y )ξ = k(η(Y )X − η(X)Y ), Math. Technical Reports, University of Ioannina, Greece, 1992.##[3] N. Basu, A. Bhattacharyya, Conformal Ricci soliton in Kenmotsu manifold, Global J. Adv. Res. on Class. Mod. Geom., 4 (2015) 1521.##[4] D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Math., 203, Birkhauser, 2010.##[5] D. E. Blair, T. Koufogiorgos, B. J. Papantoniou, Contact metric manifolds satisfying a nullity condition, Israel J. Math., 91 (1995) 189214.##[6] H. D. Cao, B. Chow, Recent developments on the Ricci flow, Bull. Amer. Math. Soc., 36 (1999) 5974.##[7] U. C. De, Certain results on N(k)contact metric manifolds, Tamkang J. Math., 49 (2018) 205220.##[8] U. C. De, A. Yildiz, S. Ghosh, On a class of N(k)contact metric manifolds, Math. Reports, 16 (2004) 207217.##[9] S. Deshmukh, Geometry of conformal vector fields, Arab J. Math. Sci, 23 (2017) 4473.##[10] A. E. Fischer, An introduction to conformal Ricci flow, Class. Quantum Grav., 21 (2004) 171218.##[11] A. Ghosh, Holomorphically planar conformal vector fields on contact metric manifolds, Acta Math. Hungarica, 129 (2010) 357367.##[12] R. S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986, 237 262.) Contemp. Math. 71, American Math. Soc., 1988.##[13] J. B. Jun, U. C. De, On N(k)contact metric manifolds satisfying some curvature conditions, Kyungponk Math. J., 34 (2011) 457468.##[14] H. G. Nagaraja, K. Venu, fKenmotsu metric as conformal Ricci soliton, Ann. Univ. Vest. Timisoara. Ser. Math. Inform., LV (2017) 119127.##[15] M. Okumura, Some remarks on space with certain contact structures, Tohoku Math. J., 14 (1962) 135145.##[16] M. Okumura, On infinitesimal conformal and projective transformations of normal contact spaces, Tohoku Math. J., 14 (1962),398412.##[17] C. Ozgur, S. Sular, On N(k)contact metric manifolds satisfying certain conditions, SUT J. Math., 44 (2008) 8999.##[18] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159v1.##[19] S. Pigola, M. Rigoli, M., Rimoldi, A. G. Setti, Ricci almost solitons, Ann. Sc. Norm. Super Pisa Cl. Sci., 10 (2011) 757799. [20] R. Sharma, D. E. Blair, Conformal motion of contact manifolds with characteristic vector field in the knullity distribution, Illinois J. Math., 40 (1996) 553563.##[21] R. Sharma, Holomorphically planar conformal vector fields on almost Hermitian manifolds, Contemp. Math., 337 (2003) 145154.##[22] R. Sharma, L. Vrancken, Conformal classification of (k, µ)Contact manifolds, Kodai Math. J., 33 (2010) 267282.##[23] R. Sharma, Certain results on Kcontact and (k, µ)contact metric manifolds, J. Geom., 89 (2008) 138147.##[24] R. Sharma, Conformal and projective characterizations of an odd dimensional unit sphere , Kodai Math. J., 42 (2019) 160169.##[25] S. Tanno, Some transformations on manifolds with almost contact and contact metric structures, Tohoku Math. J., 15 (1963) 140147.##[26] K. Yano, Integral formulas in Riemannian geometry, Marcel Dekker, New York, 1970##]
1

On the reversible geodesics of a Finsler space endowed with a special deformed (α, β)metric
https://ajmc.aut.ac.ir/article_4270.html
10.22060/ajmc.2021.19522.1048
1
The scope of this paper is twofold. On the one hand, we will investigate the reversible geodesics of a Finsler space endowed with the deformed newly introduced (α,β)metric begin{equation}F_{ε}(α,β)=frac{β^{2}+α^{2}(a+1)}{α}+εβend{equation}where ε is a real parameter with ε<2√a+1 and ain(¼,∞); and on the other hand, we will investigate the Ttensor for this metric.
0

73
80


LaurianIoan
Pişcoran
Department of Mathematics and Computer Science, Victoriei 76, 430122 Baia Mare, Romania
Romania
plaurian@yahoo.com


Ali
Akram
Departamento de MatematicaICE, Universidade Federal de AmazonasUFAM, 69080900 Manaus
Brazil
akramali@ufam.edu.br


Cătălin
Barbu
"Vasile Alecsandri" National College, str. Vasile Alecsandri nr. 37, Bacau, Romania
Romania
kafka_mate@yahoo.com
Finsler (α,β)metric
deformation of an (α,β)metric
Ttensor
[[1] M. Crampin, Randers spaces with reversible geodesics, Publ. Math. Debrecen, 67 (2005), 401409.##[2] M. Crasmareanu, New tools in Finsler geometry: stretch and Ricci solitons, Math. Rep. (Bucur.) 16 (66)(1) (2014) 8393.##[3] M. Crasmareanu, A gradienttype deformation of conics and a class of Finslerian flows, An. Stiint. Univ. Ovidius Constanta, Ser. Mat., 25 (2017), no. 2, 8599.##[4] M. Crasmareanu, A complex approach to the gradienttype deformations of conics, Bull. Transilv. Univ. Brasov, Ser. III, Math. Inform. Phys., 10(59) (2) (2017) 5962.##[5] S. Elgendi, L. Kozma, (α, β)metrics satisfying the Tcondition or the σT condition, The Journal of Geometric Analysis, http://doi.org/10.1007/s12220020005553, 2020.##[6] I. Masca, S. V. Sab˘au, H. Shimada, Reversible geodesics for (α, β)metric, Int. J. Math., 21 (2010), 10711094.##[7] M. Matsumoto, On three dimensional Finsler spaces satisfying the T and Bp conditions, Tensor N.S., 29 (1975), 1320.##[8] L. I. Pi¸scoran, V. N. Mishra, Projectivelly flatness of a new class of (α, β)metrics, Georgian Mathematical Journal 26 (1) (2019) 133139.##[9] L. I. Pi¸scoran, V. N. Mishra, Scurvature for a new class of (α, β)metrics, RACSAM, doi:10.1007/s13398016 03583, 2017. [10] P. LaurianIoan, B. Najafi, C. Barbu, T. Tabatabaeifar, The deformation of an (α, β)metrics, International Electronic Journal of Geometry, 2021, Accepted.##[11] S. V. Sab˘au, H. Shimada, Finsler manifolds with reversible geodesics, Rev. Roumaine Math. Pures Appl., 57, (2012), 91103. ##[12] S. S. Chen, Z. Shen, RiemannFinsler geometry, Singapore: World Scientific, 2005.##[13] A. Tayebi, H. Sadeghi, H. Peyghan, On Finsler metrics with vanishing Scurvature, Turkish J. Math., 38 (2014) 154165.##]
1

Ensuring software maintainability at software architecture level using architectural patterns
https://ajmc.aut.ac.ir/article_4272.html
10.22060/ajmc.2021.19232.1044
1
Software architecture is known to be an effective tool with regards to improving software quality attributes. Many quality attributes such as maintainability are architecture dependent, and as such, using an appropriate architecture is essential in providing a sound foundation for the development of highly maintainable software systems. An effective way to produce a wellbuilt architecture is to utilize standard architectural patterns. Although the use of a particular architectural pattern cannot have a preserving effect on software maintainability, the mere conformance of a system to any architecture cannot guarantee the system’s high maintainability. The use of an inappropriate architecture can seriously undermine software maintainability at lower levels. In this article, the effect of standard architectural patterns on software maintainability quality attributes is investigated. We develop a quality model for maintainability quality attributes, which is later used to compare various standard architectural patterns. We finish by investigating two realworld experiences regarding the application of a particular pattern to two different existing architectures, exploring the effect of the change in architecture on maintainability quality attributes.
0

81
102


Mohammad
Tanhaei
Department of Engineering, Ilam University
Iran
m.tanhaei@ilam.ac.ir


Zahed
Rahmati
Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran
Iran
zrahmati@aut.ac.ir
Patterns
Software Architecture
Maintainability
[[1] F. Bachmann, R. L. Nord, I. Ozkaya, Architectural tactics to support rapid and agile stability, CrossTalk (SEI), May/June2012: 2025.##[2] L. Bass, P. Clements, R. Kazman, Software Architecture in Practice, 2nd Edition, AddisonWesley Longman Publishing Co., Inc., Boston, MA, USA, 2003.##[3] P. Bengtsson, Towards maintainability metrics on software architecture: An adaptation of objectoriented metrics, in: First Nordic Workshop on Software Architecture (NOSA’98), Ronneby, 1998.##[4] P. Bengtsson, N. Lassing, J. Bosch, H. van Vliet, Architecturelevel modifiability analysis (alma), Journal of Systems and Software 69 (1) (2004) 129147.##[5] B. W. Boehm, J. R. Brown, M. Lipow, Quantitative evaluation of software quality, in: Proceedings of the 2nd international conference on Software engineering, IEEE Computer Society Press, (1976) 592605.##[6] S. Bohner, Impact analysis in the software change process: a year 2000 perspective, in: Software Maintenance 1996, Proceedings., International Conference on Software Maintenance, (1996) 4251.##[7] F. Buschmann, R. Meunier, H. Rohnert, P. Sommerlad, M. Stal, Patternoriented Software Architecture: A System of Patterns, John Wiley & Sons, Inc., New York, NY, USA, 1996.##[8] P. Clements, D. Garlan, L. Bass, J. Stafford, R. Nord, J. Ivers, R. Little, Documenting software architectures: views and beyond, Pearson Education, 2002.##[9] J. Deacon, Modelviewcontroller (mvc) architecture, Online][Citado em: 10 de mar¸co de 2006.] http://www. jdl. co. uk/briefings/MVC. pdf (2009).##[10] R. G. Dromey, A model for software product quality, IEEE Transactions Software Engineering 21 (2) (1995) 146162.##[11] R. Dromey, Cornering the chimera [software quality], Software, IEEE 13 (1) (1996) 3343.##[12] D. G. Firesmith, Common concepts underlying safety security and survivability engineering, Tech. rep., Carnegie Mellon Software Engineering Institute  Technical Note CMU/SEI2003TN033 (2003).##[13] M. Fowler, Refactoring: improving the design of existing code, AddisonWesley Professional, 1999.##[14] J. E. Gaffney Jr, Metrics in software quality assurance, in: Proceedings of the ACM’81 conference, ACM (1981) 126130.##[15] J. Garc´ıaMart´ın, M. SutilMartin, Virtual machines and abstract compilerstowards a compiler pattern language, in: In Proceeding of EuroPlop 2000, Irsee, Citeseer, 2000.##[16] D. Garlan, M. Shaw, An introduction to software architecture, Advances in software engineering and knowledge engineering 1 (1993) 140.##[17] R. B. Grady, Practical Software Metrics for Project Management and Process Improvement, PrenticeHall, Inc., Upper Saddle River, NJ, USA, 1992.##[18] P. A. Grubb, A. A. Takang, Software maintenance: concepts and practice, World Scientific, 2003.##[19] B. HayesRoth, A blackboard architecture for control, Artificial intelligence 26 (3) (1985) 251321.##[20] P. Heged˝us, D. B´an, R. Ferenc, T. Gyim´othy, Myth or reality? analyzing the effect of design patterns on software maintainability, in: Computer Applications for Software Engineering, Disaster Recovery, and Business Continuity, Springer, (2012) 138145.##[21] I. Heitlager, T. Kuipers, J. Visser, A practical model for measuring maintainability, in: Quality of Information and Communications Technology, 2007. QUATIC 2007. 6th International Conference on the, IEEE, (2007) 3039.##[22] F. Hoffman, Architectural software patterns and maintainability: A case study, Ph.D. thesis, University of Sk¨ovde (2001). [23] ISO/IEC, Iso standard 9126: Software engineering  product quality, parts 1, 2 and 3 (2001 (part 1), 2003 (parts 2 and 3)). [24] I. Jacobson, G. Booch, J. Rumbaugh, The Unified Software Development Process, AddisonWesley Longman Publishing Co., Inc., Boston, MA, USA, 1999.##[25] H. Kabaili, R. K. Keller, F. Lustman, Cohesion as changeability indicator in objectoriented systems, in: Proceedings of the Fifth European Conference on Software Maintenance and Reengineering, CSMR ’01, IEEE Computer Society, Washington, DC, USA, (2001) 3946.##[26] P. Kruchten, Architectural blueprints  the ”4+1” view model of software architecture, Tutorial Proceedings of TriAda 95 (1995) 540555.##[27] M. Leotta, F. Ricca, G. Reggio, E. Astesiano, Comparing the maintainability of two alternative architectures of a postal system: Soa vs. nonsoa, in: Software Maintenance and Reengineering (CSMR), 2011 15th European Conference on, IEEE, (2011) 317320.##[28] N. L´evy, F. Losavio, A. Matteo, Comparing architectural styles: Broker specializes mediator, in: Proceedings of the Third International Workshop on Software Architecture, ISAW ’98, ACM, New York, NY, USA, (1998) 9396.##[29] B. P. Lientz, E. B. Swanson, Software Maintenance Management, AddisonWesley Longman Publishing Co., Inc., Boston, MA, USA, 1980.##[30] L. Liu, X.D. Zhu, X.L. Hao, Maintainability metrics of software architecture based on symbol connector, in: Quality, Reliability, Risk, Maintenance, and Safety Engineering (QR2MSE), 2013 International Conference on, IEEE Computer Society Press, (2013) 15641567.##[31] J. A. McCall, P. K. Richards, G. F. Walters, Factors in software quality. volumeiii. preliminary handbook on software quality for an acquisition manager, Tech. rep., DTIC Document, 1977.##[32] S. McConnell, Code complete, O’Reilly Media, Inc., 2004.##[33] R. T. Monroe, A. Kompanek, R. Melton, D. Garlan, Architectural styles, design patterns, and objects, Software, IEEE 14 (1) (1997) 4352.##[34] S. Muthanna, K. Kontogiannis, K. Ponnambalam, B. Stacey, A maintainability model for industrial software systems using design level metrics, in: Reverse Engineering, 2000. Proceedings. Seventh Working Conference on, IEEE Computer Society Press, (2000) 248256.##[35] Y. Ping, K. Kontogiannis, T. C. Lau, Transforming legacy web applications to the mvc architecture, in: Software Technology and Engineering Practice, 2003. Eleventh Annual International Workshop on, IEEE, (2003) 133142.##[36] L. Prechelt, B. Unger, W. F. Tichy, P. Br¨ossler, L. G. Votta, A controlled experiment in maintenance comparing design patterns to simpler solutions, IEEE Trans. Softw. Eng. 27 (12) (2001) 11341144.##[37] R. Pressman, Software Engineering: A Practitioner’s Approach, 6th Edition, McGrawHill, Inc., New York, NY, USA, 2005.##[38] J. Sanders, E. Curran, Software Quality: A Framework for Success in Software Development and Support, ACM Press/AddisonWesley Publishing Co., New York, NY, USA, 1994.##[39] M. Shaw, P. Clements, A field guide to boxology: Preliminary classification of architectural styles for software systems, in: Computer Software and Applications Conference, 1997. COMPSAC’97. Proceedings., The TwentyFirst Annual International, IEEE, (1997) 613.##[40] M. Shaw, D. Garlan, Software Architecture: Perspectives on an Emerging Discipline, PrenticeHall, Inc., Upper Saddle River, NJ, USA, 1996.##[41] M. Shaw, Some patterns for software architectures, Pattern languages of program design 2 (1996) 255269.##[42] O. Silva, A. Garcia, C. Lucena, The reflective blackboard pattern: Architecting large multiagent systems, in: Software engineering for largescale multiagent systems, Springer, (2003) 7393.##[43] J. Vlissides, R. Helm, R. Johnson, E. Gamma, Design patterns: Elements of reusable objectoriented software, Reading: AddisonWesley 49 (1995) 120.##[44] M. V¨olter, M. Kircher, U. Zdun, Remoting Patterns: Foundations of Enterprise, Internet and Realtime Distributed Object Middleware, Wiley. com, 2005.##]
1

On some aspects of measure and probability logics and a new logical proof for a theorem of Stone
https://ajmc.aut.ac.ir/article_4273.html
10.22060/ajmc.2021.19318.1045
1
One of the functions of mathematical logic is studying mathematical objects and notions by logical means. There are several important representation theorems in analysis. Amongst them, there is a wellknown classical one which concerns probability algebras. There are quite a few proofs of this result in the literature. This paper pursue two main goals. One is to consider some aspects of measure and probability logics and expose a novel proof for the mentioned representation theorem using ideas from logic and by application of an important result from model theory. The second and even more important goal is to present more connections between two fields of analysis and logic and reveal more the strength of logical methods and tools in analysis. The paper is mostly written for general mathematicians, in particular the people who are active in analysis or logic as the main audience. It is selfcontained and includes all prerequisites from logic and analysis.
0

103
108


Alireza
Mofidi
Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran
Iran
mofidi@aut.ac.ir
Integration and probability logics
Representations and measure existence theorems
Compactness in logic
[[1] S. Bagheri, M. Pourmahdian, The logic of integration, Arch. Math. Logic 48 (2009) 465492.##[2] D. Hoover, Probability logic, Annals of Mathematical Logic 14 (1978) 287313.##[3] H. Keisler, Probability quantifiers, in: Model Theoretic Logics, edited by J. Barwise and S. Feferman, SpringerVerlag, (1985), pp. 509556.##[4] A. Mofidi, On some dynamical aspects of NIP theories., Arch. Math. Logic, 57 (12) (2018) 3771.##[5] A. Mofidi, S. M. Bagheri, Quantified universes and ultraproduct, Math. Logic Quart. 58 (2012) 6374.##[6] M. Raskovic, R. Dordevic, Probability quantifiers and operators, Vesta Company, Belgrade, 1996.##]
1

The complexity of cost constrained vehicle scheduling problem
https://ajmc.aut.ac.ir/article_4274.html
10.22060/ajmc.2021.19454.1046
1
This paper considered the cost constrained vehicle scheduling problem under the constraint that the total number of vehicles is known in advance. Each depot has a different time processing cost. The goal of this problem is to find a feasible minimum cost schedule for vehicles. A mathematical formulation of the problem is developed and the complexity of the problem when there are more than two depots is investigated. It is proved that in this case, the problem is NPcomplete. Also, it is showed that there is not any constant ratio approximation algorithm for the problem, i.e., it is in the complexity class APX.
0

109
115


Malihe
Niksirat
Department of Computer Science, Faculty of Computer and Industrial Engineering, Birjand University of Technology, Birjand, Iran
Iran
niksirat@birjandut.ac.ir


Seyed Naser
Hashemi
Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran
Iran
nhashemi@aut.ac.ir
Vehicle scheduling problem
Fixed job scheduling
NPcomplete
Approximation algorithm
APX
[[1] R. K. Ahuja, T. L. Magnanti, J. B. Orlin, Network flows: theory, algorithms and applications, PrenticeHall, Englewood Cliffs, NJ, 1993.##[2] D. P. Bertsekas, Linear Network Optimization: Algorithms and Codes, MIT Press, Cambridge, MA, 1991.##[3] L. Bodin, B. Golden, A. Assad, M. Ball, Routing and scheduling of vehicles and crews: the state of the art, Computers and Operations Research, 10 (1983) 63211.##[4] G. Carpaneto, M. DellAmico, M. Fischetti, P. Toth, A branch and bound algorithm for the multiple vehicle scheduling problem, Computers and Operations Research, 19 (1989) 531548.##[5] S. Carosi, A. Frangioni, L. Galli, L. Girardi, G. Vallese, A matheuristic for integrated timetabling and vehicle scheduling. Transportation Research Part B: Methodological, 127 (2019) 99124.##[6] M. DellAmico, M. Fischetti, P. Toth, Heuristic algorithms for the multiple depot vehicle scheduling problem, Management Science, 39 (1993) 115125.##[7] J. Desrosiers, Y. Dumas, M. M. Solomon, F. Soumis, Time constrained routing and schedulin, Handbooks in operations research and management science: Network routing, M. O. Ball, T. L. Magnanti, C. L. Monma and G. L. Nemhauser (Eds.), (1995) 35139.##[8] D. T. Eliiyi, A. Ornek, S. S. Karaku tu, A vehicle scheduling problem with fixed trips and time limitations, International Journal of Production Economics, 117 (2009) 150161.##[9] M. A. Forbes, J. N. Holt, A. M. Watts, An exact algorithm for multiple depot bus scheduling, European Journal of Operational Research, 72 (1994) 115124.##[10] R. Freling, A. P. M. Wagelmans, J. M. P. Paixo, Models and Algorithms for SingleDepot Vehicle Scheduling, Transportation Science, 35 (2001) 165180.##[11] M. R. Garey, D. S. Johnson, Computer and Intractability, A Guide to the Theory of NPCompleteness, New York, NY, USA, 2000.##[12] P. C. Guedes, D. Borenstein, Column generation based heuristic framework for the multipledepot vehicle type scheduling problem, Computers & Industrial Engineering, 90 (2015), 361370.##[13] V. Guihaire, J. K. Hao, Transit network design and scheduling: A global review, Transportation Research Part A, 42 (2008), 12511273.##[14] A. Hadjar, O. Marcotte, F. Soumis, A branchandcut algorithm for the multiple depot vehicle scheduling problem, Operations Research, 54 (2006) 130149.##[15] N. Huimin, Z. Xuesong, T. Xiaopeng, Coordinating assignment and routing decisions in transit vehicle schedules: A variablesplitting Lagrangian decomposition approach for solution symmetry breaking, Transportation Research Part B: Methodological, 107 (2018) 70101.##[16] O. J. IbarraRojas, R. Giesen, Y. A. RiosSolis, An integrated approach for timetabling and vehicle scheduling problems to analyze the tradeoff between level of service and operating costs of transit networks, Transportation Research Part B, 70 (2014) 3546.##[17] B. Laurent, J.K. Hao, Iterated local search for the multiple depot vehicle scheduling problem, Computers & Industrial Engineering, 57 (2009), 277286.##[18] Q. Huang, E Lloyd, Cost Constrained Fixed Job Scheduling, Theoretical computer sciences, proceedings. Lecture notes in computer science, 2003.##[19] N. Kliewer, T. Mellouli, L. Suhl, A timespace network based exact optimization model for multidepot bus scheduling, European Journal of Operational Research, 175 (2006) 16161627.##[20] S. Kulkarni, M. Krishnamoorthy, A. Ranade, A. T. Ernst, R. Patil, A new formulation and a column generationbased heuristic for the multiple depot vehicle scheduling problem, Transportation Research Part B: Methodological, 118 (2018) 457487.##[21] M. Mesquita, J. Paixo, Multiple depot vehicle scheduling problem: a new heuristic based on quasiassignment algorithms, Lecture notes in economics and mathematical systems, Computeraided transit scheduling, M. Desrochers and J.M. Rousseau (Eds.), (1992) 167180.##[22] M. Mnich, R. van Bevern, Parameterized complexity of machine scheduling: 15 open problems. Computers & Operations Research, 100 (2018) 254261.##[23] H. Nagamochi, T. Ohnishi, Approximating a vehicle scheduling problem with time windows and handling times, Theoretical Computer Science, 393, (2008) 33146.##[24] M. Niksirat, S. M. Hashemi, M. Ghatee, Branchandprice algorithm for fuzzy integer programming problems with block angular structure, Fuzzy Sets and Systems, 296 (2016) 7096.##[25] T. Liu, A. A. Ceder, Batteryelectric transit vehicle scheduling with optimal number of stationary chargers. Transportation Research Part C: Emerging Technologies, 114 (2020) 118139.##[26] E. Levner, V. Kats, D. A. L. de Pablo, T. E. Cheng, Complexity of cyclic scheduling problems: A stateoftheart survey. Computers & Industrial Engineering, 59 (2010) 352361.##[27] A. R. Odoni, J.M. Rousseau, N. H. M. Wilson, Multiple depot vehicle scheduling problem: a new heuristic based on quasiassignment algorithms, Handbooks in operations research and management, S. M. Pollock, M. H. Rothkopf, and A. Barnett (Eds.)”, (1994) 107150.##[28] E. F. Olariu, C. Frasinaru, MultipleDepot Vehicle Scheduling Problem Heuristics, 2020, arXiv preprint arXiv:2004.14951.##[29] C.Ribeiro, F. Soumis, A column generation approach to the multiple depot vehicle scheduling problem, Operations Research, 42 (1994) 4152.##[30] J. Ren, D. Du, D. Xu, The complexity of two supply chain scheduling problems. Information Processing Letters, 113 (2013) 609612.##[31] G. Simonin, R. Giroudeau, J. C. K¨onig, Complexity and approximation for scheduling problem for a torpedo. Computers & Industrial Engineering, 61 (2011) 352356.##[32] C. Wang, H. Shi, X. Zuo, A multiobjective genetic algorithm based approach for dynamical bus vehicles scheduling under traffic congestion. Swarm and Evolutionary Computation, 54 (2020) 100667.##[33] E. Yao, T. Liu, T.Lu, Y. Yang, Optimization of electric vehicle scheduling with multiple vehicle types in public transport. Sustainable Cities and Society, 52 (2020) 101862.##[34] Y. Zheng, Y. Shang, Z. Shao, L. Jian, A novel realtime scheduling strategy with nearlinear complexity for integrating largescale electric vehicles into smart grid. Applied Energy, 217 (2018) 113.##]