Edge-coloring Vertex-weightings of Graphs 2 2 Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $n$. A $k$-vertex weightings of a graph $G$ is a mapping $w: V(G) to {1, ldots, k}$. A $k$-vertex weighting induces an edge labeling $f_w: E(G) to N$ such that $f_w(uv)=w(u)+w(v)$. Such a labeling is called an {it edge-coloring k-vertex weightings} if $f_{w}(e)not= f_{w}(e')$ for any two adjacent edges $e$ and $e'$. Denote by $mu'(G)$ the minimum $k$ for $G$ to admit an edge-coloring $k$-vertex weightings. In this paper, we determine $mu'(G)$ for some classes of graphs. 1 13 2017/02/23 1395/12/5 2021/04/10 1400/1/21 W.-Ch. Shiu W.-Ch. Shiu Hong Kong Baptist University China wcshiu@hkbu.edu.hk G.-Ch. Lau G.-Ch. Lau Universiti Teknologi MARA Malaysia Malaysia geeclau@yahoo.com H.-K. Ng H.-K. Ng San Jose State University, USA USA ho-kuen.ng@sjsu.edu Edge coloring Vertex weightings. J.A. Bondy, U.S.R. Murty, Graph theory with applications, New York, MacMillan, 1976.##G.J. Chang, C. Lu, J. Wu and Q. Yu, Vertex coloring edge-weighting of graphs, Taiwanese J. Math., 15(4), (2011) 1807-1813.##M.R. Farahani, A new vertex-coloring edge-weighting of complete graphs, J. Appl. Math. & Informatics, Vol. 32, (2014), No. 1 - 2, 1 - 6.##J.A. Gallian, A dynamic survey of graph labeling, Electronic J. Comb., 20, (2017) #DS6.##M. Kalkowski, M. Kar'onski, and F. Pfender, Vertax-Coloring Edge-weighting With Integer Weights At Most 6, Rostock. Math. Kolloq., 64, (2009), 39-43.##M. Kalkowski, M.Kar'onski, and F. Pfender, Vertex-coloring edge-weightings: Towards the 1-2-3-conjecture, J. Combin. Theory, Ser. B, 100, (2010), 347-349.##M. Kar'onski, T. Luczak, A. Thomason, Edge weights and vertex colours, J. Combin. Theory Ser. B, 91, (2004) 151¨C157.##D. Leven and Z. Galil, NP completeness of finding the chromatic index of regular graphs, J. Algorithms, 4(1), (1983), 35 - 44.##T. Wang, and Q. Yu, On vertex-coloring $13$-edge-weighting, Front. Math. China, 3(4), (2008), 1-7.## ##

A Trust-region Method using Extended Nonmonotone Technique for Unconstrained Optimization 2 2 In this paper, we present a nonmonotone trust-region algorithm for unconstrained optimization. We first introduce a variant of the nonmonotone strategy proposed by Ahookhosh and Amini cite{AhA 01} and incorporate it into the trust-region framework to construct a more efficient approach. Our new nonmonotone strategy combines the current function value with the maximum function values in some prior successful iterates. For iterates far away from the optimizer, we give a very strong nonmonotone strategy. In the vicinity of the optimizer, we have a weaker nonmonotone strategy. It leads to a medium nonmonotone strategy when iterates are not far away from or close to the optimizer. Theoretical analysis indicates that the new approach converges globally to a first-order critical point under classical assumptions. In addition, the local convergence is also studied. Extensive numerical experiments for unconstrained optimization problems are reported. 15 33 2017/02/232017/10/22 1396/7/30 2021/04/102020/03/26 1399/1/7 M. kimiaei M. kimiaei Vienna University Iran kimiaeim83@univie.ac.at H. esmaeili H. esmaeili Bu Ali University Iran esmaeili47@yahoo.com F. rahpeymaii F. rahpeymaii Payame Noor Iran rahpeyma_83@yahoo.com Unconstrained optimization Trust-region framework Nonmonotone technique Theoretical convergence M. Ahookhosh, K. Amini, An efficient nonmonotone trust-region method for unconstrained optimization,textit{Numerical Algorithms} textbf{59}(4), (2012), 523--540.##M. Ahookhosh, K. Amini, A nonmonotone trust region method with adaptive radius for unconstrained optimization problems, textit{Computers and Mathematics with Applications}, textbf{60}, (2010), 411--422.##M. Ahookhosh, K. Amini, M. Kimiaei, A globally convergent trust-region method for large-scale symmetric nonlinear systems, textit{Numerical Functional Analysis and Optimization}, textbf{36}, (2015), 830--855.##M. Ahookhosh, K. Amini, H., Nosratipour, An inexact line search approach using modified nonmonotone strategy for unconstrained optimization, textit{Numerical Algorithms}, textbf{66}, (2014), 49--78.##M. Ahookhosh, K. Amini, M.R. Peyghami, A nonmonotone trust-region line search method for large-scale unconstrained optimization, textit{Applied Mathematical Modelling}, textbf{36}, (2012), 478--487.##M. Ahookhosh, H. Esmaeili, M. Kimiaei, An effective trust-region-based approach for symmetric nonlinear systems, textit{International Journal of Computer Mathematics}, textbf{90}, (2013), 671--690.##M. Ahookhosh, S. Ghaderi, Two globally convergent nonmonotone trust-region methods for unconstrained optimization, textit{Journal of Applied Mathematics and Computing}, textbf{50}(1-2), (2016), 529--555.##N. Andrei, An unconstrained optimization test functions collection, textit{Advanced Modeling and Optimization}, textbf{10}(1), (2008), 147--161.##R. Byrd, J. Nocedal, R. Schnabel, Representation of quasi-Newton matrices and their use in limited memory methods, textit{Mathematical Programming}, textbf{63}, (1994), 129--156.##A.R. Conn, N.I.M. Gould, Ph.L. Toint, textit{Trust-Region Methods}, Society for Industrial and Applied Mathematics SIAM, Philadelphia, 2000.##N.Y. Deng, Y. Xiao, F.J. Zhou, Nonmonotonic trust region algorithm, textit{Journal of Optimization Theory and Applications}, textbf{26}, (1993), 259--285.##E.D. Dolan, J.J. Mor'{e}, Benchmarking optimization software with performance profiles, textit{Mathematical Programming}, textbf{91}, (2002), 201--213.##H. Esmaeili, M. Kimiaei, An improved adaptive trust-region method for unconstrained optimization, textit{Mathematical Modelling and Analysis}, textbf{19}, (2014), 469--490.##G. Fasano, F. Lampariello, M. Sciandrone, A truncated nonmonotone Gauss-Newton method for large-scale nonlinear least-squares problems, textit{Computational Optimization and Applications}, textbf{34}(3), 343--358, (2006).##A. Fischer, P.K. Shukla, M. Wang, On the inexactness level of robust Levenberg-Marquardt methods, textit{Optimization}, textbf{59}(2), (2010), 273--287.##N.I.M Gould, D. Orban, Ph.L. Toint, CUTEst: a Constrained and Unconstrained Testing Environment with safe threads for mathematical optimization, textit{Computational Optimization and Applications}, textbf{60}(3), (2015), 545--557.##L. Grippo, F. Lampariello, S. Lucidi, A nonmonotone line search technique for Newton's method, textit{SIAM Journal on Numerical Analysis}, textbf{23}, (1986), 707--716.##L. Grippo, F. Lampariello, S. Lucidi, A truncated Newton method with nonmonotone linesearch for unconstrained optimization, textit{Journal of Optimization,Theory and Applications}, textbf{60}(3), (1989), 401--419.##L. Grippo, F. Lampariello, S. Lucidi, A class of nonmonotone stabilization method in unconstrained optimization, textit{Numerische Mathematik}, textbf{59}, (1991), 779--805.##L. Kaufman, Reduced storage quasi-Newton trust region approaches to function optimization, textit{SIAM Journal on Optimization}, textbf{10}(1), 56--69, (1999).##bibitem{kimiaei0} M. Kimiaei, A new class of nonmonotone adaptive trust-region methods for nonlinear equations with box constraints, textit{Calcolo}, textbf{54}(3), 769--812, (2017) .##M. Kimiaei, S. Ghaderi, A new restarting adaptive Trust-Region method for unconstrained optimization, textit{Journal of the Operations Research Society of China}, textbf{5}(4), (2017), 487--507.##M. Kimiaei, F. Rahpeymaii, A new nonmonotone line-search trust-region approach for nonlinear systems, textit{TOP}, textbf{27}(2), (2019), 199--232.##L. Lukv{s}an, C. Matonoha, J. Vlv{c}ek, Modified CUTE problems for sparse unconstrained optimization. textit{Techical Report}, textbf{1081}, ICS AS CR, November, 2010.##L. Lukv{s}an, J. Vlv{c}ek, Sparse test problems for unconstrained optimization, textit{Techical Report}, textbf{1064}, ICS AS CR, November 2003.##YU. Nesterov, Modified Gauss-Newton scheme with worst case guarantees for global performance, textit{Optimization Methods and Software}, textbf{22}(3), (2007), 469--483.##J. Nocedal, S.J. Wright, textit{Numerical Optimization}, Springer, NewYork, (2006).##M.J.D. Powell, Convergence properties of a class of minimization algorithms. in Nonlinear Programming, O.L. Mangasarian, R.R. Meyer, and S.M. Robinson, eds., Academic Press, NewYork, 1--27, (1975).##D.F. Shanno, K.H. Phua, Matrix conditioning and non-linear optimization, textit{Mathematical Programming}, textbf{14}, (1978), 149--160.##T. Steihaug, The conjugate gradient method and trust regions in large scale optimization, textit{SIAM Journal on Numerical Analysis}, textbf{20}, (1983), 626--637.##W. Sun, Y. Yuan, textit{Optimization Theory and Methods}: Nonlinear Programming. Springer, Berlin, (2006).##S.W. Thomas, textit{Sequential estimation techniques for quasi-Newton algorithms}, Cornell University, 1975.##Ph.L. Toint, Numerical solution of large sets of algebraic nonlinear equations, textit{Mathematics of Computation}, textbf{46}(173), (1986), 175--189.##L. Xu, J.V. Burke, An active set $ell_{infty}-$trust region algorithm for box constrained optimization. Technical Report preprint, Departeman Mathematics, niversity of Washington, Seattle, WA 98195, U.S.A.##H.C. Zhang, W.W. Hager, A nonmonotone line search technique for unconstrained optimization, textit{SIAM journal on Optimization}, textbf{14}(4), (2004), 1043--1056.## ##

On Contact and Symplectic Lie Algeroids 2 2 In this paper, we will study compatible triples on Lie algebroids. Using a suitable decomposition for a Lie algebroid, we construct an integrable generalized distribution on the base manifold. As a result, the symplectic form on the Lie algebroid induces a symplectic form on each integral submanifold of the distribution. The induced Poisson structure on the base manifold can be represented by means of the induced Poisson structures on the integral submanifolds. Moreover, for any compatible triple with invariant metric and admissible almost complex structure, we show that the bracket annihilates on the kernel of the anchor map. 35 53 2017/02/232017/10/222017/11/26 1396/9/5 2021/04/102020/03/262018/05/26 1397/3/5 E. Nazari E. Nazari Tarbiat Modares University Iran e.nazari@modares.ac.ir A. Heydari A. Heydari Tarbiat Modares University Iran aheydari@modares.ac.ir Lie algebroid Symplectic Lie algebroid Contact Lie algebroid Poisson structure P. Antunes, J. M. Nunes da Costa, Hypersymplectic structures with torsion on Lie algebroids, J. Geom. Phys., 104, (2016), 39-53.##M. Boucetta, Riemannian Geometry of Lie Algebroids, Journal of the Egyptian Mathematical Society, 19 (2011) 57-70.##M. Crainic, R. Fernandes, Integrability of Lie brackets, Ann. of Math., 157 (2003) 575-620.##E. de Leon, J. C. Marrero, E. Martinez, Lagrangian submanifolds and dynamics on Lie algebroids. J. Phys. A, 38, (2005), 241-308.##Gh. Fasihi Ramandi, N. Boroojerdian, Forces Uni cation in The Framework of Transitive Lie Algebroids, Int. J. Theor. Phys., 54 (2015), 1581-1593.##C. Ida, P. Popescu, On almost complex Lie algebroids, Mediterr. J. Math., 13, (2016), 803-824.##C. Ida, P. Popescu, Contact structures on Lie algebroids, arXiv:1507.01110, 2015 - arxiv.org##D. Iglesias, J. Marrero, D. Martin de Diego, E. Martinez, E, Padron, Reduction of Symplectic Lie Algebroids by a Lie Subalgebroid and a Symmetry Lie Group, SIGMA, 3 (2007) 049, 28 pp.##Y. Kosmann-Schwarzbach, Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey, SIGMA, 4 (2008) 005, 30 pp.##K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, Cambridge University Press, 2005.##N. Neumaier, Waldmann, Stefan Deformation quantization of Poisson structures associated to Lie algebroids. SIGMA 5 (2009), Paper 074, 29 pp.##R. Nest, B. Tsygan, Deformations of symplectic Lie algebroids, deformations of holomorphic symplectic structures, and index theorems Asian J. Math., 5 (2001), 599-635.##L. Popescu, Geometrical structures on Lie algebroids, Publ. Math. Debrecen, 72 (2008), no. 1, 1-15.##L. Popescu, Lie algebroids framework for distributional systems. An. tiin. Univ. Al. I. Cuza Iai. Mat (N.S.), 55 (2009), no. 2, 257-274.##L. Popescu, A note on Poisson-Lie algebroids, I. Balkan J. Geom. Appl, 14 (2009), no. 2, 79-89.##L. Popescu, Symmetries of second order differential equations on Lie algebroids. J. Geom. Phys, 117 (2017), 84-98.##P. Popescu, Poisson structures on almost complex Lie algebroids, Int. J. Geom. Methods Mod. Phys 11 (2014), no. 8, 1450069, 22 pp.##Y. Vorobiev, On Poisson realizations of transitive Lie algebroids.J. Nonlinear Math. Phys, 11 (2004), suppl., 43-48.## ##

Wijsman Statistical Convergence of Double Sequences of Sets 2 2 In this paper, we study the concepts of Wijsman statistical convergence, Hausdorff statistical convergence and Wijsman statistical Cauchy double sequences of sets and investigate the relationship between them. 55 64 2017/02/232017/10/222017/11/262017/10/5 1396/7/13 2021/04/102020/03/262018/05/262018/07/25 1397/5/3 E. Dundar E. Dundar Department of Mathematics, Faculty of Science and Literature, Afyon Kocatepe University Turkey edundar@aku.edu.tr F. Nuray F. Nuray Department of Mathematics, Faculty of Science and Literature, Afyon Kocatepe University Turkey fnuray@aku.edu.tr U. Ulusu U. Ulusu Department of Mathematics, Faculty of Science and Literature, Afyon Kocatepe University Turkey ulusu@aku.edu.tr Statistical convergence Double sequence of sets Wijsman convergence Hausdorff convergence. B. Altay, F. Bac sar, Some new spaces of double sequences, textit{J. Math. Anal. Appl.}, textbf{309} (1), (2005), 70--90.##J.-P. Aubin, and H. Frankowska, textit{Set-valued analysis}, Birkhauser, Boston, 1990.##M. Baronti, and P. Papini, Convergence of sequences of sets, textit{Methods of functional analysis in approximation theory},textbf{76}, Birkhauser-Verlag, Basel,(1986), 133-155.##G. Beer, On convergence of closed sets in a metric space and distance functions, textit{Bull. Aust. Math. Soc.}, textbf{31}, (1985), 421--432.##G. Beer, Wijsman convergence: A survey, textit{Set-Valued Var. Anal.}, textbf{2}, (1994), 77--94.##J. S. Connor, The statistical and strong p-Ce{sa}ro convergence of sequences, textit{Analysis}, textbf{8}, (1988), 46--63.##C. c{c}akan, B. Altay, Statistically boundedness and statistical core of double sequences, textit{J. Math. Anal. Appl.}, textbf{317}, (2006), 690--697.##R. c{C}olak, Y. Alt{i}n, Statistical convergence of double sequences of order $alpha$, textit{Journal of Function Spaces and Applications}, textbf{2013}, (2013), 1--5.##H. Fast, Sur la convergence statistique, textit{Colloq. Math.},textbf{2}, (1951), 241--244.##A. R. Freedman, J.J. Sember, M. Raphael, Some Ces'{a}ro type summability spaces, textit{Proc. London Math. Soc.}, textbf{37}, (1978), 508--520.##J. A. Fridy, C. Orhan, Statistical limit superior and inferior, textit{Proc. Amer. Math. Soc.}, textbf{125}, (1997) 3625--3631.##J. D. Hill, On perfect summability of double sequences, textit{Bull. Amer. Math. Soc.}, textbf{46}, (1940), 327-331.##M. Ic{s}{i}k, Y. Alt{i}n, $f_{(lambda,mu)}$-statistical convergence of order $widetilde{alpha}$ for double sequences, textit{Journal of Inequalities and Applications}, textbf{2017}(246), (2017), 8 pages.##I. G. Kull, Multiplication of summable double series, textit{Uch.zap. Tartusskogo un-ta}, textbf{62}, (1958), 3--59 (in Russian).##B. V. Limayea, M. Zeltser, On the Pringsheim convergence of double series,textit{Proc. Est. Acad. Sci.}, textbf{58}, (2009), 108--121.##M. Mursaleen, O. H. H. Edely, Statistical convergence of double sequences, textit{J. Math. Anal. Appl.}, textbf{288}, (2003), 223--231.##F. Nuray, B. E. Rhoades, Statistical convergence of sequences of sets, textit{Fasc. Math.}, textbf{49}, (2012), 87--99.##F. Nuray, U. Ulusu, E. D"{u}ndar, Ces'{a}ro summability of double sequences of sets, textit{Gen. Math. Notes} textbf{25}(1), (2014), 8--18.##A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, textit{Math. Ann.}, textbf{53}, (1900), 289--321.##R.T. Rockafellar, R.J-B Wets, Variational Analysis, textit{Grundlehren der Mathematischen Wissenschaften} 317, Springer-Verlag, 2009.##E. Savac{s}, On some double lacunary sequence spaces of fuzzy numbers, textit{Mathematical and Computational Applications}, textbf{15}(3), (2010), 439--448.##I. J. Schoenberg, The integrability of certain functions and related summability methods, textit{Amer. Math. Monthly}, textbf{66},(1959), 361--375.##Y. Sever, {O}. Talo, On Statistical Convergence of Double Sequences of Closed Sets, textit{Filomat}, textbf{30}(3), (2016), 533--539, DOI 10.2298/FIL1603533S.##Y. Sever, {O}. Talo, B. Altay, On convergence of double sequences of closed sets, textit{Contemp. Anal. Appl. Math.}, textbf{3}, (2015), 30--49.##{O}. Talo, Y. Sever, F. Bac{s}ar, On statistically convergent sequences of closed set, textit{Filomat}, textbf{30}(6), (2016), 1497--1509.##U. Ulusu, F. Nuray, Lacunary statistical convergence of sequence of sets, textit{Progress in Applied Mathematics}, textbf{4}(2), (2012), 99--109.##U. Ulusu, F. Nuray, On strongly lacunary summability of sequences of sets, textit{Journal of Applied Mathematics and Bioinformatics}, textbf{3}3, (2013), 75--88.##R. A. Wijsman, Convergence of sequences of convex sets, cones and functions, textit{Bull. Amer. Math. Soc.}, textbf{70}, (1964), 186--188.##R. A. Wijsman, Convergence of Sequences of Convex sets, Cones and Functions II, textit{Trans. Amer. Math. Soc.}, textbf{123}(1), (1966), 32--45.##M. Zeltser, M. Mursaleen, S. A. Mohiuddine, emph{On almost conservative matrix methods for double sequence spaces}, Publ. Math. Debrecen, textbf{75} (2009), 1--13.## ##

One-point Goppa Codes on Some Genus 3 Curves with Applications in Quantum Error-Correcting Codes 2 2 We investigate one-point algebraic geometric codes CL(D, G) associated to maximal curves recently characterized by Tafazolian and Torres given by the affine equation yl = f(x), where f(x) is a separable polynomial of degree r relatively prime to l. We mainly focus on the curve y4 = x3 +x and Picard curves given by the equations y3 = x4-x and y3 = x4 -1. As a result, we obtain exact value of minimum distance in several cases and get many records that don’t exist in MinT tables (tables of optimal parameters for linear codes), such as codes over F72 of dimension less than 36. Moreover, using maximal Hermitian curves and their sub-covers, we obtain a necessary and sufficient condition for self-orthogonality and Hermitian self-orthogonally of CL(D, G). 65 76 2017/02/232017/10/222017/11/262017/10/52018/02/20 1396/12/1 2021/04/102020/03/262018/05/262018/07/252018/04/25 1397/2/5 R. Mohammadi R. Mohammadi Department of Mathematics, Tarbiat Modares University. Iran rasool.mohammadi@modares.ac.ir Algebraic geometric codes Maximal curves Minimum distance Goppa bound Quantum error-correcting codes 1. A. Ashikhim, E. Knill, Non-binary quantum stabilizer codes, IEEE Trans. Inf. Theory, 47, (2001), 3065-3072.##2. D. Bartoli, L. Quoos, G. Zini, Algebraic Geometric Codes on Many Points from Kummer Extensions, arXiv, 1606.04143, (2016).##3. A.R. Calderbank, P.W. Shor, Good quantum error-correcting codes exist, Physical Review, A 54, (1996), 1098-1105.##4. A.S. Castellanos, A.M. Masuda, L. Quoos, One- and Two-Point Codes Over Kummer Extensions, IEEE Trans. Inf. Theory, 62, (2016), 4867-4872.##5. H. Chen, Some good quantum error-correcting codes from algebraic geometry codes, IEEE Trans. Inf.Theory, 47, (2001), 2059-2061.##6. Y. Edel, Some good quantum twisted codes, http://www.mathi.uni-heidelberg.de/~yves/Matrizen/QTBCH/QTBCHindex.html.##7. C. Galindo, F. Hernando, Quantum codes from affine variety codes and their subfield subcodes, Designs, Codes and Cryptography, 76 (1), (2015), 89-100.##8. O. Geil, C. Munuera, D. Ruano, F. Torres, On the order bound for one-point codes, Advances in Mathematics of Communication, 5, (2011), 489-504.##9. V.D. Goppa, Codes on algebraic curves, Dokl. Akad. NAUK, SSSR, 259, (1981), 1289-1290.##10. V.D. Goppa, Algebraic geometric codes, Izv. Akad. NAUK, SSSR, 46, (1982), 75-91.##11. D.Hankerson, A.Menezes, S.Vanstone, Guide to Elliptic Curve Cryptography, Springer Professional Computing, Springer-Verlag, (2004).##12. T. Hasegawa, Some remarks on superspecial and ordinary curves of low genus, Math. Nachr, 286, (2013), 17-33.##13. C. Hu, S.Yang, Multi-point codes over Kummer extensions, Designs, Codes and Cryptography, 86 (1), (2018), 211-230.##14. L. Jin, Quantum stabilizer codes from maximal curves, IEEE Trans. Inf. Theory, 60 (1), (2014), 313-316.##15. L. Jin, C.P. Xing, Euclidean and Hermitian self-orthogonal Algebraic Geometry codes and their application to Quantum codes, IEEE Trans. Inf. Theory, 58 (8), (2012), 5484-5489.##16. A. Kazemifard, S. Tafazolian, A note on some Picard curves over finite fields, Finite Fields and Their Applications, 34, (2015), 107-122.##17. J. Kim, J. Walker, Non-binary quantum error-correcting codes from algebraic curves, Discrete Mathematics, 308, (2008), 3115-3124.##18. Magma Computational Algebra System, http://magma.maths.usyd.edu.au/magma/.##19. G.L. Matthews, Weierstrass semigroups and codes from a quotient of the Hermitian curve, Designs, Codes and Cryptography, 37, (2005), 473-492.##20. MinT, Tables of optimal parameters for linear codes, Univ. Salzburg, Salzburg. Austria, (2009), http://mint.sbg.ac.at/.##21. C. Munuera, R. Pellikaan, Equality of geometric Goppa codes and equivalence of divisors, J. Pure Appl. Algebra, 90 (1993), 229-252.##22. C. Munuera, W. Tenrio, F. Torres, Quantum error-correcting codes from algebraic geometry codes of Castle type, Quantum Information Processing, 16 (10), (2016), 4071-4088.##23. E.M. Rains, Non-binary quantum codes, IEEE Trans. Inform. Theory, 45, (1999), 18271832.##24. P.K. Sarpevalli, A. Klappenecker, Non-binary quantum codes from Hermitian curves, Applied algebra, algebraic algorithms and error-correcting codes, Lecture Notes in Computer Science 3857, Springer, Berlin, (2006), 136-143.##25. H. Stichtenoth, A note on Hermitian codes over GF(q2), IEEE Trans. Inf. Theory, 34, (1988), 1345-1348.##26. H. Stichtenoth, Algebraic Function Fields and Codes. Second edition. Graduate Texts in Mathematics, Springer-Verlag, Berlin, 254, (2009).##27. S. Tafazolian, F. Torres, On the curve yn = xm +x over finite fields, J. Number Theory, 45, (2014), 51-66.##28. Y. Takizawa, Some remarks on the Picard curves over a finite field, Math. Nachr, 280, (2007), 802-811.## ##

Viewing Some Ordinary Differential Equations from the Angle of Derivative Polynomials 2 2 In the paper, the authors view some ordinary differential equations and their solutions from the angle of (the generalized) derivative polynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius-Euler polynomials, the Euler numbers and polynomials, in terms of the Stirling numbers of the first and second kinds. 77 95 2017/02/232017/10/222017/11/262017/10/52018/02/202018/02/6 1396/11/17 2021/04/102020/03/262018/05/262018/07/252018/04/252018/05/26 1397/3/5 B. -N. Guo B. -N. Guo Henan Polytechnic University China bai.ni.guo@hotmail.com F. Qi F. Qi Tianjin Polytechnic University China qifeng618@gmail.com Viewpoint Ordinary differential equation Solution Derivative polynomial Identity Stirling numbers Bernoulli number Bernoulli polynomial Frobenius-Euler polynomial K. N. Boyadzhiev, A note on the higher derivatives of the function $1/(exp(x)-1)$, emph{Adv. 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On Eulerianity and Hamiltonicity in Annihilating-ideal Graphs 2 2 Let $R$ be a commutative ring with identity, and $ mathrm{A}(R) $ be the set of ideals with non-zero annihilator. The annihilating-ideal graph of $ R $ is defined as the graph $AG(R)$ with the vertex set $ mathrm{A}(R)^{*}=mathrm{A}(R)setminuslbrace 0rbrace $ and two distinct vertices $ I $ and $ J $ are adjacent if and only if $ IJ=0 $. In this paper, conditions under which $AG(R)$ is either Eulerian or Hamiltonian are given. 97 104 2017/02/232017/10/222017/11/262017/10/52018/02/202018/02/62017/12/30 1396/10/9 2021/04/102020/03/262018/05/262018/07/252018/04/252018/05/262019/07/9 1398/4/18 A. Kourehpaz A. Kourehpaz Department of Mathematics, Jundi-Shapur University of Technology Iran asma_korehpaz@jsu:ac:ir R. Nikandish R. Nikandish Department of Mathematics, Jundi-Shapur University of Technology Iran r.nikandish@jsu.ac.ir Annihilating-ideal graph Eulerian graphs Hamiltonian graphs G. Aalipour, S. Akbari, R. Nikandish, M. J. Nikmehr, F. Shaveisi, On the coloring of the annihilating-ideal graph of a commutative ring, Discrete Math. 312 (2012) 2620--2626.##S. Akbari, A. Alilou, J. Amjadi, S. M. Sheikholeslami, The Co-annihilating-ideal graphs of commutative rings, Canad. Math. Bull. 60(2017), 3--11.##D. F. Anderson, A. Badawi, On the total graph of a commutative ring without the zero element, J. Algebra Appl. 11, 1250074 (2012) [18 pages].##M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Company (1969).##A. Badawi, On the dot product graph of a commutative ring, Comm. Algebra 43 (2015) 43--50.##M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10 (2011) 727--739.##R. Nikandish, H. R. Maimani, H. Izanloo, The annihilating-ideal graph of $mathbb{Z}_{n}$ is weakly perfect, Contribituins to Discrete Mathematics 11 (2016) 16--21.##F. Shaveisi, The central vertices and radius of the regular graph of ideals, Transactions on Combinatorics (TOC) 6 (2017) 1--13.##D. B. West, Introduction to Graph Theory, 2nd ed., Prentice Hall, Upper Saddle River (2001).##H. Y. Yu, T. Wu, Commutative rings $R$ whose $C(mathbb{AG}(R))$ consists only of triangles, Comm. Algebra 43 (2015) 1076--1097.## ##

Graph Clustering by Hierarchical Singular Value Decomposition with Selectable Range for Number of Clusters Members 2 2 Graphs have so many applications in real world problems. When we deal with huge volume of data, analyzing data is difficult or sometimes impossible. In big data problems, clustering data is a useful tool for data analysis. Singular value decomposition(SVD) is one of the best algorithms for clustering graph but we do not have any choice to select the number of clusters and the number of members in each cluster. In this paper, we use hierarchical SVD to cluster graphs with it's adjacency matrix. In this algorithm, users can select a range for the number of members in each cluster. The results show in hierarchical SVD algorithm, clustering measurement parameters are more desirable and clusters are as dense as possible. The complexity of this algorithm is less than the complexity of SVD clustering method. 105 121 2017/02/232017/10/222017/11/262017/10/52018/02/202018/02/62017/12/302018/02/4 1396/11/15 2021/04/102020/03/262018/05/262018/07/252018/04/252018/05/262019/07/92019/08/8 1398/5/17 A. Sadeghian A. Sadeghian Yazd University Iran a_sadeghian@stu.yazd.ac.ir S. A.l Shahzadeh Fazeli S. A.l Shahzadeh Fazeli Yazd University Iran fazeli@yazd.ac.ir S. M. Karbassi S. M. Karbassi Yazd University Iran smkarbassi@yazd.ac.ir Graph Clustering Singular Value Decomposition Hierarchical Clustering Selectable Clusters Number. T. P. Cason, P. A. Absil, P. Van Dooren, Iterative methods for low rank approximation of graph similarity matrices textit{ Linear Algebra and its Applications},textbf{1}(438),(2013), 1863-1882.##A. K. Cline, S. Dhillon, Computation of the Singular Value Decomposition, 14, The University of Texas at Austin, 2007.##B.Datta, textit{Numerical linear algebr and Applications}, Second edition, SIAM, 2010.##J. Demmel, Accurate singular value decomposition of structured matrices textit{ SIAM},textbf{21}(2),(1997), 562-580.##J. Dongarra, Accuracy of computed singular values, textit{SIAM},textbf{1}(4), (1983), 712-719 .##E. P. Douglas, textit{Clustering datasets with singular value decomposition}, College of Charleston, 2008.##G. H. Golub, C. Reinsch, Singular value decomposition and least squares solutions, textit{Number Math}, textbf{1}(14), (1973), 403-420.##L. Rokach, O. Maimon, Data Mining and Knowledge Discovery Handbook chapter: Clustering Methods,textit{ Springer US},(2005), 321-352.##S. E. Schaeffer, Survey: Graph clustering, textit{ Computer Science Review},textbf{1}(1), (2007), 27-64.##G. W. Stewart, Error and perturbation bounds for subspaces associated with certain eigenvalue problems textit{SIAM}, textbf{1}(15) (1973), 727-764.##X. Zhou, SVD-based incremental approaches for recommender systems, textit{ Computer and System Sciences},textbf{1}(81), (2015), 717-733.## ##

Surfaces Generated by Translation Surfaces of Type 1 in I^1_3 2 2 In this paper, we classify surface at a constant distance from the edge of regression on translation surfaces of Type 1 in the three dimensional simply isotropic space I^1_3 satisfying some algebraic equations in terms of the coordinate functions and the Laplacian operators with respect to the first, the second and the third fundamental form of the surface. We also give explicit forms of these surfaces. 123 135 2017/02/232017/10/222017/11/262017/10/52018/02/202018/02/62017/12/302018/02/42018/01/10 1396/10/20 2021/04/102020/03/262018/05/262018/07/252018/04/252018/05/262019/07/92019/08/82020/08/15 1399/5/25 M. Karacan M. Karacan Usak University Turkey murat.karacan@usak.edu.tr A. Çakmak A. Çakmak Bitlis Eren University Turkey acakmak@beu.edu.tr S. Kızıltuğ S. Kızıltuğ Erzincan University Turkey skiziltug@erzincan.edu.tr H. Es H. Es Gazi Universiy Turkey hasanes@gazi.edu.tr Simply isotropic space Translation surfaces Surface at a constant distance from the edge of regression on a surface. M.E.Aydin, A generalization of translation surfaces with constant curvature in the isotropic space, J. Geom. 107 (3), (2016), 603-615.##Ch. Baba-Hamed, M. Bekkar and H. Zoubir, Translation surfaces in the three dimensional Lorentz-Minkowski space satisfying ri = Ari, Int. Journal of Math. Analysis 4(17), (2010), 797-808.##M.Bekkar, B. Senoussi, Translation surfaces in the 3-dimensional space satisfying III ri = iri; J. Geom., 103, (2012), 367-374.##B.Bukcu, D.W.Yoon and M.K.Karacan, Translation surfaces in the 3-dimensional simply isotropic space I^1_3 satisfying IIIxi = ixi, Konuralp Journal of Mathematics 4(1),(2016), 275-281.##A.Cakmak, O.Tarakci, Surfaces at a constant distance from the Edge of Regression on a Surface of Revolution in E3, Applied Mathematical Sciences, 10(15), (2016), 707-719.##A. Cakmak, O. Tarakci, The image curves on surfaces at a constant distance from the edge of regression on a surface of revolution, International Journal of Mathematics and Computation, 1, (2016), 74-85.##B.Y. Chen, A report on submanifold of nite type, Soochow J. Math., 22, (1996), 117-337.##F. Dillen, J. Pas and L. Vertraelen, On surfaces of nite type in Euclidean 3-space, Kodai Math. J., 13, (1990), 10-21.##F. Dillen, J. Pas and L. Vertraelen, On the Gauss map of surfaces of revolution, Bull. Inst. Math. Acad. Sinica 18, (1990), 239-246.##O. J. Garay, An extension of Takahashi's theorem, Geom. Dedicata, 34, (1990), 105-112.##G. Kaimakamis, B. Papantoniou, K. Petoumenos, Surfaces of revolution in the 3-dimensional Lorentz-Minkowski space satisfying III r = Ar, Bull.Greek Math. Soc. 50, (2005), 75-90.##M.K.Karacan, D.W.Yoon and B.Bukcu, Translation surfaces in the three dimensional simply isotropic space I^1_3, Int. J. Geom. Methods Mod. Phys., 13, (2016), 1650088.##H. Sachs, Isotrope geometrie des raumes, Vieweg Verlag, Braunschweig, 1990.##D. Saglam, O. B. Kalkan, Surfaces at a constant distance from the edge of regression on a surface in E3, Di erential Geometry-Dynamical Systems, 12, (2010), 187-200.##B.Senoussi, M. Bekkar, Helicoidal surfaces with J r = Ar in 3-dimensional Euclidean space, Stud. Univ. Babes-Bolyai Math. 60(13), (2015), 437-448.##Z.M. Sipus, Translation Surfaces of constant curvatures in a simply Isotropic space, Period Math. Hung. 68 (2014), 160-175.##K. Strubecker, Di erentialgeometrie des Isotropen raumes III, Flachentheorie, Math. Zeitsch. 48, (1942), 369-427.##T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380-385.##O. Tarakc , H.H Hac salihoglu, Surfaces at a constant distance from the edge of regression on a surface, Appl. Mathematics and Computation, 155, (2004), 81-93.##D.W.Yoon, On the Gauss map of translation surfaces in Minkowski 3-space, Taiwanese Journal of Mathematics, 6(3), (2002), 389-398.##D.W.Yoon, Some classi cation of translation surfaces in Galilean 3-Space, Int. Journal of Math. Analysis, 6(28), (2012), 1355 - 1361.##S. Yurttancikmaz, O. Tarakci, The relationship between focal surfaces and surfaces at a constant distance from the edge of regression on a surface, Advances in Mathematical Phys., (2015), Article ID 397126, 1-6.## ##

Recognition of $L_{2}(q)$ by the Main Supergraph 2 2 Let $G$ be a finite group. The main supergraph $mathcal{S}(G)$ is a graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if and only if $o(x) mid o(y)$ or $o(y)mid o(x)$. In this paper, we will show that $Gcong L_{2}(q)$ if and only if $mathcal{S}(G)cong mathcal{S} (L_{2}(q))$, where $q$ is a prime power. This work implies that Thompson's problem holds for the simple group $L_{2}(q)$. 137 144 2017/02/232017/10/222017/11/262017/10/52018/02/202018/02/62017/12/302018/02/42018/01/102018/02/2 1396/11/13 2021/04/102020/03/262018/05/262018/07/252018/04/252018/05/262019/07/92019/08/82020/08/152018/07/21 1397/4/30 S. S. Salehi Amiri S. S. Salehi Amiri Islamic Azad University, Babol Iran salehisss@baboliau.ac.ir A.R. Khalili Asboei A.R. Khalili Asboei Farhangian University Iran khaliliasbo@yahoo.com Graph Main supergraph Thompson's problem P. J. Cameron, The power graph of a finite group, II,textit{J. Group Theory}, textbf{13} (2010), 779-783.##I. Chakrabarty, S. Ghosh, M. K. Sen, Undirected power graphs of semigroups, textit{Semigroup Forum}, textbf{78} (2009), 410-426.##G. Y. Chen, On Frobenius and $2$-Frobenius group, textit{J.Southwest China Normal Univ}, textbf{20} (1995), 485-487. (in Chinese)##G. Y. Chen, Further reflections on Thompson's conjecture,textit{J. Algebra}, textbf{218} (1999), 276-285.##G. Y. Chen, On Thompson's conjecture, textit{J. Algebra},textbf{185}(1) (1996), 184--193.##J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R.A. Wilson, textit{Atlas of finite groups}, Clarendon, Oxford, 1985.##G. Frobenius, Verallgemeinerung des Sylow'schen Satzes.textit{Berl. Ber}, (1895), 981--993. (In German)##A. Hamzeh, A. R. Ashrafi, Automorphism groups of supergraphs of the power graph of a finite group, textit{European J. Combin}, textbf{60} (2017), 82-88.##B. Huppert, textit{Endliche Gruppen}, I, Springer, Berlin,1967.##N. Iiyori, H. Yamaki, Prime graph components of the simple groups of Lie type over the field of even characteristic, textit{J. Algebra}, textbf{155}(2) (1993), 335-343.##A. Khalili, A. Iranmanesh, A characterization of linear group $L_{2}(p)$, textit{Czechoslovak Math. J}, textbf{64} (139) (2014),459-464.##A. Khalili, S. S. Salehi, Some alternating and symmetric groups and related graphs, textit{Beitr Algebra Geom}, textbf{59} (2018),21-24.##A. Khalili, S. S. Salehi, Some results on the main supergraph of finite groups, textit{Algebra Discrete Math}, textbf{30}(2), (2020), 172-178.##A. Khalili, S. S. Salehi, The small Ree group $^{2}G_{2}(3^{2n+1})$ and related graph, textit{Comment. Math. Univ. Carolin}, textbf{59}(3) (2018), 271--276.##A. Khalili, S. S. Salehi, Recognizability of finite groups by Suzuki group, textit{Arch. Math., Brno, }textbf{55} (2019),225-228.##V. D. Mazurov, E. I. Khukhro, textit{Unsolved problems in group theory}, The Kourovka Notebook, (English version), ArXiv e-prints,(18), January 2014. Available at http://arxiv.org/abs/1401.0300v6.##A. S. Kondtratev, V. D. Mazurove, Recognition of alternating groups of prime degree from their element orders, textit{Sib. Math.J},textbf{41}(2) (2000), 294-302.##J. S. Williams, Prime graph components of finite groups, textit{J. Algebra}, textbf{69}(2) (1981), 487--513.## ##

Application of Tau Approach for Solving Integro-Differential Equations with a Weakly Singular Kernel 2 2 In this work, the convection-diffusion integro-differential equation with a weakly singular kernel is discussed. The Legendre spectral tau method is introduced for finding the unknown function. The proposed method is based on expanding the approximate solution as the elements of a shifted Legendre polynomials. We reduce the problem to a set of algebraic equations by using operational matrices. Also the convergence analysis for shifted Legendre polynomials and error estimation for tau method have been discussed and approved with the exact solution. Finally, several numerical examples are given to demonstrate the high accuracy of the method. 145 168 2017/02/232017/10/222017/11/262017/10/52018/02/202018/02/62017/12/302018/02/42018/01/102018/02/22018/01/27 1396/11/7 2021/04/102020/03/262018/05/262018/07/252018/04/252018/05/262019/07/92019/08/82020/08/152018/07/212020/02/19 1398/11/30 R. Pourgholi R. Pourgholi School of Mathematics and Computer Science, Iran pourgholi@du.ac.ir A. Tahmasbi A. Tahmasbi School of Mathematics and Computer Science, Iran tahmasbi@du.ac.ir R. Azimi R. Azimi School of Mathematics and Computer Science, Iran r.azimi@std.du.ac.ir Shifted Legendre tau method Weakly singular kernel Integro-differential equation Convection-diffusion equation. S. Abbasbandy, E. Taati, Numerical solution of the system of nonlinear Volterra integro-differential equations with nonlinear differential part by the operational Tau method and error estimation, J. Comput. Appl. Math. 231, 106–113, (2009).##M. Atabakzadeh, M. Akrami, G. Erjaee, Chebyshev operational matrix method for solving multiorder fractional ordinary differential equations. Appl. Math. Model. 37(20), 8903–8911 (2013).## P. Bar-Yoseph, E. Moses, U. Zrahia, A.L. Yarin, spacetime spectral element methods for onedimensional nonlinear advectiondiffusion problems, J. Comput. Phys. 119, 62–74 (1995) .## S. Behiry, Solution of nonlinear fredholm integro-differential equations using a hybrid of block pulse functions and normalized bernstein polynomials. J. Comput. Appl. Math. 260, 258–265 (2014).## A. H. Bhrawy, A. Alofi, The operational matrix of fractional integration for shifted chebyshev polynomials. Appl. Math. Lett. 26(1), 25–31 (2013).##J. P. Boyd, Chebyshev and Fourier Spectral Methods, second ed, Dover, Mineola, NY, (2001).##C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics,Springer-Verlag, New York, (1988).## C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods Fundamentals in Single Domains. Springer, Berlin (2006).##H. Danfu, S. Xufeng, Numerical solution of integro-differential equations by using cas wavelet operational matrix of integration. Appl. Math 194(2), 460–466 (2007).##M. K. El-Daou, E. L. Ortiz, The weighted subspace of the Tau method and orthogonal collocation.J. Math. Anal. Appl. 326, 622–631 (2007).##F. Ghoreishi, M. Hadizadeh, Numerical computation of the Tau approximation for the VolterraHammerstein integral equations. Numer. Algorithms 52, 541–559 (2009).##D. Gottlieb, S.A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, in:CBMS-NSF Monograph, No. 26, Soc. Indus. Appl. Math, Philadelphia, (1977).##S. M. Hosseini, The adaptive operational Tau method for system of ODEs. J. Comput. Appl. Math.231, 24–38 (2009).## S. M. Hosseini, S. Shahmorad, Tau numerical solution of Fredholm integro-differential equations with arbitrary polynomial bases. Appl. Math. Model. 27, 145–154 (2003).## S. M. Hosseini, S. Shahmorad, Numerical solution of a class of integro-differential equations by the Tau method with an error estimation. Appl. Math. Comput. 136, 559–570 (2003).## W. Labecca, O. Guimaraes, J. R.C. Piqueira, Diracs formalism combined with complex fourier operational matrices to solve initial and boundary value problems. Commun. Nonlinear Sci. Numer. Simul. 19(8), 2614–2623 (2014).## P. Mokhtary, F. Ghoreishi, The L2-convergence of the Legendre spectral Tau matrix formulation for nonlinear fractional integro-differential equations. Numer. Algor. 58(4), 475–496 (2011).## E.L. Ortiz, The Tau method, SIAM. J. Numer. Anal. 6, 480–492, (1969).## E. L. Ortiz, H. Samara, An operational approach to the Tau method for the numerical solution of nonlinear differential equations, Computing 27, 15–25, (1981).## E. L. Ortiz, H. Samara, Numerical solution of differential eigenvalue problems with an operational approach to the Tau method. Computing 31, 95–103 (1983).## E. L. Ortiz, H. Samara, Numerical solution of partial differential equations with variable coefficients with an operational approach to the Tau method. Comput. Math. Appl. 10, 5–13 (1984).## A. Saadatmandi, Bernstein operational matrix of fractional derivatives and its applications. Appl.Math. Model 38(4), 1365–1372 (2014).##J. Shen, T. Tang, L. L. Wang, Spectral Methods, Algorithms, Analysis and Applications, first ed,Springer, New York, (2011).## S. S. Siddiqi , S. Arshed, Numerical solution of convection-diffusion integro-differential equations with a weakly singular kernel, J. Basic. Appl. Sci. Res., 3(11), 106–120, (2013).##J-G Tang, H-P Ma, A Legendre spectral method in time for first-order hyperbolic equations, Appl. Numer. Math. 57, 1–11, (2007).##S. Yousefi, M. Behroozifar, Operational matrices of bernstein polynomials and their applications. Int. J. Syst. Sci. 41(6), 709–716 (2010).##U. Zrahia, P. Bar-Yoseph, Space–time spectral element method for solution of second-order hyperbolic equations, Comput. ethods Appl. Mech. Engrg. 116, 135–146 (1994).## ##

Tame Loci of Generalized Local Cohomology Modules 2 2 Let $M$ and $N$ be two finitely generated graded modules over a standard graded Noetherian ring $R=bigoplus_{ngeq 0} R_n$. In this paper we show that if $R_{0}$ is semi-local of dimension $leq 2$ then, the set $hbox{Ass}_{R_{0}}Big(H^{i}_{R_{+}}(M,N)_{n}Big)$ is asymptotically stable for $nrightarrow -infty$ in some special cases. Also, we study the torsion-freeness of graded generalized local cohomology modules $H^{i}_{R_{+}}(M,N)$. Finally, the tame loci $T^{i}(M,N)$ of $(M,N)$ will be considered and some sufficient conditions are proposed for the openness of these sets in the Zariski topology. 169 180 2017/02/232017/10/222017/11/262017/10/52018/02/202018/02/62017/12/302018/02/42018/01/102018/02/22018/01/272017/05/7 1396/2/17 2021/04/102020/03/262018/05/262018/07/252018/04/252018/05/262019/07/92019/08/82020/08/152018/07/212020/02/192020/10/24 1399/8/3 F. Dehghani Zadeh F. Dehghani Zadeh Islamic Azad University, Yazd branch Iran dehghanizadeh@iauyazd.ac.ir M. Jahangiri M. Jahangiri Kharazmi university Iran mjahangiri@ipm.ir Graded modules Generalized local cohomology modules Associated prime ideals Tame loci. M. P. Brodmann, Asymptotic behaviour of cohomology:tameness, support and associated primes, in commutative algebra and##algebraic geometry, S. Ghorpade, H. Srinivasan and J. Verma, eds.,contemp. Math 390 (2005) 31-61.## M. P. Brodmann, A cohomological stability result for projective schemes over surfaces, J. Reine angew. Math, 606,(2007) 179-92.## M. P. Brodmann, S. Fumasoli and C. S. Lim, Low-codimensional associated primes of graded components of local cohomology modules, J. Alg. 275 (2004) 867-882.## M. P. Brodmann and M. Jahangiri, Tame loci of certain local cohomology modules, J. Commut. Alg. 4(1),(2012) 79-100.## M. P. Brodmann and R. Y. Sharp, Local cohomology -An Algebraic introduction with geometric applications, (Cambridge Studies in Advanced Mathematics 60, Cambridge University Press (1998).##W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge studies in Advanced Mathematics 39, Revised edition, Cambridge University Press (1998).##S. D. Cutkosky and J. Herzog, Failure of tameness for local cohomology, J. Pure Appl. Alg. 211 (2007) 428-432.## F. Dehghani-Zadeh and H. Zakeri, Some Results on Graded Generalized Local Cohomology Modules, J. Math. Ext 5(1),(2010) ,9-73.##K. Divaani-Azar and A. Hajikarimi, Cofiniteness of Generalized Local Cohomology Modules for One-Dimensional Ideals,Canad. Math. Bull (2011) 1-7.##J. Herzog, Komplexe, Aufl"{o}sungen und Dualit"{a}t in der Lokalen Algebra, Habilitationsschrift, Universit"{a}t Regensburg,1974.##M. Jahangiri, N. Shirmohammadi and sh. Tahamtan,Tameness and Artinianness of graded generalized local cohomology modules, Alg. Colloq. 22(1) (2015) 131-146.## K. Khashyarmanesh, Associated primes of graded components of generalized local cohomology modules, Comm. Alg. 33,(2005),3081-3090.##D. Kirby, Artinian modules and Hilbert polynomials,Q. J. Math 24(2) (1973) 17-57.C. S. Lim, Graded local cohomology modules and their associated primes: the Cohen-Macauly case, J. Pure Appl. Alg.185(2003) 225-238.##H. Matsumura, Commutative Ring Theory, Cambridge, UK:Cambridge University Press (1986).##L. Melkersson,Properties of cofinite modules and applications to local cohomology , Math. Proc. Camb. Phil. Soc. 125, (1999), 417-423.##J. J. Rotman, An Introduction to Homological Algebra, Academic Press, Orlando (1979).## C. Rotthaus and L. M. Sega, Some properties of graded local cohomology modules, J. Algebra, 283, (2005), 232- 247.## N. Suzuki, On the generalized local cohomology and its duality J. Math. Kyoto. Univ 18 (1978) 71-85.##S. Yassemi, Generalized section functors, J. Pure. Appl. Alg. 95 (1994) 103-119.## N. Zamani, On graded generalize local cohomology, Arch.Math. 86 (2006) 321-330.## ##

Relative non-Normal Graphs of a Subgroup of Finite Groups 2 2 Let G be a ﬁnite group and H,K be two subgroups of G. We introduce the relative non-normal graph of K with respect to H , denoted by NH,K, which is a bipartite graph with vertex sets HHK and KNK(H) and two vertices x ∈ H HK and y ∈ K NK(H) are adjacent if xy / ∈ H, where HK =Tk∈K Hk and NK(H) = {k ∈ K : Hk = H}. We determined some numerical invariants and state that when this graph is planar or outerplanar. 181 189 2017/02/232017/10/222017/11/262017/10/52018/02/202018/02/62017/12/302018/02/42018/01/102018/02/22018/01/272017/05/72017/11/3 1396/8/12 2021/04/102020/03/262018/05/262018/07/252018/04/252018/05/262019/07/92019/08/82020/08/152018/07/212020/02/192020/10/242019/07/12 1398/4/21 M. Ziaaddini M. Ziaaddini Department of Pure Mathematics, Ferdowsi University of Mashhad Iran ma.ziyaaddini@stu.um.ac.ir A. Erfanian A. Erfanian Department of Pure Mathematics and the Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad Iran erfanian@um.ac.ir Non-normal graph Relative Non-normal graph Normality degree Outer planar. A. Abdollahi, Engel graph associated with a group, J. Algebra, 318 (2007), 680-691.##A. Abdollahi and A. Mohammadi Hassanabadi, Non-cyclic graph of a group, Comm. in Algebra, 35 (2007), 2057-2081.##A. Abdollahi, S. Akbari and H. R. Maimani, Non-commuting graph of a group, J. Algebra, 298 (2006), 468-492.##M. Bodirsky, O. Gimenez, M. Kang and M. Noy, Enumeration and limit laws of seriesparallel graphs, European Journal of Combinatorics, 28, (2005), 2091-2105.##J. A. Bondy and J. S. R. Murty, Graph Theory with Applications, Elsevier, (1977).##G. Chartrand and P. Zhang, Chromatic Graph Theory Taylor & Francis, 2009.##A. Erfanian, M. Farrokhi D.G., and B. Tolue, Non-normal graphs of finite groups, J. Algebra Appl, 12 (2013) [9 pages] Doi:10.1142/S0219498812501939.##F. Saeedi, M. Farrokhi D. G. and S. H. Jafari, Subgroup normality degrees of finite groups I, Arch. Math., 96 (2011), 215-224.## ##

Nearly Rational Frobenius Groups 2 2 In this paper, we study the structure of nite Frobenius groups whose non-rational or non-real irreducible characters are linear. 191 194 2017/02/232017/10/222017/11/262017/10/52018/02/202018/02/62017/12/302018/02/42018/01/102018/02/22018/01/272017/05/72017/11/32018/02/28 1396/12/9 2021/04/102020/03/262018/05/262018/07/252018/04/252018/05/262019/07/92019/08/82020/08/152018/07/212020/02/192020/10/242019/07/122018/10/8 1397/7/16 S. M. Robati S. M. Robati Department of Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran. Iran sajjad.robati@gmail.com Frobenius groups Rational groups Real groups. 1. B. G. Basmaji, Rational Non-Linear Characters of Metabelian Groups, Proc. Amer. Math. Soc, 85, (1982), 175-180.##Y. Berkovich and E. Zhmud, Characters of finite groups, Part I, American mathematical society, (1998).##D. Chillag and A. Mann, Nearly odd-order and nearly real finite groups, Comm. Algebra, 26(7), (1998), 2041-2064.##M. R. Darafsheh and H. Sharifi, Frobenius Q-groups, Arch. Math. (Basel), 83(2), (2004), 102-105.##M. R. Darafsheh, A. Iranmanesh, and S. A. Moosavi, Groups whose nonlinear irreducible characters are rational valued, Arch. Math. (Basel), 94, (2010), 411-418.##S. Dolfi, G. Navarro, and P. H. Tiep, Primes dividing the degrees of the real characters, Math. Z., 259, (2008), 755-774.##L. Dornhofi, Group representation theory. Part A: Ordinary representation theory, Marcel Dekker (New York, 1971).##G. Navarro and P. H. Tiep, Degrees of rational characters of finite groups, Adv. Math., 224, (2010), 1121-1142.##M. Norooz-Abadian and H. Sharifi, Frobenius Q1-groups, Arch. Math. (Basel), 105(6), (2015), 509-517.##D. S. Passman, Permutation groups, New York, (1968).## ##

Some Weighted Integral Inequalities for Generalized Conformable Fractional Calculus 2 2 In this paper, we have obtained weighted versions of Ostrowski, Čebysev and Grüss type inequalities for conformable fractional integrals which is given by Katugompola. By using the Katugampola definition for conformable calculus, the present study confirms previous findings and contributes additional evidence that provide the bounds for more general functions. 195 212 2017/02/232017/10/222017/11/262017/10/52018/02/202018/02/62017/12/302018/02/42018/01/102018/02/22018/01/272017/05/72017/11/32018/02/282017/12/8 1396/9/17 2021/04/102020/03/262018/05/262018/07/252018/04/252018/05/262019/07/92019/08/82020/08/152018/07/212020/02/192020/10/242019/07/122018/10/82018/05/8 1397/2/18 H. Budak H. Budak Department of Mathematics, Faculty of Science and Arts, Düzce University Turkey hsyn.budak@gmail.com F. Usta F. Usta Department of Mathematics, Faculty of Science and Arts, Düzce University Turkey fuatusta@duzce.edu.tr M. Z. Sarikaya M. Z. Sarikaya Department of Mathematics, Faculty of Science and Arts, Düzce University Turkey sarikaymz@gmail.com Ostrowski inequality Čebysev inequality Grüss inequality Conformable fractional integrals. T. Abdeljawad, On conformable fractional calculus,textit{Journal of Computational and Applied Mathematics}, textbf{279}, (2015), 57--66.##D. R. Anderson, textit{Taylor's formula and integral inequalities for conformable fractional derivatives}, Contributions in Mathematics and Engineering, in Honor of Constantin Caratheodory, Springer,to appear.##A. Atangana, D. Baleanu, and A. Alsaedi, New properties of conformable derivative, textit{Open Math.}, textbf{13}, (2015), 889-898.##P. L. v{C}ebyv{s}ev, Sur less expressions approximatives des integrales definies par les autres prises entre les memes limites,textit{Proc. Math. Soc. Charkov }, textbf{2}, (1882), 93-98.##R. Gorenflo, F. Mainardi, textit{Fractional calculus:integral and differential equations of fractional order}, Springer Verlag,Wien, 223-276, 1997.##G. Gruss, {U}ber das maximum des absoluten Betrages textit{ }$frac{1}{b-a}int limits_{a}^{b}f(x)g(x)dx-frac{1}{(b-a)^{2}}int limits_{a}^{b}f(x)dxint limits_{a}^{b}g(x)dx$, textit{Math. Z.}, textbf{39}, (1935), 215-226.##Abu Hammad, R. Khalil, Abel s formula and wronskian for conformable fractional differential equations, textit{International Journal of Differential Equations and Applications}, textbf{13}(3), (2014), 177-183.##O. S. Iyiola and E. R.Nwaeze, Some new results on the new conformable fractional calculus with application using D Alambert approach, textit{Progr. Fract. Differ. Appl.}, textbf{2}(2), (2016), 115-122.##U. Katugampola, A new fractional derivative with classical properties, ArXiv:1410.6535v2.##R. Khalil, M. Al horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, textit{Journal of Computational Applied Mathematics}, textbf{264}, (2014), 65-70.##A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, textit{Theory and Applications of Fractional Differential Equations}, North-Holland Mathematics Studies, 204, Elsevier Sci. B.V., Amsterdam, 2006.##S. Miller and B. Ross, textit{An introduction to the Fractional Calculus and Fractional Differential Equations}, John Wiley Sons, USA, 1993.##D. S. Mitrinvi'{c}, J. E. Pecari'{c} and A. M. Fink, textit{Classical and New Inequalities in Analysis}, Kluwer Academic Publishers, Dordrecht, 1993.##D. S. Mitrinovi{c}, J. E. Pecari{c} and A. M. Fink, textit{Inequalities involving functions and their integrals and derivatives}, Springer Science & Business Media, 2012.##A. M. Ostrowski, textit{{U}ber die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert}, textit{Comment. Math.Helv.}, textbf{10}, (1938), 226-227.##B.G. Pachpatte, textit{Analytic Inequalities}. Atlantis Press, Paris, 2012.##I. Podlubni, textit{Fractional Differential Equations},Academic Press, San Diego, 1999.##M. Z. Sarikaya, On the Ostrowski type integral inequality, textit{Acta Math. Univ. Comenianae}, textbf{LXXIX}(1), (2010), 129-134.##M. Z. Sarikaya and H. Budak, New inequalities of Opial type for conformable fractional integrals, textit{Turk. J. Math}, textbf{41}(5), (2017), 1164 - 1173.## ##

ABSTRACTS IN PERSIAN Vol.16, No.1 2 2 Please see the full text contains the pesian abstracts of this volume. 213 228 2017/02/232017/10/222017/11/262017/10/52018/02/202018/02/62017/12/302018/02/42018/01/102018/02/22018/01/272017/05/72017/11/32018/02/282017/12/82021/08/2 1400/5/11 2021/04/102020/03/262018/05/262018/07/252018/04/252018/05/262019/07/92019/08/82020/08/152018/07/212020/02/192020/10/242019/07/122018/10/82018/05/82021/08/2 1400/5/11 The Name of Authors in this Volume The Name of Authors in this Volume Academic Center for Education, Culture and Research (ACECR) Iran fatemeh.bardestani@gmail.com ABSTRACTS PERSIAN Vol. 16 No. 1 All references of the papers in Vol16,No1## ##