2016
42
2
2
0
Radical of $cdot$ideals in $PMV$algebras
2
2
In this paper, we introduce the notion of the radical of a $PMV$algebra $A$ and we charactrize radical $A$ via elements of $A$. Also, we introduce the notion of the radical of a $cdot$ideal in $PMV$algebras. Several characterizations of this radical is given. We define the notion of a semimaximal $cdot$ideal in a $PMV$algebra. Finally we show that $A/I$ has no nilpotent elements if and only if $I$ is a semimaximal $cdot$ideal of $A$.
1

233
246


F.
Forouzesh
Faculty of Mathematics and computing, Higher Education complex of Bam, Bam, Iran.
Faculty of Mathematics and computing,
Iran
frouzesh@bam.ac.ir
$PMV$algebra
$cdot$ideal
$cdot$prime ideal
radical
[A. Bigard, K. Keimel and S. Wolfenstein, Groupes et Anneaux Reticules, Lecture Notes in Math., 608, SpringerVerlag, BerlinHeidelbergNew York, 1977. ##C. C. Chang, Algebraic analysis of many valued logic, Trans. Amer. Math. Soc. 88 (1958) 467490. ##R. Cignoli, I. M. L. D'Ottaviano and D. Mundici, Algebraic Foundations of ManyValued Reasoning, Kluwer Academic Publishers, Dordrecht, 2000. ##A. Di Nola and A. Dvurecenskij, Product MV algebras, Mult.Valued Log. 6 (2001), no. 12, 193215. ##A. Di Nola, P. Flondor and I. Leustean, MV modules, J. Algebra, 267 (2003), no. 1, 2140. ##A. Dvurecenskij, On partial addition in pseudo MV algebras, Information technology (Bucharest, 1999), 952960, Inforec, Bucharest, 1999. ##A. Filipoiu, G. Georgescu and , A. Lettieri, Maximal MV algebras, Mathware Soft Comput. 4 (1997), no. 1, 5362. ##F. Forouzesh, E. Eslami and A. Borumand Saeid, On prime Aideals in MV modules, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 76 (2014), no. 3, 181198. ##F. Forouzesh, E. Eslami and A. Borumand Saeid, Radical of Aideals in MV modules, An. Stiint. Univ. Al. I. Cuza Iasi Inform., Accepted. ##F. Forouzesh, Some results in PMV algebras, U.P.B. Sci. Bull., Series A, Submitted. ##A. Iorgulescu, Algebras of logic as BCK algebras, Academy of economic studies Bucharest, Romania, 2008. ##F. Montagna, An algebraic approach to propositional fuzzy logic, J. Logic Lang. Inform. 9 (2000), no. 1, 91124. ##D. Mundici, Interpretation of AFCalgebras in Lukasiewicz sentential calculus, J. Funct. Anal. 65 (1986), no. 1, 1563. ##D. Piciu, Algebras of fuzzy logic, Ed. Universitaria, Craiova, 2007.##]
Existence of solutions of boundary value problems for Caputo fractional differential equations on time scales
2
2
In this paper, we study the boundaryvalue problem of fractional order dynamic equations on time scales, $$ ^c{Delta}^{alpha}u(t)=f(t,u(t)),;;tin [0,1]_{mathbb{T}^{kappa^{2}}}:=J,;;1<alpha<2, $$ $$ u(0)+u^{Delta}(0)=0,;;u(1)+u^{Delta}(1)=0, $$ where $mathbb{T}$ is a general time scale with $0,1in mathbb{T}$, $^c{Delta}^{alpha}$ is the Caputo $Delta$fractional derivative. We investigate the existence and uniqueness of solution for the problem by Banach's fixed point theorem and Schaefer's fixed point theorem. We also discuss the existence of positive solutions of the problem by using the Krasnoselskii theorem.
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247
262


R. A.
Yan
School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, P R China
School of Mathematical Sciences, University
China (P. R. C.)
yanrian89@163.com


S. R.
Sun
School of Mathematical Sciences, University of Jinan, Jinan,
Shandong 250022, P R China
School of Mathematical Sciences, University
China (P. R. C.)
sshrong@163.com


Z. L.
Han
School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, P R China
School of Mathematical Sciences, University
China (P. R. C.)
hanzhenlai@163.com
Fractional differential equation
Time scales
Boundaryvalue problem
Fixedpoint theorem
[R. P. Agarwal and M. Boner, Basic calculus on time scales and some of its applications, Results Math. 35 (1999), no. 12, 322. ##R. P. Agarwal and D. O'Regan and S. Stanek, Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, J. Math. Anal. Appl. 371 (2010), no. 1, 5768. ##R. P. Agarwal, M. Bohner, A. Peterson and D. O'Regan, Dynamic equations on time scales: a survey, Dynamic equations on time scales, J. Comput. Appl. Math. 141 (2002), no. 12, 126. ##G. A. Anastassiou, Principles of delta fractional calculus on time scales and inequalities, Math. Comput. Modelling 52 (2010), no. 34, 556566. ##B. Aulbach and S. Hilger, Linear dynamic processes with inhomogeneous time scale, Nonlinear dynamics and quantum dynamical systems, 920, Math. Res., 59, AkademieVerlag, Berlin, 1990. ##A. Ahmadkhanlu and M. Jahanshahi, On the existence and uniqueness of solution of initial value problem for fractional order differential equations on time scales, Bull. Iranian Math. Soc. 38 (2012), no. 1, 241252. ##D. Baleanu and P. Agarwal, On generalized fractional integral operators and the generalized Gauss hypergeometric functions, Abstr. Appl. Anal. 2014 (2014), Article ID 630840, 8 pages. ##N. Bastos, D. Mozyrska and D. Torres, Fractional derivatives and integrals on time scales via the inverse generalized Laplace transform, Int. J. Math. Comput. 11 (2011) 19. ##M. Benchohra, S. Hamani and S. K. Ntouyas, Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal. 71 (2009), no. 78, 23912396. ##M. Bohner and A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkhauser Boston, Inc., Boston, 2001. ##M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser Boston, Inc., Boston, 2003. ##A. Cabada and G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl. 389 (2012), no. 1, 403411. ##J. Choi and P. Agarwal, Certain fractional integral inequalities involving hypergeometric operators, East Asian Math. J. 30 (2014), no. 3, 283291. ##J. Choi and P. Agarwal, Some new Saigo type fractional integral inequalities and their qanalogues, Abstr. Appl. Anal. 2014 (2014) Article ID 579260. ##M. Elshahed and J. J. Nieto, Nontrivial solutions for a nonlinear multipoint boundary value problem of fractional order, Comput. Math. Appl. 59 (2010), no. 11, 34383443. ##W. Feng, S. Sun, Z. Han and Y. Zhao, Existence of solutions for a singular system of nonlinear fractional differential equations, Comput. Math. Appl. 62 (2011), no. 3, 13701378. ##S. Hilger, Analysis on measure chainsa unified approach to continuous and discrete calculus, Results Math. 18 (1990), no. 12, 1856. ##A. A. Kilbas, H. H. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B. V., Amsterdam, 2006. ##M. A. Krasnoselskii, Positive Solutions of Operator Equations, P. Noordhoff Ltd., Groningen, 1964. ##K. B. Oldham and J. Spanier, The Fractional Calculus, Theory and applications of differentiation and integration to arbitrary orde Academic Press, New YorkLondon, 1974. ##M. Rehman and R. A. Khan, Existence and uniqueness of solutions for multipoint boundary value problems for fractional differential equations, Appl. Math. Lett. 23 (2010), no. 9, 10381044. ##S. Sun, Y. Zhao, Z. Han and Y. Li, The existence of solutions for boundary value problem of fractional hybrid differential equations, Commun. Nonlinear Sci. Numer. Simulat. 17 (2012), no. 12, 49614967. ##S. Sun, Y. Zhao, Z. Han and M. Xu, Uniqueness of positive solutions for boundary value problems of singular fractional differential equations, Inverse Probl. Sci. Eng. 20 (2012), no. 3, 299309. ##S. Sun, Q. Li and Y. Li, Existence and uniqueness of solutions for a coupled system of multiterm nonlinear fractional differential equations, Comput. Math. Appl. 64 (2012), no. 10, 33103320. ##P. A. Williams, Fractional calculus on time scales with taylor's theorem, Fract. Calc. Appl. Anal. 15 (2012), no. 4, 616638. ##G. Wang, P. Agarwal and M. Chand, Certain Gruss type inequalities involving the generalized fractional integral operator, J. Inequal. Appl. 2014 (2014) 8 pages. ##Z. Wei, C. Pang and Y. Ding, Positive solutions of singular Caputo fractional differential equations with integral boundary conditions, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), no. 8, 31483160. ##P. A. Williams, Unifying fractional calculus with time scales, PhD thesis, Department of Mathematics and Statistics, The University of Melbourne, 2012. ##R. Yan, S. Sun, Y. Sun and Z. Han, Boundary value problems for fractional differential equations with nonlocal boundary conditions, Adv. Difference Equ. 2013 (2013) 12 pages. ##S. Zhang, The existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl. 252 (2000), no. 2, 804812. ##S. Zhang, Existence of positive solution for some class of nonlinear fractional differential equations, J. Math. Anal. Appl. 278 (2003), no. 1, 136148. ##X. Zhang and C. Zhu, Cauchy problem for a class of fractional differential equations on time scales, Int. J. Comput. Math. 91 (2014), no. 3, 527538. ##S. Zhang, Positive solutions for boundaryvalue problems of nonlinear fractional differential equations, Electron. J. Differential Equations 36 (2006) 12 pages. ##Y. Zhao, S. Sun, Z. Han and M. Zhang, Positive solutions for boundary value problems of nonlinear fractional differential equations, Appl. Math. Comput. 217 (2011), no. 16, 69506958. ##Y. Zhao, S. Sun, Z. Han and Q. li, Positive solutions to boundary value problems of nonlinear fractional differential equations, Abs. Appl. Anal. 2011 (2011), Article ID 390543, 16 pages. ##Y. Zhao, S. Sun, Z. Han and Q. Li, The existence of multiple positive solutions for bound ary value problems of nonlinear fractional differential equations, Commun. Nonlinear. Sci. Numer. Simulat. 16 (2011), no. 4, 20862097. ##Y. Zhao, S. Sun, Z. Han and Q. Li, Theory of fractional hybrid differential equations, Comput. Math. Appl. 62 (2011), no. 3, 13121324. ##J. Zhao, P. Wang and W. Ge, Existence and nonexistence of positive solutions for a class of third order BVP with integral boundary conditions in Banach spaces, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), no. 1, 402413. ##Y. Zhou, F. Jiao and J. Li, Existence and uniqueness for ptype fractional neutral differential equations, Nonlinear Anal. 71 (2009), no. 78, 27242733.##]
Locally GCD domains and the ring $D+XD_S[X]$
2
2
An integral domain $D$ is called a emph{locally GCD domain} if $D_{M}$ is a GCD domain for every maximal ideal $M$ of $D$. We study some ringtheoretic properties of locally GCD domains. E.g., we show that $D$ is a locally GCD domain if and only if $aDcap bD$ is locally principal for all $0neq a,bin D$, and flat overrings of a locally GCD domain are locally GCD. We also show that the tclass group of a locally GCD domain is just its Picard group. We study when a locally GCD domain is Pr"{u}fer or a generalized GCD domain. We also characterize locally factorial domains as domains $D$ whose minimal prime ideals of a nonzero principal ideal are locally principal and discuss conditions that make them Krull domains. We use the $D+XD_{S}[X]$ construction to give some interesting examples of locally GCD domains that are not GCD domains.
1

263
284


G. W.
Chang
Department of Mathematics Education, Incheon National University,
Incheon 406772, Republic of Korea.
Department of Mathematics Education,
South Korea
whan@incheon.ac.kr


T.
Dumitrescu
Facultatea de Matematica si Informatica, University of Bucharest, 14 Academiei Str., Bucharest, RO 010014, Romania
Facultatea de Matematica si Informatica,
Romania
tiberiu@fmi.unibuc.ro


M.
Zafruhhah
Department of Mathematics, Idaho State University, Pocatello, ID 83209, USA
Department of Mathematics, Idaho
United States of America
mzafrullah@usa.net
Locally GCD domain
generalized GCD domain
$D+XD_S[X]$
[M. M. Ali, Generalized GCD rings IV, Beitrage Algebra Geom. 55 (2014), no. 2, 371 386. ##D. D. Anderson, domains, overrings, and divisorial ideals, Glasg. Math. J. 19 (1978), no. 2, 199203. ##D. D. Anderson and D. F. Anderson, Generalized GCD domains, Comment. Math. Univ. St. Pauli 28 (1980), no. 2, 215221. ##D. D. Anderson, Globalization of some local properties in Krull domains, Proc. Amer. Math. Soc. 85 (1982), no. 2, 141145. ##D. D. Anderson, D. F. Anderson, and M. Zafrullah, Factorization in integral domains, II, J. Algebra 152 (1992), no. 1, 7893. ##D. D. Anderson, D. F. Anderson and M. Zafrullah, A generalization of unique factorization, Boll. Un. Mat. Ital. A (7) 9 (1995), no. 2, 401413. ##D. D. Anderson, D. F. Anderson and M. Zafrullah, The ring D+XDS[X] and tsplitting sets, Arab. J. Sci. Eng. Sect. C Theme Issues 26 (2001), no. 1, 316. ##D. D. Anderson and G. W. Chang and M. Zafrullah, Integral domains of finite tcharacter, J. Algebra 396 (2013) 169183. ##D. D. Anderson and B. G. Kang, PseudoDedekind domains and divisorial ideals in R[X]T , J. Algebra 122 (1989), no. 2, 323336. ##D. D. Anderson and M. Zafrullah, Weakly factorial domains and groups of divisibility, Proc. Amer. Math. Soc. 109 (1990), no. 4, 907913. ##D. D. Anderson and M. Zafrullah, On a theorem of Kaplansky, Boll. Un. Mat. Ital. A (7) 8 (1994), no. 3, 397402. ##D. F. Anderson, The class group and local class group of an integral domain, NonNoetherian commutative ring theory, 3355, Math. Appl., 520, Kluwer Acad. Publ., Dordrecht, 2000. ##J. T. Arnold and R. Matsuda, An almost Krull domain with divisorial height one primes, Canad. Math. Bull. 29 (1986), no. 1, 5053. ##A. Bouvier, Le groupe des classes d'un anneau integre, 107eme Congres National des Societes Savantes, Brest, 1982, fasc. IV, 8592. ##J. Brewer and W. Heinzer, Associated primes of principal ideals, Duke Math. J. 41 (1974) 17. ##P. M. Cohn, Bezout rings and their subrings, Proc. Cambridge Philos. Soc. 64 (1968) 251264. ##D. Costa, J. Mott and M. Zafrullah, The construction D + XDS[X], J. Algebra 53 (1978), no. 2, 423439. ##P. Eakin and J. Silver, Rings which are almost polynomial rings, Trans. Amer. Math. Soc. 174 (1972) 425449. ##M. Fontana, E. Houston and T. Lucas, Factoring ideals in integral domains, Lecture Notes of the Unione Matematica Italiana, 14. Springer, Heidelberg, Bologna, 2013. ##M. Fontana and M. Zafrullah, On vdomains: A survey, Commutative algebraNoetherian and nonNoetherian Perspectives, 145179, Springer, New York, 2011. ##R. Gilmer, Integral domains which are almost Dedekind, Proc. Amer. Math. Soc. 15 (1964) 813818. ##R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, Inc., New York, 1972. ##R. Gilmer and T. Parker, Divisibility properties in semigroup rings, Michigan Math. J. 21 (1974) 6586. ##F. HalterKoch, Ideal Systems  An Introduction to Multiplicative Ideal Theory, Marcel Dekker, Inc., New York, 1998. ##J. R. Hedstrom and E. Houston, Some remarks on staroperations, J. Pure Appl. Algebra 18 (1980), no. 1, 3744. ##W. Heinzer and J. Ohm, An essential ring which is not a vmultiplication ring Canad. J. Math. 25 (1973) 856861. ##E. Houston and M. Zafrullah, Integral domains in which each tideal is divisorial, Michigan Math. J. 35 (1988), no. 2, 291300. ##B. G. Kang, On the converse of a wellknown fact about Krull domains, J. Algebra 124 (1989), no. 2, 284299. ##I. Kaplansky, Commutative Rings, Polygonal Publishing House, New Jersey, 1994. ##M. D. Larsen and P. J. McCarthy, Multiplicative Theory of Ideals, 43, Academic Press, New YorkLondon, 1971. ##S. Malik, J. Mott and M. Zafrullah, On tinvertibility, Comm. Algebra 16 (1988), no. 1, 149170. ##J. Mott and M. Zafrullah, On Prufer vmultiplication domains, Manuscripta Math. 35 (1981), no. 12, 126. ##N. Nakano, Idealtheorie in einem speziellen unendlichen algebraischen Zahlkorper, J. Sci. Hiroshima Univ. Ser. A. 16 (1953) 425439. ##J. Querre, Intersections d'anneaux integres, J. Algebra 43 (1976), no. 1, 5560. ##M. Zafrullah, On a result of Gilmer, J. London Math. Soc. (2) 16 (1977), no. 1, 1920. ##M. Zafrullah, On finite conductor domains, Manuscripta Math. 24 (1978), no. 2, 191203. ##M. Zafrullah, On generalized Dedekind domains, Mathematika 33 (1986), no. 2, 285296. ##M. Zafrullah, On a property of preSchreier domains, Comm. Algebra 15 (1987), no. 9, 18951920. ##M. Zafrullah, The D+XDS[X] construction from GCD domains, J. Pure Appl. Algebra 50 (1988), no. 1, 93107. ##M. Zafrullah, Ascending chain conditions and star operations, Comm. Algebra 17 (1989), no. 6, 15231533. ##M. Zafrullah, Flatness and invertibility of an ideal, Comm. Algebra 18 (1990), no. 7, 21512158. ##M. Zafrullah, Putting tinvertibility to use, NonNoetherian commutative ring theory, 429457, Math. Appl., 520, Kluwer Acad. Publ., Dordrecht, 2000. ##http://people.fas.harvard.edu/char126relaxamathew/chfactorization.pdf.##]
Sufficiency and duality for a nonsmooth vector optimization problem with generalized $alpha$$d_{I}$typeI univexity over cones
2
2
In this paper, using Clarke’s generalized directional derivative and dIinvexity we introduce new concepts of nonsmooth KαdIinvex and generalized type I univex functions over cones for a nonsmooth vector optimization problem with cone constraints. We obtain some sufficient optimality conditions and MondWeir type duality results under the foresaid generalized invexity and type I coneunivexity assumptions.
1

285
295


H.
Jiao
School of Mathematics and Statistics, Yangtze Normal University, Chongqing 408100, P. R. China.
School of Mathematics and Statistics, Yangtze
China (P. R. C.)
jiaohh361@126.com
Vector optimization
Type I univexity
Cones
Optimality
duality
[T. Antczak, Optimality conditions and duality for nondifferentiable multiobjective programming problems involving drtype I functions, J. Comput. Appl. Math. 225 (2009), no. 1, 236250. ##M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming, Theory and Algorithms, third edition, WileyInterscience [John Wiley & Sons], Hoboken, 2006. ##L. Batista dos Santos, R. OsunaGomez, M. A. RojasMedar, and A. RuanLizana, Preinvex functions and weak efficient solutions for some vectorial optimization problem in Banach spaces, Comput. Math. Appl. 48 (2004), no. 56, 885895. ##B. D. Craven, Control and Optimization, Chapman & Hall, London, 1995. ##F. H. Clarke, Optimization and Nonsmooth Analysis, A WileyInterscience Publication, John Wiley & Sons, Inc., New York, 1983. ##M. Hachimi and B. Aghezzaf, Sufficiency and duality in nondifferentiable multiobjective programming involving generalized type I functions, J. Math. Anal. Appl. 319 (2006), no. 1, 110123. ##M. A. Hanson and B. Mond, Necessary and sufficient conditions in constrained optimization, Math. Programming 37 (1987), no. 1, 5158. ##M. A. Hanson, R. Pini, and C. Singh, Multiobjective programming under generalized typeI invexity. J. Math. Anal. Appl. 261 (2001), no. 2, 562577. ##A. Jayswal, On sufficiency and duality in multiobjective programming problem under generalized dVtype I univexity, J. Global Optim. 46 (2010), no. 2, 207216. ##A. Jayswal and R. Kumar, Some nondifferentiable multiobjective programming under generalized dVtypeI univexity, J. Comput. Appl. Math. 229 (2009), no. 1, 175182. ##O. L. Mangasarian, Nonlinear Programming, McGrawHill Book Co., New YorkLondonSydney, 1969. ##S. K. Mishra, S. Y. Wang, and K. K. Lai, Nondifferentiable multiobjective programming under generalized dunivexity, European J. Oper. Res. 160 (2005), no. 1, 218226. ##M. A. Noor, On generalized preinvex functions and monotonicities, JIPAM. J. Inequal. Pure Appl. Math. 5 (2004), no. 4, Article 110, 9 pages. ##H. Slimani and M. S. Radjef, Nondifferentiable multiobjective programming under generalized dI invexity, European J. Oper. Res. 202 (2010), no. 1, 3241. ##S. K. Suneja and M. K. Srivastava, Optimality and duality in nondifferentiable multiobjective optimization involving dtype I and related functions, J. Math. Anal. Appl. 206 (1997), no. 2, 465479. ##S. K. Suneja, S. Khurana and M. Bhatia, Optimality and duality in vector optimization involving generalized type I functions over cones, J. Global Optim. 49 (2011), no. 1, 2335. ##N. D. Yen and P. H. Sach, On locally Lipschitz vectorvalued invex functions, Bull. Austral. Math. Soc. 47 (1993) 259271. ##G. L. Yu and S. Y. Liu, Some vector optimization problems in Banach spaces with generalized convexity, Comput. Math. Appl. 54 (2007), no. 1112, 14031410.##]
A new approach for solving the firstorder linear matrix differential equations
2
2
Abstract. The main contribution of the current paper is to propose a new effective numerical method for solving the firstorder linear matrix differential equations. Properties of the Legendre basis operational matrix of integration together with a collocation method are applied to reduce the problem to a coupled linear matrix equations. Afterwards, an iterative algorithm is examined for solving the obtained coupled linear matrix equations. Numerical experiments are presented to demonstrate the applicably and efficiency of our method.
1

297
314


A.
Golbabai
School of Mathematics, Iran
University of Science and Technology, P.O. Box 1684613114,
Tehran, Iran.
School of Mathematics, Iran
Un
Iran
golbabai@iust.ac.ir


S.
P. A. Beik
School of Mathematics, Iran
University of Science and Technology, P.O. Box 1684613114,
Tehran, Iran
School of Mathematics, Iran
Un
Iran
panjehali@iust.ac.ir


D.
K. Salkuyeh
Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
Faculty of Mathematical Sciences, Univer
Iran
salkuyeh@gmail.com
Linear matrix differential equation
Legendre polynomials
Coupled linear matrix equations
Iterative algorithm
[A. Üsküplü Altnbaşak and M. Demiralp, Solutions to linear matrix ordinary differential equations via minimal, regular, and excessive space extension based universalization, J. Math. Chem. 48 (2010), no. 2, 253265. ##P. W. Atkins and J. C. de Paula, Physical Chemistry, 7th Edition, Oxford University Press, Oxford, 2002. ##S. Barnett, Matrices in Control Theory, Van Nostrand Reinhold Co., LondonNew York Toronto, 1971. ##D. S. Bernstein, Matrix Mathematics: theory, facts, and formulas, Second edition, Princeton University Press, Princeton, 2009. ##A. Canada, P. Drbek, A. Fonda (eds.), Handbook of Differential Equations: Ordinary Differential Equations, ElsevierNorthHolland, Amsterdam, 2006. ##R. Y. Chang and M. L. Wang, Shifted Legendre direct method for variational problems, J. Optim. Theory Appl. 39 (1983), no. 2, 299307. ##T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New YorkLondonParis, 1978. ##E. Defez, A. Hervás, J. Ibanez and M. M. Tung, Numerical solutions of matrix differential models using higherorder matrix splines, Mediterr. J. Math. 9 (2012), no. 4, 865882. ##E. Defez, A. Hervás, L. Soler and M. M. Tung, Numerical solutions of matrix differential models cubic spline II, Math. Comput. Modelling. 46 (2007) 657669. ##E. Defez, L. Soler, A. Hervás and C. Santamaria, Approximating and computing nonlinear matrix differential models, Math. Comput. Modelling. 55 (2012), no. (78), 20122022. ##E. Defez, M. M. Tung, J. J. Ibanez and J. Sastre, Approximating and computing nonlinear matrix differential models, Math. Comput. Modelling. 55 (2012), no. 78, 20122022. ##L. D. Faddeyev, The inverse problem in the quantum theory of scattering, J. Mathematical Phys. 4 (1963), no. 1, 72104. ##T. M. Flett, Differential Analysis, Cambridge University Press, CambridgeNew York, 1980. ##F. FakharIzadi and M. Dehghan, The spectral methods for parabolic Volterra integrodifferential equations, J. Comput. Appl. Math. 235 (2011), no. 14, 40324046. ##G. H. Golub and C. F. V. Loan, Matrix Computations, second ed., The Johns Hopkins University Press, Baltimore, 1989. ##E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, Inc., 1978. ##A. J. Laub, Matrix Analysis for Scientists & Engineers, SIAM, Philadelphia, 2005. ##A. Lotfi, M. Dehghan and S. A. Yousefi, A numerical technique for solving fractional optimal control problems, Comput. Math. Appl. 62 (2011), no. 3, 10551067. ##D. A. McQuarrie, Quantum Chemistry, 2nd edn., University Science Books, California, 2008. ##A. C. Norris, Computational Chemistry, An Introduction to Numerical Methods, John Wiley & Sons, Ltd., Chichester, 1981. ##W. T. Reid, Riccati differential equations, Mathematics in Science and Engineering, 86, Academic Press, New YorkLondon, 1972. ##Y. Saad, Iterative Methods for Sparse linear Systems, PWS Press, New York, 1995. ##J. J. Zhang, A note on the iterative solutions of the general coupled matrix equation, App. Math. Comput. 217 (2011), no. 22, 93809386. ##H. Zheng and W. Han, On some discretization methods for solving a linear matrix ordinary differential equation, J. Math. Chem. 49 (2011), no. 5, 10261041.##]
An analytic solution for a nonlocal initialboundary value problem including a partial differential equation with variable coefficients
2
2
This paper considers a nonlocal initialboundary value problem containing a first order partial differential equation with variable coefficients. At first, the nonselfadjoint spectral problem is derived. Then its adjoint problem is calculated. After that, for the adjoint problem the associated eigenvalues and the subsequent eigenfunctions are determined. Finally the convergence of series solution and the uniqueness of this solution will be proved.
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315
326


M.
Jahanshahi
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran.
Department of Mathematics, Azarbaijan Shahid
Iran
jahanshahi@azaruniv.edu


M.
Darabadi
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
Department of Mathematics, Azarbaijan Shahid
Iran
m.darabadi@azaruni.edu
Partial Differential Equation
Boundary Value Problem
Self Adjoint Problem
NonSelf Adjoint Operators
NonLocalBoundary Conditions
[N. Aliev, M. Jahanshahi and S. M. Hosseini, An analyticalnumerical method for investigation and solving three dimensional steady state NavierStokes equations, II, Int. J. Differ. Equ. Appl. 10 (2005), no. 2, 135145. ##G. Avalishvili and M. Avalishvili, On a nonlocal problem with integral boundary conditions for a multidimensional elliptic equation, Appl. Math. Lett 24 (2011), no. 4, 566571. ##G. Berikelashvili and N. Khomeriki, On the convergence of difference schemes for one nonlocal boundaryvalue problem, Lith. Math. J. 52 (2012), no. 4, 353362. ##A. Burchard and M. Chugunova, On computing the instability index of a nonselfadjoint differential operator associated with coating and rimming ows, SIAM J. Math. Anal. 43 (2011), no. 1, 367388. ##R. Carlson, Adjoint and selfadjoint differential operators on graphs, Electron. J. Differential Equations (1998), no. 6, 10 pages. ##R. Ciupaila, M. Sapagovas and O. Stikonien, Numerical solution of nonlinear elliptic equation with nonlocal condition, Nonlinear Anal. Model. Control 18 (2013), no. 4, 412426. ##G. Evans, J. Blackledge and P. Yardley, Analytic Methods for Partial Differential Equations, Springer Undergraduate Mathematics Series, SpringerVerlag London, Ltd., London, 2000. ##G. Freiling and A. Yarco, Inverse problem for SturmLiouville equation with boundary conditions, J. Inverse Problem 20 (2010). ##M. G. Gasymov and G. S. Guseĭnov, Determination of a diffusion operator from spectral data, Akad. Nauk Azerbaĭdzhan. SSR Dokl. 37 (1981), no. 2, 1923. ##D. G. Gordeziani and G. A. Avalishvili, Timenonlocal problems for Schrodinger type equations, II, Results for specific problems, Differ. Equ. 41 (2005), no. 6, 852859. ##M. Jahanshahi and A. Ahmadkhanloo, The wave equation in nonclassic cases: nonself adjoint with nonlocal and nonperiodic boundary conditions, Iran. J. Math. Sci. Inform. 9 (2014), no. 1, 112, 100. ##M. Jahanshahi, N. Aliev and S. M. Hosseini, An analytic method for investigation and solving twodimensional steady state navierstokes equations I, Southeast Asian Bull. Math. 33 (2009), no. 6, 10751089. ##M. Jahanshahi and N. Aliev, Reduction of linearized BenjaminOno equation to the Schrdinger equation, Int. Math. Forum 2 (2007), no. 11, 543549, ##M. Jahanshahi and M. Darabadi, Existence and uniqueness of solution of a nonselfadjoint initialboundary value problem for partial differential equation with nonlocal boundary conditions, Vietnam J. Math., Accepted. ##M. Jahanshahi and M. Fatehi, Analytic solution for the cauchyriemann equation with nonlocal boundary conditions in the first quarter, Int. J. Pure Appl. Math. 46 (2008), no. 2, 245249, ##M. Jahanshahi and M. Sajjadmanesh, A new method for investigation and recognizing of selfadjoint boundary value problems, Journal of Science (Kharazmi University) 12 (2012), no. 1, 295304. ##T. MyintU and L. Debnath, Linear partial Differential Equations for Scientists and Engineers, Birkhäuser, Boston, 2007.##]
Trivially related lax pairs of the SawadaKotera equation
2
2
We show that a recently introduced Lax pair of the SawadaKotera equation is not a new one but is trivially related to the known old Lax pair. Using the socalled trivial compositions of the old Lax pairs with a differentially constrained arbitrary operators, we give some examples of trivial Lax pairs of KdV and SawadaKotera equations.
1

327
330


D.
Talati
Sama Technical and Vocational Training College, Islamic Azad university, Urmia Branch, Urmia, Iran.
Sama Technical and Vocational Training College,
Iran
talati@eng.ankara.edu.tr
SawadaKotera equation
Lax pair
integrability
[M. Hickman, W. Hereman, J. Larue and U. Göktaş, Scaling invariant Lax pairs of nonlinear evolution equations, Appl. Anal. 91 (2012), no. 2, 381402. ##S. Sakovich, A note on Lax pairs of the SawadaKotera equation, J. Math. (2014), Article ID 906165, 4 pages. ##P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968) 467490. ##R. K. Dodd and J. D. Gibbon, The prolongation structure of a higher order KortewegdeVries equation, Proc. R. Soc. Lond. A 358 (1978), no. 1694, 287296. ##K. Sawada and T. Kotera, A method for finding Nsoliton solutions of the K.d.V. equation and K.d.V.like equation, Prog. Theoret. Phys. 51 (1974) 13551367.##]
On Silverman's conjecture for a family of elliptic curves
2
2
Let $E$ be an elliptic curve over $Bbb{Q}$ with the given Weierstrass equation $ y^2=x^3+ax+b$. If $D$ is a squarefree integer, then let $E^{(D)}$ denote the $D$quadratic twist of $E$ that is given by $E^{(D)}: y^2=x^3+aD^2x+bD^3$. Let $E^{(D)}(Bbb{Q})$ be the group of $Bbb{Q}$rational points of $E^{(D)}$. It is conjectured by J. Silverman that there are infinitely many primes $p$ for which $E^{(p)}(Bbb{Q})$ has positive rank, and there are infinitely many primes $q$ for which $E^{(q)}(Bbb{Q})$ has rank $0$. In this paper, assuming the parity conjecture, we show that for infinitely many primes $p$, the elliptic curve $E_n^{(p)}: y^2=x^3np^2x$ has odd rank and for infinitely many primes $p$, $E_n^{(p)}(Bbb{Q})$ has even rank, where $n$ is a positive integer that can be written as biquadrates sums in two different ways, i.e., $n=u^4+v^4=r^4+s^4$, where $u, v, r, s$ are positive integers such that $gcd(u,v)=gcd(r,s)=1$. More precisely, we prove that: if $n$ can be written in two different ways as biquartic sums and $p$ is prime, then under the assumption of the parity conjecture $E_n^{(p)}(Bbb{Q})$ has odd rank (and so a positive rank) as long as $n$ is odd and $pequiv5, 7pmod{8}$ or $n$ is even and $pequiv1pmod{4}$. In the end, we also compute the ranks of some specific values of $n$ and $p$ explicitly.
1

331
340


K.
Nabardi
Department of
Mathematics, Azarbaijan Shahid Madani University,
Tabriz 5375171379, Iran.
Department of
Mathematics, Azarbaijan Shahid
Iran
k.nabardi@urmia.ac.ir


F.
Izadi
Department of
Mathematics, Azarbaijan Shahid Madani University, P. O. Box 5375171379,
Tabriz , Iran.
Department of
Mathematics, Azarbaijan Shahid
Iran
izadi@azaruniv.edu
Silverman's Conjecture
Elliptic Curve
Quadratic Twist
rank
Parity Conjecture
[B. J. Birch and N. M. Stephens, The parity of the rank of the MordellWeil group, Topology 5 (1966) 295299. ##H. Cohen, Number Theory volume I, Tools and Diophantine Equations, Springer, New York, 2007. ##A. Choudhry, The Diophantine equation A4 + B4 = C4 + D4, Indian J. Pure Appl. Math. 22 (1991), no. 1, 911. ##D. A. Cox, Primes of the Form x2 + ny2: Fermat, Class Field Theory, and Complex Multiplication, Pure and Applied Mathematics, John Wiley & Sons, 2011. ##J. Cremona, MWRANK Program, available from, http://www.maths.nottingham.ac.uk/personal/jec/ftp/progs/. ##L. E. Dickson, History of the Theory of numbers, 2, Reprinted by chelsea, New York, 1971. ##L. Euler, Novi Comm. Acad Petrop., v. 17, p. 64. 1772. ##L. Euler, Nova Acta Acad, Petrop., v. 13, ad annos 17956, 1802, 1778. ##L. Euler, Mem. Acad. Sc. St. Petersb., v.11, 1830. ##G. H. Hardy and E. M. Wright, An introduction to the theory of numbers. Sixth edition. Revised by D. R. HeathBrown and J. H. Silverman, With a foreword by Andrew Wiles, Oxford University Press, Oxford, 2008. ##F. A. Izadi, F. Khoshnam and K. Nabardi, Sums of two biquadrates and elliptic curves of rank ≥4, Math. J. Okayama Univ. 56 (2014) 5163. ##G. H. Hardy, Ramanujan: Twelve Lectures on Subjected by His Life and Work, 3rd ed New York, Chelsea, 1999. ##L. J. Lander, and T. R. Parkin, Equal sums of biquadrates, Math. Comp. 20 (1966) 450451. ##L. J. Lander, T. R. Parkin, and J. L. Selfridge, A Survey of equal sums of like powers, Math. Comp. 21 (1967) 446459. ##P. Monsky, Mock Heegner Points and Congruent Numbers, Math. Z. 204 (1990), no. 1, 4567. ##K. Ono, Twists Of Elliptic Curves, Compositio Math. 106 (1997), no. 3, 349360. ##K. Ono and T. Ono, Quadratic Form And Elliptic Curves III, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), no. 9, 204205. ##P. Serf, Congruent Numbers and Elliptic Curves, Computational Number Theory (Debrecen, 1989), 227238, de Gruyter, Berlin, 1991. ##SAGE Software, Version 4.3.5, http://sagemath.org. ##J. H. Silverman, The Arithmetic of Elliptic curves, SpringerVerlag, New York, 1986. ##J. H. Silverman, A Friendly Introduction to Number Theory, SpringerVerlag, New York, 2001. ##A. Zajta, Solutions of the Diophantine equation A4 + B4 = C4 + D4, Math. Comp. 41 (1983), no. 164, 635659.##]
Every class of $S$acts having a flatness property is closed under directed colimits
2
2
Let $S$ be a monoid. In this paper, we prove every class of $S$acts having a flatness property is closed underdirected colimits, it extends some known results. Furthermore this result implies that every $S$act has a flatness cover if and only if it has a flatness precover.
1

341
351


H.
Qiao
College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, P. R. China.
College of Mathematics and Statistics, Northwest
China (P. R. C.)
qiaohs@nwnu.edu.cn


L.
Wang
College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, P. R. China.
College of Mathematics and Statistics, Northwest
China (P. R. C.)
wanglm@nwnu.edu.cn


X.
Ma
College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, P. R. China.
College of Mathematics and Statistics, Northwest
China (P. R. C.)
maxin263@126.com
Flatness property
colimit
closed
[A. Bailey and J. Renshaw, Covers of acts over monoids and pure epimorphisms, Proc. Edinb. Math. Soc. (2) 57 (2014), no. 3, 589617. ##A. Bailey, J. Renshaw, Covers of acts over monoids II, Semigroup Forum 87 (2013), no. 1, 257274. ##S. BulmanFleming, M. Kilp and V. Laan, Pullbacks and atness properties of acts, Part II, Comm. Algebra 29 (2001), no. 2, 851878. ##A. Golchin, Flatness and Coproducts, Semigroup Forum 72 (2006), no. 3, 433440. ##M. Howie, Fundamentals of Semigroup Theory, London Mathematical Society Monographs, New Series, 12, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995. ##M. Kilp, U. Knauer and A. Mikhalev, Monoids, acts and categories, With applications to wreath products and graphs, A handbook for students and researchers, de Gruyter Expositions in Mathematics, 29, Walter de Gruyter & Co., Berlin, 2000. ##V. Laan, Pullbacks and atness properties of acts, Part I, Comm. Algebra 29 (2001), no. 2, 829850. ##P. Normak, On Equalizerat and Pullbackat acts, Semigroup Forum 36 (1987), no. 3, 293313. ##J. Renshaw, Extension and Amalgamation in Monoids and semigroups, Proc. London Math. Soc. (3) 52 (1986), no. 3, 119141. ##B. Stenstrom, Flatness and localization over monoids, Math. Nachr. 48 (1971) 315334.##]
Partial proof of Graham Higman's conjecture related to coset diagrams
2
2
Graham Higman has defined coset diagrams for PSL(2,ℤ). These diagrams are composed of fragments, and the fragments are further composed of two or more circuits. Q. Mushtaq has proved in 1983 that existence of a certain fragment γ of a coset diagram in a coset diagram is a polynomial f in ℤ[z]. Higman has conjectured that, the polynomials related to the fragments are monic and for a fixed degree, there are finite number of such polynomials. In this paper, we consider a family Ϝ of fragments such that each fragment in Ϝ contains one vertex fixed byF_v [(〖xy〗^(1) )^(s_1 ) (xy)^(s_2 ) (〖xy〗^(1) )^(s_3 ),(xy)^(q_1 ) (〖xy〗^(1) )^(q_2 ) (xy)^(q_3 ) ]where s₁,s₂,s₃,q₁,q₂,q₃∈ℤ⁺, and prove Higman's conjecture for the polynomials obtained from the fragments in Ϝ.
1

353
369


Q.
Mushtaq
Vice Chancellor, The Islamia University of Bahawalpur, Pakistan.
Vice Chancellor, The Islamia University of
Pakistan
pir_qmushtaq@yahoo.com


A.
Razaq
Department of Mathematics, Govt. Post Graduate College Jauharabad, Pakistan.
Department of Mathematics, Govt. Post Graduate
Pakistan
makenqau@gmail.com
Modular group
Coset diagrams
projective line over finite field
[B. Everitt, Alternating quotients of the (3; q; r) triangle groups, Comm. Algebra 26 (1997), no. 6, 18171832. ##G. Higman and Q. Mushtaq, Coset diagrams and relations for PSL(2; Z), Arab Gulf J. Sci. Res. 1 (1983), no. 1, 159164 . ##Q. Mushtaq, A condition for the existence of a fragment of a coset diagram, Quart. J. Math. Oxford Ser. (2) 39 (1988), no. 153, 8195. ##Q. Mushtaq, Coset diagrams for the modular group, D. Phil. Thesis, University of Oxford, Oxford, 1983. ##Q. Mushtaq, Parametrization of all homomorphisms from PGL(2;Z) into PGL(2; q), Comm. Algebra 20 (1992), no. 4, 10231040. ##Q. Mushtaq and G. C. Rota, Alternating groups as quotients of two generator groups, Adv. Math. 96 (1993), no. 1, 113121. ##Q. Mushtaq and H. Servatius, Permutation representation of the symmetry groups of regular hyperbolic tessellations, J. Lond. Math. Soc. (2) 48 (1993), no. 1, 7786. ##Q. Mushtaq and A. Razaq, Equivalent pairs of words and points of connection, Sci. World J. 2014 (2014), Article ID 505496, 8 pages. ##A. Torstensson, Coset diagrams in the study of finitely presented groups with an application to quotients of the modular group, J. Commut. Algebra 2 (2010), no. 4, 501514.##]
Toroidalization of locally toroidal morphisms of 3folds
2
2
A toroidalization of a dominant morphism $varphi: Xto Y$ of algebraic varieties over a field of characteristic zero is a toroidal lifting of $varphi$ obtained by performing sequences of blow ups of nonsingular subvarieties above $X$ and $Y$. We give a proof of toroidalization of locally toroidal morphisms of 3folds.
1

371
405


R.
Ahmadian
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 193955746, Tehran, Iran.
School of Mathematics, Institute for Research
Iran
ahmadian@ipm.ir
Toroidalization
resolution of morphisms
principalization
[D. Abramovich, K. Karu, K. Matsuki and J. Wlodarczyk, Torification and factorization of birational maps, J. Amer. Math. Soc. 15 (2002), no. 3, 531572. ##W. Bruns and J. Herzog, CohenMacaulay Rings, Cambridge University Press, Cambridge, 1993. ##E. Bierstone and P. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997) no. 2, 207302. ##A. Bravo, S. Encinas and O. Villamayor, A simplified proof of desingularization and applications, Rev. Mat. Iberoamericana 21 (2005), no. 2, 349458. ##S. D. Cutkosky, Local monomialization and factorization of morphisms, Asterisque 260 1999. ##S. D. Cutkosky, Monomialization of morphisms from 3folds to surfaces, Lecture Notes in Mathematics, 1786, SpringerVerlag, Berlin, 2002. ##S. D. Cutkosky, Resolution of Singularities, Graduate Studies in Mathematics, 63, American Mathematical Society, Providence, 2004. ##S. D. Cutkosky, Toroidalization of dominant morphisms of 3folds, Mem. Amer. Math. Soc. 190 (2007), no. 890, vi+222 pages. ##S. D. Cutkosky, A simpler proof of toroidalization of morphisms from 3folds to surfaces, Ann. Inst. Fourier 63 (2013), no. 3, 865922. ##S. D. Cutkosky, Introduction to Algebraic Geometry, Preprint. ##S. D. Cutkosky and O. Kascheyeva, Monomialization of strongly prepared morphisms from nonsingular nfolds to surfaces, J. Algebra 275 (2004), no. 1, 275320. ##D. Cox, J. Little and D. O′Shea, Ideals, Varieties, and Algorithms, An Introduction to Computational Algebraic Geometry and Commutative Algebra, Third ed. UTM, Springer, New York, 2007. ##S. D. Cutkosky and O. Piltant, Monomial resolution of morphisms of algebraic surfaces, Comm. Algebra 28 (2000) no. 12, 59355959. ##S. Ensinas and H. Hauser, Strong resolution of singularities in characteristic zero, Comment Math. Helv. 77 (2002), no. 4, 821845. ##K. Hanumanthu, Toroidalization of locally toroidal morphisms from nfolds to surfaces, J. Pure Appl. Algebra 213 (2009), no. 3, 349359. ##H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math. (2) 79 (1964) 109326. ##G. Kempf, F. Knudsen, D. Mumford and B. SaintDonat, Toroidal embeddings I, Lecture Notes in Mathematics, 339, SperingerVerlag, BerlinNew York, 1973. ##B. Teissier, Valuations, deformations and toric geometry, Valuation theory and its applications II, F.V. Kuhlmann, S. Kuhlmann and M. Marshall, editors, Fields Institute Communications 33, Amer. Math. Soc., Providence, 361459. ##O. Zariski, Local uniformization on algebraic varieties, Ann. Math. (2) 41 (1940) 852896.##]
Finite groups with $X$quasipermutable subgroups of prime power order
2
2
Let $H$, $L$ and $X$ be subgroups of a finite group $G$. Then $H$ is said to be $X$permutable with $L$ if for some$xin X$ we have $AL^{x}=L^{x}A$. We say that $H$ is emph{$X$quasipermutable } (emph{$X_{S}$quasipermutable}, respectively) in $G$ provided $G$ has a subgroup $B$ such that $G=N_{G}(H)B$ and $H$ $X$permutes with $B$ and with all subgroups (with all Sylow subgroups, respectively) $V$ of $B$ such that $(H, V)=1$. In this paper, we analyze the influence of $X$quasipermutable and $X_{S}$quasipermutable subgroups on the structure of $G$. Some known results are generalized.
1

407
416


X.
Yi
Department of Mathematics, Zhejiang SciTech University, 310018, Hangzhou, P. R. China.
Department of Mathematics, Zhejiang SciTech
China (P. R. C.)
yxlyixiaolan@163.com


X.
Yang
Department of Mathematics, Zhejiang SciTech University, 310018, Hangzhou, P. R. China.
Department of Mathematics, Zhejiang SciTech
China (P. R. C.)
yangxue0222@126.com
$X$quasipermutable subgroup
Sylow subgroup
$p$soluble group
$p$supersoluble group
$p$nilpotent group
[K. Doerk and T. Hawkes, Finite Soluble Groups, Walter de Gruyter, BerlinNew York, 1992. ##W. Guo, Finite groups with seminormal Sylow subgroups, Acta Math. Sinica, English Series 24 (2008), no. 10, 17511758. ##W. Guo and A. N. Skiba, On FΦ*hypercentral subgroups of finite groups, J. Algebra 372 (2012) 285292. ##W. Guo, A. N. Skiba and K. P. Shum, Xpermutable subgroups of finite groups, Siberian Math. J. 48 (2007), no. 4, 593605. ##W. Guo, K. P. Shum and A. N. Skiba, Xsemipermutable subgroups of finite groups, J. Algebra 315 (2007), no. 1, 3141. ##B. Huppert, Endliche Gruppen I, SpringerVerlag, BerlinNew York, 1967. ##I. M. Isaacs, Finite Group Theory, American Mathematical Society, Providence, 2008. ##I. M. Isaacs, Semipermutable πsubgroups, Arch. Math. (Basel) 102 (2014), no. 1, 16. ##O. H. Kegel, Zur Struktur mehrfach faktorisierter endlicher Gruppenn, Math. Z. 87 (1965) 409434. ##S. Li, Z. Shen, J. Liu and X. Liu, The inuence of SSquasinormality of some subgroups on the structure of finite groups, J. Algebra 319 (2008), no. 10, 42754287. ##V. V. Podgornaya, Seminormal subgroups and supersolubility of finite groupds, Vesti NAN Belarus, Ser. Phys.Math. Sciences 4 (2000) 2225. ##X. Yi and A.N. Skiba, Some new characterizations of PSTgroups, J. Algebra 399 (2014) 3954.##]
The augmented Zagreb index, vertex connectivity and matching number of graphs
2
2
Let $Gamma_{n,kappa}$ be the class of all graphs with $ngeq3$ vertices and $kappageq2$ vertex connectivity. Denote by $Upsilon_{n,beta}$ the family of all connected graphs with $ngeq4$ vertices and matching number $beta$ where $2leqbetaleqlfloorfrac{n}{2}rfloor$. In the classes of graphs $Gamma_{n,kappa}$ and $Upsilon_{n,beta}$, the elements having maximum augmented Zagreb index are determined.
1

417
425


A.
Ali
Department of Mathematics, National University of Computer and
Emerging Sciences, BBlock, Faisal Town, Lahore, Pakistan.
Department of Mathematics, National University
Pakistan
akbarali.maths@gmail.com


A.
Bhatti
Department of Mathematics, National University of Computer and
Emerging Sciences, BBlock, Faisal Town, Lahore, Pakistan.
Department of Mathematics, National University
Pakistan
akhlaq.ahmad@nu.edu.pk


Z.
Raza
Department of Mathematics, National University of Computer
and Emerging Sciences, BBlock, Faisal Town, Lahore, Pakistan.and Department of Mathematics, College of Sciences, University
of Sharjah, Sharjah, United Arab Emirates.
Department of Mathematics, National University
Pakistan
zahid.raza@nu.edu.pk
augmented Zagreb index
vertex connectivity
matching number
spanning subgraph
[A. Ali, A. A. Bhatti and Z. Raza, Further inequalities between vertexdegreebased topological indices, arXiv:1401.7511 [math.CO]. ##A. Ali, Z. Raza and A. A. Bhatti, On the augmented Zagreb index, arXiv:1402.3078 [math.CO]. ##J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier Publishing Co., Inc., New York, 1976. ##J. Chen, J. Liu and X. Guo, Some upper bounds for the atombond connectivity index of graphs, Appl. Math. Lett. 25 (2012), no. 7, 1077{1081. ##K. C. Das, I. Gutman and B. Furtula, On atombond connectivity index, Filomat 26 (2012), no. 4, 733{738. ##J. Devillers and A. T. Balaban (Eds.), Topological Indices and Related Descriptors in QSAR and QSPR, Gordon and Breach, Amsterdam, 1999. ##D. Dimitrov, Efficient computation of trees with minimal atombond connectivity index, Appl. Math. Comput. 224 (2013), no. 1, 663{670. ##E. Estrada, Atombond connectivity and the energetic of branched alkanes, Chem. Phys. Lett. 463 (2008), no. 6, 422{425. ##E. Estrada, L. Torres, L. Rodrهguez and I. Gutman, An atombond connectivity index: modelling the enthalpy of formation of alkanes, Indian J. Chem. A 37 (1998), no. 10, 849{855. ##B. Furtula, I. Gutman, M. Ivanovic and D. Vukicevic, Computer search for trees with minimal ABC index, Appl. Math. Comput. 219 (2012), no. 2, 767{772. ##B. Furtula, A. Graovac and D. Vukicevic, Augmented Zagreb index, J. Math. Chem. 48 (2010), no. 2, 370{380. ##I. Gutman, B. Furtula, M. B. Ahmadi, S. A. Hosseini, P. S. Nowbandegani and M. Zarrinderakht, The ABC index conundrum, Filomat 27 (2013), no. 6, 1075{1083. ##I. Gutman and B. Furtula (Eds.), Novel Molecular Structure DescriptorsTheory and Applications vols. III, Univ. Kragujevac, Kragujevac, 2010. ##I. Gutman and J. Tosovic, Testing the quality of molecular structure descriptors: Vertex degreebased topological indices, J. Serb. Chem. Soc. 78 (2013), no. 6, 805{810. ##I. Gutman, J. Tosovic, S. Radenkovic and S. Markovic, On atombond connectivity index and its chemical applicability, Indian J. Chem. A 51 (2012), no. 5, 690{694. ##F. Harary, Graph Theory, AddisonWesley, Philippines, 1969. ##S. A. Hosseini, M. B. Ahmadi and I. Gutman, Kragujevac trees with minimal atombond connectivity index, MATCH Commun. Math. Comput. Chem. 71 (2014), no. 1, 5{20. ##Y. Huang, B. Liu and L. Gan, Augmented Zagreb index of connected graphs, MATCH Commun. Math. Comput. Chem. 67 (2012), no. 2, 483494. ##W. Karcher and J. Devillers, Practical Applications of Quantitative StructureActivity Relationships (QSAR) in Environmental Chemistry and Toxicology, Springer, Heidelberg, 1990. ##W. Lin, T. Gao, Q. Chen and X. Lin, On the minimal ABC index of connected graphs with given degree sequence, MATCH Commun. Math. Comput. Chem. 69 (2013), no. 3, 571578. ##L. Lovasz and M. D. Plummer, Matching theory [M], Ann. Discrete Math. 29, Amsterdam, 1986. ##R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, WileyVCH, Weinheim, 2000. ##R. Todeschini and V. Consonni, Molecular Descriptors for Chemoinformatics, WileyVCH, Weinheim, 2009. ##D.Wang, Y. Huang and B. Liu, Bounds on augmented Zagreb index, MATCH Commun. Math. Comput. Chem. 68 (2012), no. 1, 209216.##]
The unit sum number of Baer rings
2
2
In this paper we prove that each element of any regular Baer ring is a sum of two units if no factor ring of $R$ is isomorphic to $Z_2$ and we characterize regular Baer rings with unit sum numbers $omega$ and $infty$. Then as an application, we discuss the unit sum number of some classes of group rings.
1

427
434


N.
Ashrafi
Semnan UniversityFaculty of Mathematics, Statistics and
Computer Science,
Semnan University, Semnan, Iran.
Semnan UniversityFaculty of Mathematics,
Iran
nashrafi@semnan.ac.ir


N.
Pouyan
Faculty of Mathematics, Statistics and Computer Science,
Semnan
University, Semnan, Iran.
Faculty of Mathematics, Statistics and Computer
Iran
neda.pouyan@gmail.com
unit sum number
regular Baer ring
$pi$regular Baer ring
right perpetual ideal
[N. Ashrafi, The unit sum number of some projective modules, Glasg. Math. J. 50 (2008), no. 1, 7174. ##N. Ashrafi and N. Pouyan, The unit sum number of discrete modules. Bull. Iranian Math. Soc. 37 (2011), no. 4, 243249. ##N. Ashrafi and P. Vamos, On the unit sum number of some rings, Q. J. Math. 56 (2005), no. 1, 112. ##S. K. Berberian, Baer Rings and Baer *rings, The University of Texas, Austin, 1988. ##I. G. Connell, On the group ring, Canad. J. Math. 15 (1963) 650685. ##B. Goldsmith, S. Pabst and A. Scott, Unit sum number of rings and modules, Qcuart. J. Math. Oxford (2) 49 (1998), no. 195, 331344. ##X.J. Guo and K.P. Shum, Reduced p.p.rings without identity, Int. J. Math. Math. Sci. 2006 (2006), Article ID 93890, 7 pages. ##X. J. Guo and K. P. Shum, Baer semisimple modules and Baer rings, Algebra Discrete Math. 2008 (2008) no. 2, 4249. ##D. Handelman, Perspectivity and cancellation in regular rings, J. Algebra 48 (1977), no. 1, 116. ##M. Henriksen, Two classes of rings generated by their units, J. Algebra 31 (1974) 182193. ##I. Kaplansky, Topological representation of algebras,II, Trans. Amer. Math. Soc. 68 (1950), 6275. ##I. Kaplansky, Rings of operators, W. A. Benjamin, Inc., New YorkAmsterdam, 1968. ##D. Khurana and A. K. Srivastava, Right Selfinjective Rings in Which Each Element is Sum of Two Units, J. Algebra Appl. 6 (2007), no. 2, 281286. ##D. Khurana and S. Ashish, Unit sum numbers of right selfinjective rings, Bull. Austral. Math. Soc. 75 (2007), no. 3, 355360. ##T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, 189, SpringerVerlag, New York, 1999. ##J. Y. Kim and J. K. Park, On Regular Baer rings, Trends in Math. 1 (1998), no. 1, 3740. ##S. H. Mohamed and B. J. Muller, Continuous and Discrete Modules, London Mathematical Society Lecture Note Series, 147, Cambridge University Press, Cambridge, 1990. ##S. T. Rizvi and C. S. Roman, Baer and quasiBaer modules, Comm. Algebra 32 (2004), no. 1, 103123. ##L. A. Skornyakov, Complemented Modular Lattices and Regular Rings, Oliver & Boyd, EdinburghLondon, 1964. ##P. Vamos, 2Good Rings, Q. J. Math. 56 (2005), no. 3, 417430. ##K. G.Wolfson, An ideal theoretic characterization of the ring of all linear transformation, Amer. J. Math. 75 (1953) 358386. ##Z. Yi and Y. Q. Zhou, Baer and quasiBaer properties of group rings, J. Aust. Math. Soc. 83 (2007), no. 2, 285296. ##D. Zelinsky, Every linear transformation is a sum of nonsingular ones, Proc. Amer. Math. Soc. 5 (1954) 627630.##]
Existence of ground states for approximately inner twoparameter $C_0$groups on $C^*$algebras
2
2
In this paper, we generalize the definitions of approximately inner $C_0$groups and their ground states to the two parameter case and study necessary and sufficient conditions for a state to be ground state. Also we prove that any approximately inner two parameter $C_0$group must have at least one ground state. Finally some applications are given.
1

435
446


R.
Abazari
Department of
Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran.
Department of
Mathematics, Mashhad Branch,
Iran
rasoolabazari@gmail.com


A.
Niknam
Department of
Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran.
Department of
Mathematics, Mashhad Branch,
Iran
niknam@um.ac.ir
Twoparameter group
Approximately inner dynamical system
Tensor product
Ground state
[R. Abazari, A. Niknam and M. Hassani, Approximately Inner Twoparameter C0group of Tensor Product of C*algebras, Aust. J. Basic & Appl. Sci. 5 (2011), no. 9, 21202126. ##E. M. Alfsen and F. W. Shultz, State spaces of operator algebras: Basic theory, orientations, and C*products, Bull. Amer. Math. Soc. 41 (2004), no. 4, 535539. ##V. Castern, Generators of strongly continuous semigroups, Pitman Advanced Publishing, Research Note, 1985. ##N. Dunford and I. E. Segal, Semigroups of operators and the Weierstrass theorem, Bull. Amer. Math. Soc. 52 (1946) 911914. ##K. Engel and R. Nagel, OneParameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, SpringerVerlag, New York, 2000. ##B. Helffer, Spectral theory and its applications. Cambridge Studies in Advanced Mathematics, 139. Cambridge University Press, Cambridge, 2013. ##E. Hille and R. S. Phillips, Functional analysis and semigroups, Third printing of the revised edition of 1957. American Mathematical Society Colloquium Publications, 31, Amer. Math. Soc., Providence, 1974. ##O. A. Ivanova, Certain theorems on an nparametric semigroup of bounded linear operators and their application in the theory of functions (Russian), Toer. Funkcii Funkcional. Anal. i Prilozeu, Vyp 2 (1966) 3441. ##M. Janfada, On twoparameter regularized semigroups and the Cauchy problem, Abstr. Appl. Anal. (2009), Article ID 415847, 15 pages. ##M. Janfada and A. Niknam, Two parameter Dynamical system and application, J. Sci. Islam. Repub. Iran 15 (2004), no. 2, 163169, 192. ##M. Khanehgir, M. Janfada and A. Niknam, Inhomogeneous twoparameter abstract Cauchy problem, Bull. Iranian Math. Soc. (2005), no. 2, 7181, 88. ##M. Khanehgir, M. Janfada and A. Niknam, Twoparameter integrated semigroups and twoparameter abstract Cauchy problems, J. Inst. Math. Comput. Sci. Math. Ser. 18 (2005), no. 1, 112. ##A. G. Miamee and H. Salehi, Harmonizability, V boundedness and stationary dilation of stochastic processes, Indiana Univ. Math. J. 27 (1978), no. 1, 3750. ##M. A. Naimark, Normed Rings, Noordhoff N. V., Groningen, 1964. ##A. Niknam, Infinitesimal generators of C*Algebras, Potential Anal. 6 (1997), no. 1, 19. ##A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, SpringerVerlag, New York, 1983. ##R. T. Powers, S. Sakai, Existence of ground states and KMS states for approximately inner dynamics, Comm. Math. Phys. 39 (1975) 273288. ##S. Sakai, C*algebra and W*algebra, SpringerVerlag, New YorkHeidelberg, 1971. ##S. Sakai, Operator Algebras in Dynamical Systems, Cambridge University Press, Cambridge, 1991. ##L. Schwartz, Theorie des distrbutions, Paris: Hermann. Part I (1957), part II (1959).##]
Remarks on microperiodic multifunctions
2
2
It is well known that a microperiodic function mapping a topological group into reals, which is continuous at some point is constant. We introduce the notion of a microperiodic multifunction, defined on a topological group with values in a metric space, and study regularity conditions implying an analogous result. We deal with Vietoris and Hausdorff continuity concepts.Stability of microperiodic multifunctions is considered, namely we show that an approximately microperiodic multifunction is close to a constant one, provided it is continuous at some point. As a consequence we obtain stability result for an approximately microperiodic singlevalued function.
1

447
459


J.
Olko
AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30059 Kraków, Poland.
AGH University of Science and Technology,
Poland
olko@agh.edu.pl
multifunction
microperiodic function
functional inequality
functional inclusion
[T. Aoki, On the stability of the linear transformation in Banach spaces, Math. Soc. Japan 2 (1950) 6466. ##J. P. Aubin and H. Frankowska, SetValued Analysis, Birkhauser, Boston, Basel, Berlin, 1990. ##C. Berge, Topological Spaces: Including a Treatment of MultiValued Functions, Vector Spaces and Convexity, Dover Books, 2010. ##N. BrillouetBelluot, J. Brzdęk and K. Cieplinski, On some recent developments in Ulam's type stability Abstr. Appl. Anal. (2012), Article ID 716936, 41 pages. ##J. Brzdęk, On Functions Satisfying some Inequalities, Abh. Math. Sem. Hamburg 63 (1993) 277281. ##J. Brzdęk, Generalizations of some results concerning microperiodic mappings, Manuscripta Math. 121 (2006), no. 2, 265276. ##J. Brzdęk, On approximately microperiodic mappings, Acta. Math. Hungar. 117 (2007), no. 12, 179186. ##A. Bahyrycz and J. Brzdęk, Remarks on some systems of two simultaneous functional inequalities, Ann. Univ. Sci. Budapest. Sect. Comput. 40 (2013) 8193. ##C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Math. 580, Springer, BerlinNew York, 1977. ##S. Hu and N. S. Papageorgiu, Handbook of Multivalued Analysis, Vol. I: Theory, Mathematics and Its Applications, 419, Kluwer Academic Publishers, Dordrecht, Boston, London, 1977. ##D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A 27 (1941) 222224. ##D. H. Hyers, G. Isac and T. M. Rassias, Stability of functional equations in several variables, Progress in Nonlinear Differential Equations and their Applications, 34, Birkäuser Boston, Inc., Boston, 1998. ##S. M. Jung, HyersUlamRassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York, 2011. ##D. Krassowska, J. Matkowski, A pair of functional inequalities of iterative type related to a Cauchy functional equation, In: (Th.M. Rassias ed.) Functional Equations, Inequalities and Applications, pp. 7389, Kluwer Academic Publishers, Dordrecht, 2003. ##D. Krassowska and J. Matkowski, A pair of linear functional inequalities and a characterization of Lpnorm, Ann. Polon. Math. 85 (2005), no. 1, 111. ##D. Krassowska and J. Matkowski, A simultaneous system of functional inequalities and mappings which are weakly of a constant sign, JIPAM J. Inequal. Pure Appl. Math. 8 (2007), no. 2, 9 pages. ##D. Krassowska, T. Ma lolepszy, J. Matkowski, A pair of functional inequalities characterizing polynomials and Bernoulli numbers, Aequationes Math. 75 (2008), no. 3, 276288. ##M. Kuczma, Functional Equations in a Single Variable, Monograe Matematyczne, Tom 46 Pastwowe Wydawnictwo Naukowe, Warsaw, 1968. ##M. Kuczma, Note on microperiodic functions, Rad. Mat. 5 (1989), no. 1, 127140. ##M. Laczkovich, Operators commuting with translations, and systems of difference equations, Colloq. Math. 80 (1999), no. 1, 122. ##P. Montel, Sur le proprietes periodiques des fonctions, C.R. Acad. Sci. Paris 251 (1960) 21112112. ##T. Popoviciu, Remarques sur la defnition fonctionnelle d'un polynome d'une variable relle, Mathematica, Cluj 12 (1936) 512. ##T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297300.##]
On cycles in intersection graphs of rings
2
2
Let $R$ be a commutative ring with nonzero identity. We describe all $C_3$ and $C_4$free intersection graph of nontrivial ideals of $R$ as well as $C_n$free intersection graph when $R$ is a reduced ring. Also, we shall describe all complete, regular and $n$clawfree intersection graphs. Finally, we shall prove that almost all Artin rings $R$ have Hamiltonian intersection graphs. We show that such graphs are indeed pancyclic.
1

461
470


N.
Hoseini
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran.
Department of Pure Mathematics, Ferdowsi
Iran
nesa.hoseini@gmail.com


A.
Erfanian
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran.
Department of Pure Mathematics, Ferdowsi
Iran
erfanian@math.um.ac.ir


A.
Azimi
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran.
Department of Pure Mathematics, Ferdowsi
Iran
ali.azimi61@gmail.com


M.
Farrokhi D. G.
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran.
Department of Pure Mathematics, Ferdowsi
Japan
m.farrokhi.d.g@gmail.com
Intersection graph
cycle
claw
Hamiltonian
pancyclic
[A. Akbari, R. Nikandish and M. J. Nikmehr, Some results on the intersection graphs of ideals of rings, J. Algebra Appl. 12 (2013), no. 4, 13 pages. ##J. Bos_ak, The graphs of semigroups, Theory of Graphs and Application, 119125, Publ. House Czechoslovak Acad. Sci., Prague, 1964. ##I. Chakrabarty, S. Ghosh, T. K. Mukherjee and M. K. Sen, Intersection graphs of ideals of rings, Discrete Math. 309 (2009), no. 17, 53815392. ##B. Cs_ak_eany and G. Poll_ak, The graph of subgroups of a finite group, Czechoslovak Math. J. 19 (1969) 241247. ##M. Herzog, P. Longobardi and M. Maj, On a graph related to the maximal subgroups of a group, Bull. Aust. Math. Soc. 81 (2010), no. 2, 317328. ##S. H. Jafari and N. Jafari Rad, Domination in the intersection graphs of rings and modules, Ital. J. Pure Appl. Math. 28 (2011) 1922. ##S. H. Jafari and N. Jafari Rad, Planarity of intersection graphs of ideals of rings, Int. Electron. J. Algebra 8 (2010) 161166. ##B. R. Macdonald, Finite Rings with Identity, Marcel Dekker, Inc., New York, 1974. ##T. A. McKee and F. R. McMorris, Topic in Intersection Graph Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1999. ##R. Shen, Intersection graphs of subgroups of finite groups, Czech. Math. J. 60(135) (2010), no. 4, 945950. ##B. Zelinka, Intersection graphs of finite abelian groups, Czechoslovak Math. J. 25 (1975) 171174.##]
On linear preservers of sgutmajorization on $textbf{M}_{n,m}$
2
2
Let $textbf{M}_{n,m}$ be the set of $n$by$m$ matrices with entries in the field of real numbers. A matrix $R$ in $textbf{M}_{n}=textbf{M}_{n,n}$ is a generalized row substochastic matrix (grow substochastic, for short) if $Releq e$, where $e=(1,1,ldots,1)^t$. For $X,$ $Y in textbf{M}_{n,m}$, $X$ is said to be sgutmajorized by $Y$ (denoted by $X prec_{sgut} Y$) if there exists an $n$by$n$ upper triangular grow substochastic matrix $R$ such that $X=RY$. This paper characterizes all linear preservers and strong linear preservers of $prec_{sgut}$ on $mathbb{R}^{n}$ and $textbf{M}_{n,m}$ respectively.
1

471
481


A.
Ilkhanizadeh Manesh
Department of
Mathematics, ValieAsr
University of Rafsanjan, P.O. Box 7713936417, Rafsanjan, Iran.
Department of
Mathematics, ValieAsr
University
Iran
a.ilkhani@vru.ac.ir
Linear preserver
Strong linear preserver
grow substochastic matrices
sgut majorization
[T. Ando, Majorization, doubly stochastic matrices, and comparison of eigenvalues, Linear Algebra Appl. 118 (1989) 163248. ##A. Armandnejad, Right gwmajorization on Mn;m, Bull. Iranian math. Soc. 35 (2009), no. 2, 6976. ##A. Armandnejad and H. Heydari, Linear preserving gdmajorization functions from Mn;m to Mn;k, Bull. Iranian math. Soc. 37 (2011), no. 1, 215224. ##A. Armandnejad and A. Ilkhanizadeh Manesh, Gutmajorization and its linear preservers, Electron. J. Linear Algebra 23 (2012) 646654. ##A. Armandnejad and A. Salemi, On linear preservers of lgwmajorization on Mn;m, Bull. Malays. Math. Soc. (2) 35 (2012), no. 3, 755764. ##A. Armandnejad and A. Salemi, The structure of linear preservers of gs majorization, Bull. Iranian Math. Soc. 32 (2006), no. 2, 3142. ##H. Chiang and C. K. Li, Generalized doubly stochastic matrices and linear preservers, Linear Multilinear Algebra 53 (2005), no. 1, 111. ##A. M. Hasani and M. Radjabalipour, On linear preservers of (right) matrix majorization, Linear Algebra Appl. 423 (2007), no. 23, 255261. ##A. M. Hasani and M. Radjabalipour, The structure of linear operators strongly preserving majorizations of matrices, Electron. J. Linear Algebra 15 (2006) 260268. ##A. Ilkhanizadeh Manesh, Linear functions preserving sutmajorization on Rn, Iran. J. Math. Sci. Inform., (submission). ##A. Ilkhanizadeh Manesh, Right gutmajorization on Mn;m, Electron. J. Linear Algebra, Accepted. ##A. Ilkhanizadeh Manesh and A. Armandnejad, Utmajorization on Rn and its Linear Preservers, Operator theory, operator algebras and applications, 253259, Oper. Theory Adv. Appl., 242, BirkhäuserSpringer, Basel, 2014. ##A. W. Marshall, I. Olkin, and B. C. Arnold, Inequalities: Theory of majorization and its applications, Springer, New York, 2011. ##B. Y. Wang, Foundations of Majorization Inequalities, Beijing Normal Univ. Press, Beijing, 1990.##]
Examples of nonquasicommutative semigroups decomposed into unions of groups
2
2
Decomposability of an algebraic structure into the union of its substructures goes back to G. Scorza's Theorem of 1926 for groups. An analogue of this theorem for rings has been recently studied by A. Lucchini in 2012. On the study of this problem for nongroup semigroups, the first attempt is due to Clifford's work of 1961 for the regular semigroups. Since then, N.P. Mukherjee in 1972 studied the decomposition of quasicommutative semigroups where, he proved that: a regular quasicommutative semigroup is decomposable into the union of groups. The converse of this result is a natural question. Obviously, if a semigroup $S$ is decomposable into a union of groups then $S$ is regular so, the aim of this paper is to give examples of nonquasicommutative semigroups which are decomposable into the disjoint unions of groups. Our examples are the semigroups presented by the following presentations: $$pi_1 =langle a,bmid a^{n+1}=a, b^3=b, ba=a^{n1}brangle,~(ngeq 3),$$ $$pi_2 =langle a,bmid a^{1+p^alpha}=a, b^{1+p^beta}=b, ab=ba^{1+p^{alphagamma}}rangle$$where, $p$ is an odd prime, $alpha, beta$ and $gamma$ are integers such that $alpha geq 2gamma$, $beta geq gamma geq 1$ and $alpha +beta > 3$.
1

483
487


N.
Hosseinzadeh
Department of
Mathematics, Tehran Science and Research Branch, Islamic Azad
University, P.O. Box 14515/1775, Tehran, Iran.
Department of
Mathematics, Tehran Science
Iran
narges.hosseinzadeh@gmail.com


H.
Doostie
Department of
Mathematics, Tehran Science and Research Branch, Islamic Azad
University, P.O. Box 14515/1775, Tehran, Iran.
Department of
Mathematics, Tehran Science
Iran
doostih@khu.ac.ir
quasicommutative semigroups
finitely presented semigroups
decomposition
[A. Arjomandfar, C.M. Campbell and H. Doostie, Semigroups related to certain group presentations, Semigroup Forum 85 (2012), no. 3, 533539. ##C. M. Campbell, E. F. Robertson, N. Ruskuc and R. M. Thomas, Semigroup and group presentations, Bull. London Math. Soc. 27 (1995), no. 1, 4650. ##A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, I, Amer. Math. Soc. Surveys 7, Providence, 1961. ##J. Howie, Fundamentals of Semigroup Theory, London Mathematical Society, New Series, Oxford University Press, New York, 1995. ##A. Lucchini and A. Maroti, Rings as the union of proper subrings, Algebr. Represent. Theory 15 (2012), no. 6, 10351047. ##N. P. Mukherjee, Quasicommutative semigroups I, Czechoslovak Math. J. 22(97) (1972) 449453. ##E. F. Robertson and Y. Ünlü, On semigroup presentations, Proc. Edinburgh Math. Soc. (2) 36 (1993), no. 1, 5568. ##G. Scorza, I gruppi che possono pensarsi come somma di tre loro sottogruppi, Boll. Un. Mat. Ital. 5 (1926) 216218.##]
Pseudo Ricci symmetric real hypersurfaces of a complex projective space
2
2
Pseudo Ricci symmetric real hypersurfaces of a complex projective space are classified and it is proved that there are no pseudo Ricci symmetric real hypersurfaces of the complex projective space CPn for which the vector field ξ from the almost contact metric structure (φ, ξ, η, g) is a principal curvature vector field.
1

489
497


S. k.
Hui
Department of Mathematics, Sidho Kanho Birsha University, Purulia723104, West Bengal, India.newline
Department of Mathematics, Bankura University, Bankura722155, West Bengal, India.
Department of Mathematics, Sidho Kanho Birsha
India
shyamal_hui@yahoo.co.in


Y.
Matsuyama
Department of Mathematics, Chuo University, Faculty of Science and Engineering, 11327 Kasuga, Bunkyoku, Tokyo 1128551, Japan.
Department of Mathematics, Chuo University,
Japan
matuyama@math.chuou.ac.jp
real hypersurface
complex projective space
pseudo Ricci symmetric
[K. Arslan, R. Ezentas, C. Murathan and C. Ozgur, On pseudo Ricci symmetric manifolds, Balkan J. Geom. and Appl. 6 (2001), no. 2, 15. ##A. Bejancu and S. Deshmukh, Real hypersurfaces of CPn with nonnegative Ricci curvature, Proc. Amer. Math. Soc. 124 (1996), no. 1, 269274. ##T. E. Cecil and P. J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans Amer. Math. Soc. 269 (1982), no. 2, 481499. ##M. C. Chaki, On pseudo Ricci symmetric manifolds, Bulgar. J. Phys. 15 (1988), no. 6, 526531. ##M. C. Chaki and S. K. Saha, On pseudoprojective Ricci symmetric manifolds, Bulgar. J. Phys. 21 (1994), no. 12, 17. ##J. T. Cho and U.Hang Ki, Real hypersurfaces of a complex projective space in terms of the Jacobi operators, Acta Math. Hungar. 80 (1998), no. 12, 155167. ##S. Deshmukh, Real hypersurfaces of a complex projective space, Proc. Indian Acad. Math. Sci. 121 (2011), no. 2, 171179. ##R. Deszcz, On Riccipseudosymmetric warped products, Demonstratio Math. 22 (1989), no. 4, 10531065. ##T. Hamada, On real hypersurfaces of a complex projective space with ηrecurrent second fundamental tensor, Nihonkai Math. J. 6 (1995), no. 2, 153163. ##T. Hamada, On real hypersurfaces of a complex projective space with recurrent second fundamental tensor, J. Ramanujan Math. Soc. 11 (1996), no. 2, 103107. ##T. Hamada, On real hypersurfaces of a complex projective space with recurrent Ricci tensor, Glasg. Math. J. 41 (1999), no. 3, 297302. ##S. K. Hui and Y. Matsuyama, On real hypersurfaces of a complex projective space with pseudo parallel second fundamental tensor, Kobe J. Math., Accepted. ##K. Ikuta, Real hypersurfaces of a complex projective space, J. Korean Math. Soc. 36 (1999), no. 4, 725736. ##U.Hang Ki, Real hypersurfaces with parallel Ricci tensor of a complex space form, Tsukuba J. Math. 13 (1989), no. 1, 7381. ##M. Kimura, Real hypersurfaces of a complex projective space, Bull. Austral. Math. Soc. 33 (1986) no. 3, 383387. ##M. Kimura, Real hypersurfaces and complex submanifolds in a complex projective space, Trans. Amer. Math. Soc. 296 (1986), no. 1, 137149. ##M. Kimura, Sectional curvatures of holomorphic planes on a real hypersurface in CPn, Math. Ann. 27 (1987), no. 3, 487497. ##M. Kimura, Some real hypersurfaces of a complex projective space, Saitama Math. J. 5 (1987), 15. correction in 10 (1992) 3334. ##M. Kimura and S. Maeda, On real hypersurfaces of a complex projective space, Math. Z. 202 (1989), no. 3, 299311. ##S. Maeda, Real hypersurfaces of complex projective space, Math. Ann. 263 (1983), no. 4, 473478. ##S. Maeda, Ricci tensors of real hypersurfaces in a complex projective space, Proc. Amer. Math. Soc. 122 (1994), no. 4, 12291235. ##Y. Maeda, On real hypersurfaces of a complex projective space, J. Math. Soc. Japan 28 (1976), no. 3, 529540. ##Y. Matsuyama, A characterization of real hypersurfaces in complex projective space, J. Institute Sci. and Eng., Chuo Univ. 2 (1996) 1113. ##Y. Matsuyama, A characterization of real hypersurfaces in complex projective space II, J. Institute Sci. and Eng., Chuo Univ. 3 (1997) 13. ##Y. Matsuyama, A characterization of real hypersurfaces in complex projective space III, Yokohama Math. J. 46 (1999) 119126. ##M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212 (1975) 355364. ##F. Ozen, On pseudo Mprojective Ricci symmetric manifolds, Int. J. Pure Appl. Math. 72 (2011), no. 2, 249258. ##E. M. Patterson, Some theorems on Riccirecurrent spaces, J. London Math. Soc. 27 (1952) 287295. ##J. D. Perez and F. G. Santos, On the Lie derivative of structure Jacobi operator of real hypersurfaces in complex projective space, Publ. Math. Debrecen 66 (2005), no. 34, 269282. ##J. D. Perez and F. G. Santos, Real hypersurfaces in complex projective space with recurrent structure Jacobi operator, Differential Geom. Appl. 26 (2008), no. 2, 218223. ##J. D. Perez, F. G. Santos and Y. J. Suh, Real hypersurfaces in complex projective space whose structure Jacobi operator is Lie ζparallel, Differential Geom. Appl. 22 (2005), no. 2, 181188. ##J. D. Perez, F. G. Santos and Y. J. Suh, Real hypersurfaces in complex projective space whose structure Jacobi operator is Dparallel, Bull. Belg. Math. Soc. Simon Stevin 13 (2006), no. 3, 459469. ##Z. I. Szabo, Structure theorems on Riemannian spaces satisfying R(X; Y ) .R = 0. I. The local version, J. Differential Geom. 17 (1982), no. 4, 531582. ##R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10 (1973) 495506. ##R. Takagi, Real hypersurfaces in a complex projective space with constant principal curvatures, J. Math. Soc. Japan 27 (1975) 4353. ##R. Takagi, Real hypersurfaces in complex projective space with constant principal curvatures II, J. Math. Soc. Japan 27 (1975), no. 4, 507516. ##Q. M.Wang, Real hypersurfaces with constant principal curvatures in complex projective space I, Sci. Sinica Ser. A 26 (1983), no. 10, 10171024.##]