Bulletin of Iranian Mathematical Society
Volume:42 Issue: 2, 2016

ýIn this paperý, ýwe introduce the notion of the radical of a PMV -algebra A and we charactrize radical A via elements of A ý. ýAlsoý, ýwe introduce the notion of the radical of a⋅ -ideal in PMV -algebrasý. ýSeveral characterizations of this radical is givený. ýWe define the notion of a semimaximal ⋅ ⋅-ideal in a PMV-algebraý. ýFinally we show that A/I has no nilpotent elements if and only if I is a semi-maximal ⋅ ⋅ -ideal of A.

ýIn this paperý, ýwe study the boundary-value problem of fractionalý ýorder dynamic equations on time scalesý,
ý
ýýc Δ α u(t)=f(t,u(t)),t∈ýý[0,1] T κ 2 :=J,1

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 ring-theoretic properties of locally GCD domains. E.g., we show that is a locally GCD domain if and only if aDcapbD is locally principal for all 0neqa,binD , and flat overrings of a locally GCD domain are locally GCD. We also show that the t-class 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 S [X] construction to give some interesting examples of locally GCD domains that are not GCD domains.

In this paper, using Clarke’s generalized directional derivative and dI-invexity we introduce new concepts of nonsmooth K-α-dI-invex and generalized type I univex functions over cones for a nonsmooth vector optimization problem with cone constraints. We obtain some sufficient optimality conditions and Mond-Weir type duality results under the foresaid generalized invexity and type I cone-univexity assumptions.

The main contribution of the current paper is to propose a new effective numerical method for solving the first-order 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.

ýThis paper considers a non-local initial-boundary value problem containing a first order partial differential equation with variable coefficientsý. ýAt firstý, ýthe non-self-adjoint spectral problem is derivedý. ýThen its adjoint problem is calculatedý. ýAfter thatý, ýfor the adjoint problem the associated eigenvalues and the subsequent eigenfunctions are determinedý. ýFinally the convergence of series solution and the uniqueness of this solution will be provedý.

We show that a recently introduced Lax pair of the Sawada-Kotera equation is not a new one but is trivially related to the known old Lax pair. Using the so-called trivial compositions of the old Lax pairs with a differentially constrained arbitrary operators, we give some examples of trivial Lax pairs of KdV and Sawada-Kotera equations.

Let E be an elliptic curve over Q with the given Weierstrass equation y 2 =x 3 欟 . 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 2 x 3 . Let E (D) (Q) be the group of Q -rational points of E (D) . It is conjectured by J. Silverman that there are infinitely many primes p for whichE (p) (Q) has positive rank, and there are infinitely many primes q for which E (q) (Q) has rank 0 . In this paper, assuming the parity conjecture, we show that for infinitely many primes p , the elliptic curve E (p) n :y 2 =x 3 −np 2 x has odd rank and for infinitely many primes p , E (p) n (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 4 =r 4 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 (p) n (Q) has odd rank (and so a positive rank) as long as n is odd and p≡5,7(mod8) or n is even and p≡1(mod4) .
In the end, we also compute the ranks of some specific values of n and p explicitly.

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.

Graham Higman has defined coset diagrams for PSL(2,ℤ). These diagrams are composedof 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 eachfragment in Ϝ contains one vertex fixed by F_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.

A toroidalization of a dominant morphism φ:X→Y of algebraic varieties over a field of characteristic zero is a toroidal lifting of φ 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 3-folds.

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 xinX we have ALx=LxA . We say that H is emph (emph{ XS -quasipermutable}, respectively) in G provided G has a subgroup B such that G=NG(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 X -quasipermutable and XS -quasipermutable subgroups on the structure of G . Some known results are generalized.

Let Gamma n,kappa be the class of all graphs with ngeq3 vertices and kappageq2 vertex connectivity. Denote by Upsilon n,beta Upsilonn,beta the family of all connected graphs with ngeq4 ngeq4 vertices and matching number beta beta where 2leqbetaleqlfloorfracn2rfloor . In the classes of graphs Gamma n,kappa and Upsilon n,beta , the elements having maximum augmented Zagreb index are determined.

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.

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 andsufficient 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.

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 single-valued function.

ýLet R be a commutative ring with non-zero identity. ýWe describe all C 3 ý- ýand C 4 -free intersection graph of non-trivial ideals of R as well as C n -free intersection graph when R is a reduced ring. ýAlso, ýwe shall describe all complete, ýregular and n -claw-free intersection graphs. ýFinally, ýwe shall prove that almost all Artin rings R have Hamiltonian intersection graphs. ýWe show that such graphs are indeed pancyclic.

Let Mn;m be the set of n-by-m matrices with entries in the field of real numbers. A matrix R in Mn = Mn;n is a generalized row substochastic matrix (g-row substochastic, for short) if Re  e, where e = (1; 1; : : : ; 1)t. For X; Y 2 Mn;m, X is said to be sgut-majorized by Y (denoted by X sgut Y ) if there exists an n-by-n upper triangular g-row substochastic matrix R such that X = RY . This paper characterizes all linear preservers and strong linear preservers of sgut on Rn and Mn;m respectively.

Decomposability of an algebraic structure into the union of its sub-structures 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 non-group 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 non-quasicommutative semigroups which are decomposable into the disjoint unions of groups. Our examples are the semigroups presented by the following presentations:
π 1 =⟨a,b∣a n =a,b 3 =b,ba=a n−1 b⟩, (n≥3),
π 2 =⟨a,b∣a 1 α =a,b 1 β =b,ab=ba 1 α−γ ⟩
where, p is an odd prime,
α,β and γ are integers such that α≥2γ α≥2γ , β≥γ≥1 and αβ>3 .

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.