Asian-European Journal of Mathematics

The bilevel programming problem is a leader–follower game in which two players try to maximize their own objective functions over a common constraint region. This paper discusses an integer nonlinear bilevel programming problem with box constraints by exploiting the quasimonotone character of the indefinite quadratic fractional function, considered as leader's objective. By making use of the duality theory, given nonlinear bilevel programming problem is transformed into single level programming problem. Various cuts have been discussed in this paper which successively rank and scan all integer feasible points of the single level programming problem in the decreasing value of objective function. An iterative algorithm is proposed, which by making use of these cuts repeatedly, solves the problem.

In this paper, we prove that any prime irreducible representation of a left ample semigroup being eventually regular can be constructed by some irreducible representation of some groups. This result enriches and extends the related results of W. D. Munn on prime irreducible representations of an inverse semigroup.

Let (X, ≤) be a totally ordered finite set, be the full transformation semigroup on X and E be an arbitrary equivalence on X. We consider a subsemigroup of defined by

For the problem of which Jordan forms are possible for n × n complex matrices A, B and C, when A = BC, geometric multiplicity restrictions are given for the eigenvalues of the three matrices. Together with the obvious determinantal condition on the eigenvalues, these necessary conditions are shown to be sufficient for the problem when n < 4, but not for n ≥ 4. Some basic observations about the problem are given in the process.

In this paper, we introduce the concepts of primal and weakly primal ideals in lattices. Further, the diameter of the zero divisor graph of a lattice with respect to a non-primal ideal is determined.

In this paper, by constructing a contraction mapping and using the time scale calculus theory and the Lyapunov function method, we obtain some sufficient conditions for the existence and stability of almost periodic solutions for a class of Nicholson's blowflies models with patch structure and linear harvesting terms on time scales. The results of this paper are completely new. Moreover, we present an illustrative example and numerical simulation to show the effectiveness of obtained results.

We introduce the notion of nil(α, δ)-compatible rings which is a generalization of reduced rings and (α, δ)-compatible rings. In [Ore extensions of weak zip rings, Glasgow Math. J.51 (2009) 525–537] Ouyang introduces the notion of right (respectively, left) weak zip rings and proved that, a ring R is right (respectively, left) weak zip if and only if the skew polynomial ring R[x; α, δ] is right (respectively, left) weak zip, when R is (α, δ)-compatible and reversible. We extend this result to the more general situation that, when R has (α, δ)-condition and quasi-IFP, then nil(R)[x; α, δ] = nil(R[x; α, δ]); and R is right (respectively, left) weak zip if and only if the skew polynomial ring R[x; α, δ] is right (respectively, left) weak zip.

Pseudo-differential operator (p.d.o) associated with the symbol a(x, y) whose derivatives satisfy certain growth condition is defined and the Zemanian-type spaces H_μ(I) and S(I) are introduced. It is shown that the p.d.o is continuous linear map of the space H_μ(I) and S(I) into itself. An integral representation of p.d.o h_{1, μ, a} is obtained. Using the Hankel convolution it is shown that p.d.o h_{1, μ, a} satisfies a certain -norm inequality. Properties of Sobolev-type space are studied.

A right R-module N is called pseudo M-injective if for any submodule A of M, every monomorphism from A to N, can be extended to a homomorphism from M to N. Module M is called pseudo injective if M is pseudo M-injective. Some characterizations of classes modules and its applications to classical rings are studied. In this paper, we consider some generalizations pseudo injective modules under monomorphism of their closed submodules. Their properties are studied.

We study a family of second-order differential operators with unbounded coefficients (and possibly singular). We show that such operators generate analytic semigroups on L²((0, w), ρ₂dx), 0 < w ≤ ∞, where dx denotes the Lebesgue measure. The choice of the weight ρ₂ will depend on the form of the considered operator and on its coefficients.

A dually normal almost distributive lattice is characterized topologically in terms of its maximal ideals and prime ideals. Some necessary and sufficient conditions for the space of maximal ideals to be dually normal are obtained.

The concepts of implicative filters and derivations are introduced in implicative n.p.o. semigroups and studied their properties in connection with each other. The relation between order filters and implicative filters is also studied. The notion of implicative congruences is introduced in implicative n.p.o. semigroups. Finally, the equivalency between the existences of idempotent derivations and implicative filters is obtained.

Let A be an ℓ-algebra and let θ and ϕ be two endomorphisms of A. The couple (θ, ϕ) is called to be separating if xy = 0 implies θ(x)ϕ(y) = 0. If in addition θ and ϕ are ring endomorphisms of A, then the couple (θ, ϕ) is said to be ring-separating. An additive mapping δ : A → A is called (θ, ϕ)-separating derivation on A if there exists a (θ, ϕ)-separating couple with δ(xy) = δ(x)θ(y) + ϕ(x)δ(y), holds for all x, y ∈ A. If an addition θ, ϕ and δ are continuous, then δ is called a continuous (θ, ϕ)-ring-separating derivation. If in addition the couple (θ, ϕ) is ring-separating then δ is called a continuous (θ, ϕ)-ring-separating derivation. An additive mapping F : A → A is called a continuous generalized (θ, ϕ)-separating derivation on A if F is continuous mapping and if there exists a derivation d : A → A such that θ and ϕ are continuous, (θ, ϕ) is a separating couple and F(xy) = F(x)θ(y) + ϕ(x)d(y), holds for all x, y ∈ A. In this paper, we give a description of continuous (θ, ϕ)-ring-separating derivations on some ℓ-algebras. This generalizes a well-known theorem by Colville, Davis, and Keimel [Positive derivations on f-rings, J. Austral. Math. Soc23 (1977) 371–375] and generalizes the results of Boulabiar in [Positive derivations on almost f-rings, Order19 (2002) 385–395], Ben Amor [On orthosymmetric bilinear maps, Positivity14(1) (2010) 123–130] and Toumi et al. in [Order bounded derivations on Archimedean almost f-algebras, Positivity14(2) (2010) 239–245]. Moreover, inspiring from [Toumi, Order-bounded generalized derivations on Archimedean almost f-algebras, Commun. Algebra38(1) (2010) 154–164], it is shown that the notion of continuous generalized (θ, ϕ)-separating derivation on an archimedean almost f-algebra A is the concept of generalized θ-multiplier, that is an additive mapping satisfying F(xyz) = F(x)θ(yz), for all x, y, z ∈ A. In the case where A is an archimedean f-algebra, the situation improves. Indeed, the collection of all continuous generalized (θ, ϕ)-separating derivation on A coincides with the concept of θ-multiplier, that is an additive mapping satisfying F(xy) = F(x)θ(y), for all x, y ∈ A. If in addition A is a Dedekind complete vector lattice and θ is a positive mapping, then the set of all order bounded generalized of the form (θ, ϕ)-separating derivations on A, under composition, is an archimedean lattice-ordered algebra.

The notion of Cartesian symmetry classes is introduced in [T. G. Lei, Notes on Cartesian symmetry classes and generalized trace functions, Linear Algebra Appl.292 (1999) 281–288]. In this paper, we discuss these classes and compute the dimensions of these classes in terms of the fixed point character of S_m. Also, we give a formula for the dimension of Cartesian symmetry class V^χ(G) in terms of the rank of an idempotent matrix M(χ). Some properties of generalized trace functions of a matrix are concluded.

Consider a variety V of acts over a left cancellative monoid S that have a ternary Maltsev operation p(〈p, S〉-algebras). Using the Magnus–Artamonov representation, the construction of the free Abelian extension of an arbitrary V-algebra was obtained and explored by the author. In the present article we prove that each IF-automorphism of a free k-step solvable V-algebra of a finite rank (k > 1) is pseudo-tame, i.e. it is presented as a product of elementary automorphisms of a special module over a ring with several objects.