we prove the generalized hyers--ulam stability  of n--th order linear differential equation of the form $y^{(n)}+p_{1}(x)y^{(n-1)}+ cdots+p_{n-1}(x)y^{prime}+p_{n}(x)y=f(x)$, with condition that there exists a non--zero solution of corresponding homogeneous equation. our main results extend and improve the corresponding results obtained by many authors.

Twisted Alexander invariant is defined for a finitely presentable group G and a representation of G and a surjective homomorphism of G to a free abelian group. In this talk, we introduce some examples and some properties of the twisted Alexander invariant. Moreover, as an application, we consider a partial order on the set of prime knots. Let K be a knot and G(K) the knot group. For two prime k...

We consider basic conceptual graphs, namely simple conceptual graphs (SGs), which are equivalent to the existential conjunctive positive fragment of first-order logic. The fundamental problem, deduction, is performed by a graph homomorphism called projection. The existence of a projection from a SG Q to a SG Gmeans that the knowledge represented by Q is deducible from the knowledge represented ...

Let X be a smooth projective variety defined over an algebraically closed field, and let L be an ample line bundle over X . We prove that for any smooth hypersurface D on X in the complete linear system |L|, the inclusion map D →֒ X induces an isomorphism of fundamental group schemes, provided d is sufficiently large and dimX ≥ 3. If dimX = 2, and d is sufficiently large, then the induced homomo...

Let V and W be vector spaces over a common field, and let S and T be sets of subspaces of V and W, respectively. A linear function φ : V → W is called a (Grassmann) homomorphism from S to T if φ(S) ∈ T holds for every S ∈ S. Coloring of graphs and hypergraphs and flows in graphs can be reformulated in terms of these homomorphisms. The main results are constructive characterizations of the sets ...

In this article, we formalize some basic facts of Z-module. In the first section, we discuss the rank of submodule of Z-module and its properties. Especially, we formally prove that the rank of any Z-module is equal to or more than that of its submodules, and vice versa, there exists a submodule with any given rank that satisfies the above condition. In the next section, we mention basic facts ...

We show that certain canonical realizations of the complexes Hom(G,H) and Hom+(G,H) of (partial) graph homomorphisms studied by Babson and Kozlov are in fact instances of the polyhedral Cayley trick. For G a complete graph, we then characterize when a canonical projection of these complexes is itself again a complex, and exhibit several well-known objects that arise as cells or subcomplexes of ...

Let A,B be two unital C∗−algebras. We prove that every almost unital almost linear mapping h : A −→ B which satisfies h(3uy + 3yu) = h(3u)h(y) + h(y)h(3u) for all u ∈ U(A), all y ∈ A, and all n = 0, 1, 2, ..., is a Jordan homomorphism. Also, for a unital C∗−algebra A of real rank zero, every almost unital almost linear continuous mapping h : A −→ B is a Jordan homomorphism when h(3uy + 3yu) = h...