Let f be a chain mixing continuous onto mapping from the Cantor set onto itself. Let g be an aperiodic homeomorphism on the Cantor set. We show that homeomorphisms that are topologically conjugate to g approximate f in the topology of uniform convergence if a trivial necessary condition on periodic points is satisfied. In particular, let f be a chain mixing continuous onto mapping from the Cantor set onto itself with a fixed point and g, an aperiodic homeomorphism on the Cantor set. Then, homeomorphisms that are topologically conjugate to g approximate f.

In this paper, we investigate asymptotic dimension of proper CAT(0) spaces and we show that for a proper cocompact CAT(0) space (X, d), the asymptotic dimension asdim(X, d) is greater than the covering dimension dim ∂X of the boundary of X.

In this paper, we study surfaces of revolution without parabolic points in the 3-dimensional Lorentz-Minkowski space whose Gauss map N satisfies the condition Δ^IIN = AN, where Δ^II is the Laplace operator with respect to the second fundamental form and A is a real 3 × 3 matrix. More precisely we prove that such surfaces are either pseudo-Riemannian spheres S²₁ or pseudohyperbolic spaces H²₀.

Let $\tilde{C}$ be a non-singular plane curve of degree d ≥ 8 with an involution σ over an algebraically closed field of characteristic 0 and $\tilde{P}$ a point of $\tilde{C}$ fixed by σ. Let π : $\tilde{C}$ → C = $\tilde{C}$/$/\langle\sigma\rangle $be the double covering. We set P = π($\tilde{P}$). When the intersection multiplicity at $\tilde{P}$ of the curve $\tilde{C}$ and the tangent line at $\tilde{P}$ is equal to d − 3 or d − 4, we determine the Weierstrass gap sequence at P on C using blowing-ups and blowing-downs of some rational surfaces.

It has been known that the centralizer Z_W(W_I) of a parabolic subgroup W_I of a Coxeter group W is a split extension of a naturally defined reflection subgroup by a subgroup defined by a 2-cell complex $\mathscr{Y}$. In this paper, we study the structure of Z_W(W_I) further and show that, if I has no irreducible components of type A_n with 2 ≤ n < ∞, then every element of finite irreducible components of the inner factor is fixed by a natural action of the fundamental group of $\mathscr{Y}$. This property has an application to the isomorphism problem in Coxeter groups.

In this paper, realizing an infinite simplicial complex K in the linear space R^K(0) naturally, we investigate the box topology on |K| inherited from R^K(0) that is finer than the metric topology and coarser than the Whitehead (weak) topology.

The purpose of this paper is to study a class of semilinear elliptic boundary value problems with degenerate boundary conditions which include as particular cases the Dirichlet and Robin problems. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. By making use of a variant of the Ljusternik-Schnirelman theory of critical points, we prove very exact results on the number of solutions of our problem. The results here extend earlier theorems due to Castro-Lazer to the degenerate case.

The aim of this paper is to study dense chaos and densely chaotic operators on Banach spaces. First, we prove that a dynamical system is densely δ-chaotic for some δ > 0 if and only if it is densely chaotic and sensitive. Meanwhile, we also show that for general dynamical systems, Devaney chaos and dense chaos do not imply each other. Then, by using these results, we have that for a operator defined on a Banach space, dense chaos, dense δ-chaos, generic chaos and generic δ-chaos are equivalent and they are all strictly stronger than Li-Yorke chaos.