In this paper, which is sequel to [10], we give a generalisation of the second Klein-Maskit combination theorem, the one dealing with HNN extensions, to higher dimension. We give some examples constructed as an application of the main theorem.

Let p be a prime integer, and q a power of p. The Ballico-Hefez curve is a non-reflexive nodal rational plane curve of degree q + 1 in characteristic p. We investigate its automorphism group and defining equation. We also prove that the surface obtained as the cyclic cover of the projective plane branched along the Ballico-Hefez curve is unirational, and hence is supersingular. As an application, we obtain a new projective model of the supersingular K3 surface with Artin invariant 1 in characteristic 3 and 5.

Let (M,g) be an n-dimensional Riemannian manifold and T₁¹(M) be its (1,1)-tensor bundle equipped with the rescaled Sasaki type metric ^Sg_f which rescale the horizontal part by a non-zero differentiable function f. In the present paper, we discuss curvature properties of the Levi-Civita connection and another metric connection of T₁¹(M). We construct almost product Riemannian structures on T₁¹(M) and investigate conditions for these structures to be locally decomposable. Also, some applications concerning with these almost product Riemannian structures on T₁¹(M) are presented. Finally we introduce the rescaled Sasaki type metric ^Sg_f on the (p,q)-tensor bundle and characterize the geodesics on the (p,q)-tensor bundle with respect to the Levi-Civita connection and another metric connection of ^Sg_f.

In this paper we consider the transverse instability for a nonlinear Schrödinger equation with power nonlinearity on R × T_L, where 2πL is the period of the torus T_L. There exists a critical period 2πL_ω,p such that the line standing wave is stable for L < L_ω,p and the line standing wave is unstable for L > L_ω,p. Here we farther study the bifurcation from the boundary L = L_ω,p between the stability and the instability for line standing waves of the nonlinear Schrödinger equation. We show the stability for the branch bifurcating from the line standing waves by applying the argument in Kirr, Kevrekidis and Pelinovsky [16] and the method in Grillakis, Shatah and Strauss [12]. However, at the bifurcation point, the linearized operator around the bifurcation point is degenerate. To prove the stability for the bifurcation point, we apply the argument in Maeda [18].

In this article, we study algebraic dependences of three meromorphic mappings which share few hyperplanes counting truncated multiplicities.

In this paper, we construct smooth isometric embeddings of multiple warped product manifolds in quadrics of semi-Euclidean spaces. Our main theorem generalizes previous results as given by Blanuša, Rozendorn, Henke and Azov.

Having redundancy, frames, compared with basis, can provide more robust representation of a vector in application. By introducing nonharmonic nonlinear Fourier frames, a method is established to construct such frames by perturbation. Based on a special class of nonharmonic nonlinear Fourier frames, the convergence of its corresponding frame operator is investigated, and the convergence rate, associated with the coefficient sequence of the frame operator, is estimated. Finally, we also discuss the equiconvergence of two different (nonlinear or linear) Fourier (basis or frame) series of f $\in$ L² (−π,π).

In the paper, we prove that the induction functor stemming from every morphism of group coring versus coring has a left adjoint, called ad-induction functor. The separability of the induction functor is characterized, extending some results for corings.

The rotation number of a planar closed curve is the total curvature divided by 2π. This is a regular homotopy invariant of the curve. We shall generalize the rotation number to a curve on a closed surface using conformal geometry of ambient surface. This conformal rotational number is not integral in general. We shall show the fractional part is relevant to harmonic 1-forms of the surface.

Let $X$ be a (colimit of) smooth algebraic variety over a subfield $k$ of $\mathbf C$. Let $K_{alg}^0(X)$ (resp. $K_{top}^0(X(\mathbf C)))$ be the algebraic (resp. topological) $K$-theory of $k$ (resp. complex) vector bundles over $X$ (resp. $X(\mathbf C)))$. When $K_{alg}^0(X) \cong K_{top}^0(X(\mathbf C))$, we study the differences of its three (gamma, geometrical and topological) filtrations. In particular, we consider in the cases $X = BG$ for algebraic group $G$ over algebraically closed fields $k$, and $X = \mathbf G_k/T_k$ the twisted form of flag varieties $G/T$ for non-algebraically closed field $k$.

In the paper we prove that there exists a simultaneous reduction of one-parameter family of $\mathfrak{m}_{n}$-primary ideals in the ring of germs of holomorphic functions. Moreover, we generalize the result of A. Płoski [8] on the semicontinuity of the Łojasiewicz exponent in a multiplicity-constant deformation.

In this article, we investigate the stability of Cartan-Eilenberg Gorenstein categories. To this end, we introduce and study the concept of two-degree Cartan-Eilenberg $\mathcal{W}$-Gorenstein complexes. We prove that a complex C is two-degree Cartan-Eilenberg $\mathcal{W}$-Gorenstein if and only if C is Cartan-Eilenberg $\mathcal{W}$-Gorenstein. As applications, we show that a complex C is two-degree C-E Gorenstein projective if and only if C is C-E Gorenstein projective. Moreover, we obtain similar results for some known modules such as Ω-Gorenstein projective modules and V-Gorenstein projective modules.

We investigate the existence and properties of uniform lattices in Lie groups and use these results to prove that, in dimension 5, there are exactly seven connected and simply connected contact Lie groups with uniform lattices, all of which are solvable. In particular, it is also shown that the special affine group has no uniform lattice.