1.
Lattices in contact Lie groups and 5-dimensional contact solvmanifolds - Diatta, André; Foreman, Brendan
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.

2.
A Note on Cartan-Eilenberg Gorenstein categories - Lu, Bo; Ren, Wei; Liu, Zhongkui
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.

3.
Reduction of a family of ideals - Rodak, Tomasz
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.

4.
Note on the filtrations of the K-theory - Yagita, Nobuaki
Let X be a (colimit of) smooth algebraic variety over a subfield k of C. Let K_{alg}^{0}(X) (resp. K_{top}^{0}(X(C))) be the algebraic (resp. topological) K-theory of k (resp. complex) vector bundles over X (resp. X(C))). When K_{alg}^{0}(X) $\cong$ K_{top}^{0}(X(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 = G_{k}/T_{k} the twisted form of flag varieties G/T for non-algebraically closed field k.

5.
The conformal rotation number - Kobayashi, Osamu
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.

6.
Induction functors for group corings - Chen, Quan-Guo; Wang, Ding-Guo
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.

7.
Nonharmonic nonlinear Fourier frames and convergence of corresponding frame series - Shen, Yanfeng; Li, Youfa; Yang, Shouzhi
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 $\in$ L^{2} (−π,π).

8.
Global isometric embeddings of multiple warped product metrics into quadrics - Mirandola, Heudson; Vitório, Feliciano
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.

10.
Stability of line standing waves near the bifurcation point for nonlinear Schrödinger equations - Yamazaki, Yohei
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...

11.
On the geometry of the rescaled Riemannian metric on tensor bundles of arbitrary type - Gezer, Aydin; Altunbas, Murat
Let (M,g) be an n-dimensional Riemannian manifold and T_{1}^{1}(M) be its (1,1)-tensor bundle equipped with the rescaled Sasaki type metric ^{S}g_{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 T_{1}^{1}(M). We construct almost product Riemannian structures on T_{1}^{1}(M) and investigate conditions for these structures to be locally decomposable. Also, some applications concerning with these almost product Riemannian structures on T_{1}^{1}(M) are presented. Finally we introduce the rescaled Sasaki type metric ^{S}g_{f} on the (p,q)-tensor bundle and characterize the geodesics on...

12.
On Ballico-Hefez curves and associated supersingular surfaces - Hoang, Thanh Hoai; Shimada, Ichiro
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.

13.
On Klein-Maskit combination theorem in space II - Li, Liulan; Ohshika, Ken'ichi; Wang, Xiantao
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.

14.
Convexity properties of Dirichlet integrals and Picone-type inequalities - Brasco, Lorenzo; Franzina, Giovanni
We focus on three different convexity principles for local and nonlocal variational integrals. We prove various generalizations of them, as well as their equivalences. Some applications to nonlinear eigenvalue problems and Hardy-type inequalities are given. We also prove a measure-theoretic minimum principle for nonlocal and nonlinear positive eigenfunctions.

15.
Nonexistence of positive very weak solutions to an elliptic problem with boundary reactions - Takahashi, Futoshi
We consider a semilinear elliptic problem with the boundary reaction: ¶ −Δu = 0 in Ω, $\frac{\partial u}{\partial \nu}$ + u = a(x) u^{p} + f(x) on ∂Ω, ¶ where Ω $subset$ R^{N}, N ≥ 3, is a smooth bounded domain with a flat boundary portion, p > 1, a, f $\in$ L^{1}(∂Ω) are nonnegative functions, not identically equal to zero. We provide a necessary condition and a sufficient condition for the existence of positive very weak solutions of the problem. As a corollary, under some assumption of the potential function a, we prove that the problem has no positive...

16.
A decomposition theorem of the Möbius energy I: Decomposition and Möbius invariance - Ishizeki, Aya; Nagasawa, Takeyuki
The Möbius energy, defined for closed curves embedded in R^{n}, is decomposed into three parts. It is called the Möbius energy since it is invariant under Möbius transformations. An alternative proof of the Möbius invariance is given using the decomposition. Furthermore the invariance of each part of decomposition is discussed.

17.
A note on Serrin's overdetermined problem - Ciraolo, Giulio; Magnanini, Rolando
We consider the solution of the torsion problem ¶ −Δu = N in Ω, u = 0 on ∂Ω, ¶ where Ω is a bounded domain in R^{N}. ¶ Serrin's celebrated symmetry theorem states that, if the normal derivative u_{ν} is constant on ∂Ω, then Ω must be a ball. In [6], it has been conjectured that Serrin's theorem may be obtained by stability in the following way: first, for the solution u of the torsion problem prove the estimate ¶ r_{e} − r_{i} ≤ $C_t\Bigl(\max_{\Gamma_t} u-\min_{\Gamma_t} u\Bigr)$ ¶ for some constant C_{t} depending on t, where r_{e} and r_{i}...

18.
Local solvability of a fully nonlinear parabolic equation - Akagi, Goro
This paper is concerned with the existence of local (in time) positive solutions to the Cauchy-Neumann problem in a smooth bounded domain of R^{N} for some fully nonlinear parabolic equation involving the positive part function r $\in$ R $\mapsto$ (r)_{+}: = r ∨ 0. To show the local solvability, the equation is reformulated as a mixed form of two different sorts of doubly nonlinear evolution equations in order to apply an energy method. Some approximated problems are also introduced and the global (in time) solvability is proved for them with an aid of convex analysis, an energy method and some...

19.
Interaction between fast diffusion and geometry of domain - Sakaguchi, Shigeru
Let Ω be a domain in R^{N}, where N ≥ 2 and ∂Ω is not necessarily bounded. We consider two fast diffusion equations ∂_{t}u = div(|∇u|^{p-2}∇u) and ∂_{t}u = Δu^{m}, where 1 < p < 2 and 0 < m < 1. Let u = u(x,t) be the solution of either the initial-boundary value problem over Ω, where the initial value equals zero and the boundary value is a positive continuous function, or the Cauchy problem where the initial datum equals a nonnegative continuous function multiplied by the characteristic function of the set R^{N}\Ω. Choose an open ball B in...

20.
A note on parabolic power concavity - Ishige, Kazuhiro; Salani, Paolo
We investigate parabolic power concavity properties of the solutions of the heat equation in Ω × [0,T), where Ω = R^{n} or Ω is a bounded convex domain in R^{n}.