We introduce Tate homology of complexes of finite Gorenstein flat dimension based on complete flat resolutions and give a new method of computing Tate homology in Christensen and Jorgensen's sense. We also investigate the relationship between Tate homology and Tate cohomology. As an application, a more brief proof of the main result on derived depth formula of [Vanishing of Tate homology and depth formula over local rings, J. Pure Appl. Algebra 219 (2015) 464-481] is given.

Rossby waves are generally expected to dominate the β plane dynamics in geophysics, and here in this paper we give a number theoretical observation of the resonant interaction with a Diophantine equation. The set of resonant frequencies does not have any frequency on the horizontal axis.

We consider smooth plane curves which are convex with respect to the origin. We describe centro-affine invariants (that is, GL₊ (2,R)-invariants), such as centro-affine curvature and arc length, in terms of the canonical Lorentz structure on the three dimensional space of all the ellipses centered at zero, by means of null curves of osculating ellipses. This is the centro-affine analogue of the approach to conformal invariants of curves in the sphere introduced by Langevin and O'Hara, using the canonical pseudo Riemannian metric on the space of circles.

In this paper, Herz-type Besov spaces with variable smoothness and integrability are introduced. Our scale contains variable Besov spaces as special cases. We prove several basic properties, especially the Sobolev-type embeddings.

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M, N$ finitely generated $R$-modules. Let $0 ≤ n \in \mathbf Z$. This note shows that the least integer $i$ such that $\dim \mathrm{Supp}(H^i_I(M, N)/K) ≥ n$ for any finitely generated submodule $K$ of $H^i_I(M, N)$ equal to the number $\inf\{f_{I_{\frak p}}(M_{\frak p},N_{\frak p})|{\frak p}\in \mathrm{Supp}(N/I_MN), \dim R/{\frak p} ≥ n\}$, where $f_{I_{\frak p}}(M_{\frak p},N_{\frak p})$ is the least integer $i$ such that $H^i_{I_{\frak p}}(M_{{\frak p}},N_{{\frak p}})$ is not finitely generated, and $I_M = \mathrm{ann}(M/IM)$. This extends the main result of Asadollahi-Naghipour [1] and Mehrvarz-Naghipour-Sedghi [8] for generalized local cohomology modules by a short proof.

A Kirchhoff type plate equation with memory is investigated. Under the suitable assumptions, we establish the existence of a global attractor by using the contraction function method.

In this paper we study a pointwise asymptotic behavior of the partial Euler product for the Riemann zeta-function on the right half of the critical strip. We discuss relations among the behavior of the partial Euler product, the distribution of the prime numbers and the distribution of nontrivial zeros of the Riemann zeta-function.

In this paper, we study unit tangent sphere bundles T₁M whose Ricci operator $\bar{S}$ is Reeb flow invariant, that is, L_ξ$\bar{S}$ = 0. We prove that for a 3-dimensional Riemannian manifold M, T₁M satisfies L_ξ$\bar{S}$ = 0 if and only if M is of constant curvature 1. Also, we prove that for a 4-dimensional Riemannian manifold M, T₁M satisfies L_ξ $\bar{S}$ = 0 and ℓ$\bar{S}$ξ = 0 if and only if M is of constant curvature 1 or 2, where ℓ = $\bar{R}$(·,ξ)ξ is the characteristic Jacobi operator.

In this paper, we are concerned with the uniqueness of ground states for a class of quasilinear elliptic equations which arise in the study of plasma physics. We obtain global uniqueness results in the sense that we don't require any assumptions on the parameter.

We determine all holomorphically separable complex manifolds of dimension p + q which admit smooth envelopes of holomorphy and effective general indefinite unitary group actions of size p + q. Also we give exact description of the automorphism groups of those complex manifolds. As an application we consider a characterization of those complex manifolds by their automorphism groups.

We study log étale cohomology. The goal is to prove relative purity in log étale cohomology.