We determine all the ideals of the homological Goldman Lie algebra, which reflects the structure of an oriented surface.

In this short note we provide a few results about the projectability of the φ-null Osserman condition onto the classical and the null-Osserman condition, via semi-Riemannian submersions as projection maps of principal torus bundles induced by a Lorentzian ${\mathcal S}$-manifold.

It is known that the Teichmüller distance on the universal Teichmüller space T coincides with the Kobayashi distance. For a metric subspace of T having a comparable complex structure with that of T, we can similarly consider whether or not the Teichmüller distance on the subspace coincides with the Kobayashi distance. In this paper, we give a sufficient condition for metric subspaces under which the problem above has a affirmative answer. Moreover, we introduce an example of such subspaces.

In this paper we establish several weighted norm inequalities for derivatives of products of composition and Laplace convolution of functions. Some generalizations including Yamada's and Opial-type inequalities are discussed. As applications we give L_p-weighted estimates for solutions of some integral equations.

Let (Mⁿ,g) be a gradient Yamabe soliton Rg + Hess f = λg with Ric_f1 ≥ K (see (1.3) for f₁) and λ, K $in$ R are constants. In this paper, it is showed that for gradient shrinking Yamabe solitons, the scalar curvature R > 0 unless R ≡ 0 and (Mⁿ,g) is the Gaussian soliton, and for gradient steady and expanding Yamabe solitons, R > λ unless R ≡ λ and (Mⁿ,g) is either trivial or a Riemannian product manifold. Replacing the assumptions Ric_f1 ≥ K by R ≥ λ, we also derive the corresponding scalar curvature estimates. In particular, we show that any shrinking gradient Yamabe soliton with R ≥ λ must have constant scalar curvature R ≡ λ. Moreover, the lower bounds of scalar curvature for quasi gradient Yamabe solitons are obtained.

We show that the Reeb vector field of an almost cosymplectic three-manifold is minimal if and only if it is an eigenvector of the Ricci operator. Then, we show that Reeb vector field ξ of an almost cosymplectic three-manifold M is minimal if and only if M is (κ, μ, ν)-space on an open dense subset. After, using the notion of strongly normal unit vector field introduced in [8], we study the minimality of ξ for an almost cosymplectic (2n + 1)-manifold. Finally, we classify a special class of almost cosymplectic three-manifold whose Reeb vector field is minimal.

As for Scholz' inequalities s ≤ r ≤ s + 1 with respect to the 3-rank of the ideal class group of quadratic fields, we give a criterion to be r = s + 1. From this, we give a new family of imaginary quadratic fields whose ideal class groups have 3-rank at least two.

In the present paper, we prove that an open continuous homomorphism between the absolute Galois groups of p-adic local fields is geometric [i.e., roughly speaking, arises from an embedding of fields] if and only if the homomorphism is HT-preserving [i.e., roughly speaking, satisfies the condition that the pull-back by the homomorphism of every Hodge-Tate representation is Hodge-Tate].

This is a continuation of our previous paper [4]. In the class of hyperbolic manifolds in the sense of S. Kobayashi [3], we obtained in [4] an intrinsic characterization of bounded symmetric domains in Cⁿ from the viewpoint of the holomorphic automorphism group. In connection with this, we give in this paper a structure theorem on diffeomorphisms between Siegel domains of the first kind that preserve the holomorphic automorphism groups. As an application, we obtain a well-known fact [2] that two Siegel domains of the first kind are biholomorphically equivalent if and only if they are linearly equivalent.

In this paper, we investigate some properties of planar harmonic mappings. First, we generalize the main results in [2] and [10], and then discuss the relationship between area integral means and harmonic Hardy spaces or harmonic weighted Bergman spaces. At the end, coefficient estimates of mappings in weighted Bergman spaces are obtained.

The main aim of this paper is to study Hopf group-crossed products and Hopf group-cleft extensions in the setting of Hopf group-coalgebras.

A complex C is said to be FR-injective (resp., FR-flat) if Ext¹(D,C) = 0 (resp., $\overline{\mathrm{Tor}}_1$ (C,D) = 0) for any finitely represented complex D. We prove that a complex C is FR-injective (resp., FR-flat) if and only if C is exact and Z_m(C) is FR-injective (resp., FR-flat) in R-Mod for all m $in$ Z. We show that the class of FR-injective complexes is closed under direct limits and the class of FR-flat complexes is closed under direct products over any ring R. We use this result to prove that every complex have FR-flat preenvelopes and FR-injective covers.

The quantum walks in the lattice spaces are represented as unitary evolutions. We find a generator for the evolution and apply it to further understand the walks. We first extend the discrete time quantum walks to continuous time walks. Then we construct the quantum Markov semigroup for quantum walks and characterize it in an invariant subalgebra. In the meanwhile, we obtain the limit distributions of the quantum walks in one-dimension with a proper scaling, which was obtained by Konno by a different method.

We show that if two meromorphic functions sharing four two-point sets CM, then one of them is a Möbius transform of the other.