This paper is concerned with the estimation of upper bound of the third order Hankel determinant for a unified subclass of analytic functions in the open unit disc E={z∈C:|z|<1}. Also the sharp upper bounds for the first four coefficients, Fekete-Szegö inequality, Zalcman inequality and second Hankel determinant, are established. The results are further extended for two-fold and three-fold symmetric functions.

A semilinear Newton equation of motion is derived from a semilinear Schrödinger equation in homogeneous and isotropic spacetimes by the Ehrenfest theorem. The Cauchy problem for the equation is considered, especially, on the existence of global solutions and nonexistence of global weak solutions. The effects of spatial expansion and contraction are studied.

This article is a continuation of [4] in a way. We construct pointed curves on the K3 surfaces treated in [1] which are double covers of rational elliptic surfaces. In some cases we calculate the Weierstrass semigroups of the pointed curves. These pointed curves are the first examples on K3 surfaces that are double covers of rational elliptic surfaces such that we can calculate the Weierstrass semigroups. Moreover, we give bi-elliptic curves of genus 9 or 10 on some such K3 surfaces.

The compact simply connected Riemannian 4-symmetric spaces were classified by J. A. Jiménez according to type of the Lie algebras. As homogeneous manifolds, these spaces are of the form G/H, where G is a connected compact simple Lie group with an automorphism γ˜ of order four on G and H is a fixed points subgroup Gγ of G. According to the classification by J. A. Jiménez, there exist seven compact simply connected Riemannian 4-symmetric spaces G/H in the case where G is of type E8. In the present article, we give the explicit form of automorphisms ω˜4, κ˜4 and ε˜4 of order four on E8 induced by the C-linear transformations ω4, κ4 and ε4 of the 248-dimensional C-vector space e8C, respectively. Further, we determine the structure of these fixed points subgroups (E8)ω4, (E8)κ4 and (E8)ε4 of E8. These amount to the global realizations of three spaces among seven Riemannian 4-symmetric spaces G/H above corresponding to the Lie algebras h=su(2)⊕iR⊕e6, iR⊕so(14) and h=su(2)⊕iR⊕so(12), where h=Lie(H). With this article, the all realizations of inner automorphisms of order four and fixed points subgroups by them have been completed in E8.

We classify real hypersurfaces of a non-flat complex space form of complex dimension 2 satisfying the condition that the Ricci tensor S is the transversal Killing tensor field on the direction of the structure vector field $\xi$ and $S\xi = \beta\xi$ where $\beta$ is a function.

We introduce regular prehomogeneous vector spaces associated with an arbitrary valued Dynkin graph $(\\Gamma, \\boldsymbol{v})$ having a fixed oriented modulation $(𝔐, \\Omega)$ over the ground field $K$. Here $K$ is of characteristic zero, but it may not be algebraically closed. We will construct a fundamental theory of such prehomogeneous vector spaces.\n¶ Each generic point of a regular prehomogeneous vector space corresponds to a hom-orthogonal partial tilting $\\Lambda$-module, where $\\Lambda$ is the tensor $K$-algebra of $(𝔐, \\Omega)$. We count the number of isomorphism classes of hom-orthogonal partial tilting $\\Lambda$-modules of type $\\mathbf{B}_n$, $\\mathbf{C}_n$, $\\mathbf{F}_4$ and $\\mathbf{G}_2$. As a consequence of our theorem, we estimate lower and upper bounds for the number of basic relative invariants of regular prehomogeneous vector spaces for any valued Dynkin quiver.

The compact simply connected Riemannian 4-symmetric spaces were classified by J. A. Jimenez according to type of the Lie algebras. As homogeneous manifolds, these spaces are of the form $G/H$, where $G$ is a connected compact simple Lie group with an automorphism $\tilde{\gamma}$ of order four on $G$ and $H$ is a fixed points subgroup $G^\gamma$ of $G$. According to the classification by J. A. Jimenez, there exist seven compact simply connected Riemannian 4-symmetric spaces $G/H$ in the case where $G$ is of type $E_8$. In the present article, we give the explicit form of automorphisms $\tilde{w}_4$, $\tilde{\upsilon}_4$ and $\tilde{\mu}_4$ of order four on $E_8$ induced by the $C$-linear transformations $w_4, \upsilon_4$ and $\mu_4$ of the 248-dimensional vector space $𝔢^C_8$, respectively. Further, we determine the structure of these fixed points subgroups $(E_8)^{w_4}, (E_8)^{\upsilon_4}$ and $(E_8)^{\mu_4}$ of $E_8$. These amount to the global realizations of three spaces among seven Riemannian 4-symmetric spaces $G/H$ above corresponding to the Lie algebras $𝔥 = i \boldsymbol{R} \oplus 𝔰𝔲(8)$, $i\boldsymbol{R} \oplus 𝔢_7$ and $𝔥 = 𝔰𝔲(2) \oplus 𝔰𝔲(8)$, where $𝔥 = \mathrm{Lie}(H)$.