In this paper, for the eigenvalue problem of a clamped plate problem on complex projective space with holomorphic sectional curvature c(> 0) and n(≥ 3)-dimensional noncompact simply connected complete Riemannian manifold with sectional curvature Sec satisfying − a ² ≤ Sec ≤ − b ², where a ≥ b ≥ 0 are constants, we obtain universal eigenvalue inequalities. Moreover, we deduce the estimates of the upper bounds of eigenvalues.

We investigate Weierstrass semigroups of ramification points on double covers of plane curves of degree d. Using the results we determine all the Weierstrass semigroups in the case d = 5 when the genus of the covering curve is greater than 17 and the ramification point is on a non-ordinary flex.

The following theorem has been proved by A. Schweizer [7]. If a nonconstant entire function f and its derivative f′ share their simple zeros and if every simple a-point of f is a (not necessarily simple) a-point of f′ for some nonzero constant a, then f ≡ f′. In this paper we shall prove that the above result is also true when the nonzero constant a is replaced by a meromorphic small function β( $\not\equiv$ 0, ∞).

In this paper, we shall give a lower diameter bound for compact domain manifolds of shrinking Ricci-harmonic solitons. Our result may be regarded as a generalization to Ricci-harmonic geometry of the recent works by Fernández-López and García-Río (Q. J. Math. 61, 319-327, 2010), Futaki and Sano (Asian J. Math. 17, 17-32, 2013), and Futaki et al. (Ann. Global Anal. Geom. 44, 105-114, 2013).

In this paper, the existence and uniqueness of pseudo $\mathcal{S}$-asymptotically ω-periodic mild solutions of class r for the nonlinear two-term time fractional differential equations with delay are investigated. The working tools are based on the generalization of the semigroup theory and fixed point theory. Finally, we present an application to a fractional partial differential equation with delay.

We investigate the links between the lattice Idl( R) of ideals of a commutative ring R and the lattices Idl( R′) of ideals of various new rings R′ constructed from R, in particular, the ring S ⁻¹ R of fractions and the ring R[ X] of polynomials. For any partially ordered set P, we construct another poset N( P) and show that P satisfies the ascending chain condition if and only if N( P) satisfies the ascending chain condition. As an application of this result, we give an order version proof for Hilbert's Basis Theorem.

Let ( X,L) be a polarized manifold of dimension 4. In this paper, we prove that h ⁰( K _X + 3 L) > 0 if K _X + 3 L is nef, which is a conjecture of Beltrametti-Sommese for polarized 4-folds.

Let K = Q ( $\sqrt{m}$) be a real quadratic number field, where m > 1 is a squarefree integer. Suppose that 0 < θ < π has rational cosine, say cos(θ) = s/r with 0 < | s| < r and gcd( r,s) = 1. A positive integer n is called a ( K,θ)-congruent number if there is a triangle, called the ( K,θ, n)-triangles, with sides in K having θ as an angle and nα _θ as area, where α _θ = $\sqrt{r^2-s^2}$. Consider the ( K,θ)-congruent number elliptic curve E _n,θ : y ² = x( x + ( r + s) n) ( x − ( r − s) n) defined over K. Denote the squarefree part of positive integer t by sqf( t). In this work, it is proved that if m ≠ sqf(2 r( r − s)) and mn ≠ 2, 3, 6, then n is a ( K,θ)-congruent number if and only if the Mordell-Weil group E _n,θ ( K) has positive rank, and all of the ( K, θ, n)-triangles are classified in four types.

In the present paper, we study category-theoretic properties of monomorphisms in categories of log schemes. This study allows one to give a purely category-theoretic reconstruction of the log scheme that gave rise to the category under consideration. We also obtain analogous results for categories of schemes of locally finite type over the ring of rational integers that are equipped with " archimedean structures". Such reconstructions were discussed in two previous papers by the author, but these reconstructions contained some errors, which were pointed out to the author by C. Nakayama and Y. Hoshi. These errors revolve around certain elementary combinatorial aspects of fan decompositions of two-dimensional rational polyhedral cones—i.e., of the sort that occur in the classical theory of toric varieties—and may be repaired by applying the theory developed in the present paper.

Let $\mathfrak{a}$ be an ideal of local ring ( R, $\mathfrak{m}$) and M a finitely generated R-module and n an integer. We prove some results concerning minimaxness and finiteness of formal local cohomology modules. We discuss the maximum and minimum integers such that $\mathfrak{F}_\mathfrak{a}^i$( M) is minimax and also we obtain the maximum and minimum integers such that $\mathfrak{F}_\mathfrak{a}^i$( M) is finitely gnerated.

In this paper, we study the eigenvalue problem of a system of sub-elliptic equations on abounded domain in the Heisenberg group and obtain some universal inequalities. Moreover, for the lower order eigenvalues of this eigenvalue problem, we also give some universal inequalities.

In this paper, we study the Deligne's sovereign structure theorem on the finite dimensional weak Hopf algebras, and give a necessary and sufficient condition for a finite dimensional (co)quasitriangular weak Hopf algebra H with bijective antipode to admit a (co)ribbon structure. As an application we discuss the ribbon structures over the Drinfeld doubles of some weak Hopf algebras to verify our theory.

We consider the initial value problem to a model system for water waves. The model system is the Euler-Lagrange equations for an approximate Lagrangian which is derived from Luke's Lagrangian for water waves by approximating the velocity potential in the Lagrangian. The model are nonlinear dispersive equations and the hypersurface t = 0 is characteristic for the model equations. Therefore, the initial data have to be restricted in an infinite dimensional manifold in order to the existence of the solution. Under this necessary condition and a sign condition, which corresponds to a generalized Rayleigh-Taylor sign condition for water waves, on the initial data, we show that the initial value problem is solvable locally in time in Sobolev spaces.