International Journal of Mathematics

We define spatial $L^p$ AF algebras for $p\in [1,\infty )\setminus \{2\}$, and prove the following analog of the Elliott AF algebra classification theorem. If $A$ and $B$ are spatial $L^p$ AF algebras, then the following are equivalent:

• $A$ and $B$ have isomorphic scaled preordered $K_0$-groups.
• $A\cong B$ as rings.
• $A\cong B$ (not necessarily isometrically) as Banach algebras.
• $A$ is isometrically isomorphic to $B$ as Banach algebras.
• $A$ is completely isometrically isomorphic to $B$ as matricial $L^p$ operator algebras.

As background, we develop the theory of matricial $L^p$ operator algebras, and show that there is a unique way to make a spatial $L^p$ AF algebra into a matricial $L^p$ operator algebra. We also show that any countable scaled Riesz group can be realized as the scaled preordered $K_0$-group of a spatial $L^p$ AF algebra.

We study relations between two log minimal models of a fixed lc pair. For any two log minimal models of an lc pair constructed with log MMP, we prove that there are small birational models of the log minimal models which can be connected by a sequence of flops, and the two log minimal models share some properties. We also give examples of two log minimal models of an lc pair which have different properties.

In this paper, we consider the heat flow for $p$-pseudoharmonic maps from a closed Sasakian manifold $(M^{2n+1},J,𝜃)$ into a compact Riemannian manifold $(N^m,g_{ij})$. We prove global existence and asymptotic convergence of the solution for the $p$-pseudoharmonic map heat flow, provided that the sectional curvature of the target manifold $N$ is non-positive. Moreover, without the curvature assumption on the target manifold, we obtain global existence and asymptotic convergence of the $p$-pseudoharmonic map heat flow as well when its initial $p$-energy is sufficiently small.

In this paper, regularity properties, Strichartz type estimates for solution of integral problem for linear and nonlinear abstract Schrödinger equations in vector-valued function spaces are obtained. The equation includes a linear operator $A$ defined in a Banach space $E$, in which by choosing $E$ and $A$ we can obtain numerous classis of initial value problems for Schrödinger equations which occur in a wide variety of physical systems.

In this paper, we prove the semi-continuity theorem of Diederich–Fornaess index and Steinness index under a smooth deformation of pseudoconvex domains in Stein manifolds.

We introduce the notion of asymptotically finitely generated contact structures, which states essentially that the Symplectic Homology in a certain degree of any filling of such contact manifolds is uniformly generated by only finitely many Reeb orbits. This property is used to generalize a famous result by Ustilovsky: We show that in a large class of manifolds (including all unit cotangent bundles and all Weinstein fillable contact manifolds with torsion first Chern class) each carries infinitely many exactly fillable contact structures. These are all different from the ones constructed recently by Lazarev. Along the way, the construction of Symplectic Homology is made more general. Moreover, we give a detailed exposition of Cieliebak’s Invariance Theorem for subcritical handle attaching, where we provide explicit Hamiltonians for the squeezing on the handle.

In this paper we derive both local and global geometric inequalities on general Riemannnian and Finsler manifolds and prove generalized Caffarelli–Kohn–Nirenberg type and Hardy type inequalities on Finsler manifolds, illuminating curvatures of both Riemannian and Finsler manifolds influence geometric inequalities.

The method of Frobenius is a standard technique to construct series solutions of an ordinary linear differential equation around a regular singular point. In the classical case, when the roots of the indicial polynomial are separated by an integer, logarithmic solutions can be constructed by means of perturbation of a root. The method for a regular $A$-hypergeometric system is a theme of the book by Saito, Sturmfels and Takayama. Whereas they perturbed a parameter vector to obtain logarithmic $A$-hypergeometric series solutions, we adopt a different perturbation in this paper.

In this paper, we obtain the crosscap number of any alternating knots by using our recently-introduced diagrammatic knot invariant (Theorem 1). The proof is given by properties of chord diagrams (Kindred proved Theorem 1 independently via other techniques). For non-alternating knots, we give Theorem 2 that generalizes Theorem 1. We also improve known formulas to obtain upper bounds of the crosscap number of knots (alternating or non-alternating) (Theorem 3). As a corollary, this paper connects crosscap numbers and our invariant with other knot invariants such as the Jones polynomial, twist number, crossing number, and hyperbolic volume (Corollaries 1–7). In Appendix A, using Theorem 1, we complete giving the crosscap numbers of the alternating knots with up to 11 crossings including those of the previously unknown values for $193$ knots (Tables A.1).

We explicitly write down all eigenvalues of the Rumin Laplacian on the standard contact spheres, and express the analytic torsion functions associated with the Rumin complex in terms of the Riemann zeta function. In particular, we find that the functions vanish at the origin and determine the analytic torsions.

We investigate the solution space to certain $A$-hypergeometric $\mathcal{𝒟}$-modules, which were defined and studied by Gelfand, Kapranov, and Zelevinsky. We show that the solution space can be identified with certain relative cohomology group of the toric variety determined by $A$, which generalizes the results of Huang, Lian, Yau and Zhu. As a corollary, we also prove the existence of rank one points for complete intersections in toric varieties.

One of the most important problems in Finsler geometry is to classify Finsler metrics of scalar flag curvature. In this paper, we study the classification problem of Randers metrics of scalar flag curvature. Under the condition that $\beta$ is a Killing 1-form, we obtain some important necessary conditions for Randers metrics to be of scalar flag curvature.