The Index theorem for quasi-tori

The Index theorem for holomorphic line bundles on complex tori asserts that some cohomology groups of a line bundle vanish according to the signature of the associated hermitian form. In this article, this theorem is generalized to quasi-tori, i.e. connected complex abelian Lie groups which are not necessarily compact. In view of the Remmert–Morimoto decomposition of quasi-tori as well as the Künneth formula, it suffices to consider only Cousin-quasi-tori, i.e. quasi-tori which have no non-constant holomorphic functions. The Index theorem is generalized to holomorphic line bundles, both linearizable and non-linearizable, on Cousin-quasi-tori using L²$L^2$-methods coupled with the Kazama–Dolbeault isomorphism and Bochner–Kodaira formulas.     

A gap theorem on submanifolds with finite total curvature in spheres

We study a complete noncompact submanifold Mⁿ$M^n$ in a sphere S^n+p${\mathbb{S}}^{n+p}$. We prove that there admit no nontrivial L²$L^2$-harmonic 1-forms on M if the total curvature is bounded from above by a constant depending only on n. The gap theorem is a generalized version of Carron?s, Yun?s, Cavalcante?s and the first author?s results on submanifolds in Euclidean spaces and Seo?s result on submanifolds in hyperbolic space without the condition of minimality.     

Dimensional variance inequalities of Brascamp–Lieb type and a local approach to dimensional Prékopaʼs theorem

We give a new approach, inspired by Hörmander?s L²$L^2$-method, to weighted variance inequalities which extend results obtained by Bobkov and Ledoux. It provides in particular a local proof of the dimensional functional forms of the Brunn–Minkowski inequalities. We also present several applications of these variance inequalities, including reverse Hölder inequalities for convex functions, weighted Brascamp–Lieb inequalities and sharp weighted Poincaré inequalities for generalized Cauchy measures.     

A proof for Antczak’s mean value theorem in invexity analysis

This paper provides another proof for Antczak’s mean value theorem in invexity analysis.     

Nonlinear stability of traveling wave solutions for the compressible fluid models of Korteweg type

This paper is concerned with the existence and time-asymptotic nonlinear stability of traveling wave solutions to the Cauchy problem of the one-dimensional compressible fluid models of Korteweg type, which governs the motions of the compressible fluids with internal capillarity. The existence of traveling wave solutions is obtained by the phase plane analysis, then the traveling wave solution is shown to be asymptotically stable by the elementary L²$L^2$-energy method.     

Energy measures on p.c.f. self-similar sets

On connected post critically finite (p.c.f.) self-similar sets we give a linear extension method to compute the energy measures of harmonic functions with respect to the standard energy, and as an application we also compute the L²$L^2$ dimensions of these measures on some p.c.f. self-similar sets.     

Conditionally evenly convex sets and evenly quasi-convex maps

Evenly convex sets in a topological vector space are defined as the intersection of a family of open half spaces. We introduce a generalization of this concept in the conditional framework and provide a generalized version of the bipolar theorem. This notion is then applied to obtain the dual representation of conditionally evenly quasi-convex maps, which turns out to be a fundamental ingredient in the study of quasi-convex dynamic risk measures.     

Planar interpolation with a pair of rational spirals

Spirals are curves of one-signed, monotone increasing or decreasing curvature. Spiral segments have the advantage that the minimum and maximum curvatures are at their endpoints. Two situations where the planar, two-point G²$G^2$ Hermite interpolation problem can be solved with a pair of rational spiral segments are outlined here.     

Necessary and sufficient condition for uniqueness of solution to the first boundary value problem for the diffusion equation in unbounded domains

This paper introduces a notion of regularity of t=-∞$t=-\infty$ for the diffusion (or heat) equation and establishes a necessary and sufficient condition for the existence of a unique bounded solution to the first boundary value problem for the diffusion equation in a general domain Ω⊂R^N+1$\Omega \subset {\mathbb{R}}^{N+1}$ which extends up to t=-∞$t=-\infty$.     

Some remarks on the asymptotic profiles of solutions for strongly damped wave equations on the 1-D half space

We consider a mixed problem with the Dirichlet null boundary conditions on the 1-dimensional half space (0,+∞)$(0,+\infty )$. We will derive an asymptotic profiles of the corresponding solutions in the case when the initial data belong to a weighted L^1,2(0,∞)$L^{1,2}(0,\infty )$ space by employing a method recently introduced in [7] or [8].     

G cubic transition between two circles with shape control

This paper describes a method for joining two circles with an S-shaped or with a broken back C-shaped transition curve, composed of at most two spiral segments. In highway and railway route design or car-like robot path planning, it is often desirable to have such a transition. It is shown that a single cubic curve can be used for blending or for a transition curve preserving G²$G^2$ continuity with local shape control parameter and more flexible constraints. Provision of the shape parameter and flexibility provide freedom to modify the shape in a stable manner which is an advantage over previous work by Meek, Walton, Sakai and Habib.     

Well-posedness and time-decay for compressible viscoelastic fluids in critical Besov space

In this paper, we construct local solution with highly oscillating initial velocity and then get the global strong solution in the L^p$L^p$ based Besov space which improves the result of J. Qian, Z. Zhang (2010) [25] and X. Hu, D. Wang (2011) [14]. The local existence and uniqueness lies on the Lagrange coordinate transform and the contraction mapping theorem. The global result lies on a decomposition of the system and some commutator estimates. In the last part, we prove a time-decay in the critical Besov space framework which seems to have little investigation. The proof is based on the properties of the Green's matrix and various interpolations between Besov type spaces.     

Stable categories of higher preprojective algebras

We introduce (n+1)$(n+1)$-preprojective algebras of algebras of global dimension n$n$. We show that if an algebra is n$n$-representation-finite then its (n+1)$(n+1)$-preprojective algebra is self-injective. In this situation, we show that the stable module category of the (n+1)$(n+1)$-preprojective algebra is (n+1)$(n+1)$-Calabi–Yau, and, more precisely, it is the (n+1)$(n+1)$-Amiot cluster category of the stable n$n$-Auslander algebra of the original algebra. In particular this stable category contains an (n+1)$(n+1)$-cluster tilting object. We show that even if the (n+1)$(n+1)$-preprojective algebra is not self-injective, under certain assumptions (which are always satisfied for n∈{1,2}$n\in \{1,2\}$) the results above still hold for the stable category of Cohen–Macaulay modules.     

Asymptotic stability of the rarefaction wave for the generalized KdV–Burgers–Kuramoto equation

In this paper, we will study the Cauchy problem for the generalized KdV–Burgers–Kuramoto equation, which represents a dissipative, stroboscopic and unstable system in physics. When the initial data is a small disturbance of a rarefaction wave of the inviscid Burgers equation, we prove the global existence of the solution to the corresponding Cauchy problem and asymptotic stability of the rarefaction wave. The analysis is based on a priori estimates and the L²$L^2$-energy method.     

Asymptotic stability of rarefaction waves for the generalized KdV–Burgers equation on the half-line

We investigate the asymptotic behavior of solutions of the initial boundary value problem for the generalized KdV–Burgers equation _ut+f_(u)x=_uxx−_uxxx$u_t+f{\left(u\right)}_x=u_{xx}-u_{xxx}$ on the half-line with the boundary condition u(0,t)=_u−$u(0,t)=u_{-}$. The corresponding Cauchy problems of the behaviors of weak and strong rarefaction waves have respectively been studied by Wang and Zhu [Z.A. Wang, C.J. Zhu, Stability of the rarefaction wave for the generalized KdV–Burgers equation, Acta Math. Sci. 22B (3) (2002) 309–328] and Duan and Zhao [R. Duan, H.J. Zhao, Global stability of strong rarefaction waves for the generalized KdV–Burgers equation, Nonlinear Anal. TMA 66 (2007) 1100–1117]. In the present problem, on the basis of the Dirichlet boundary conditions, the asymptotic states are divided into five cases dependent on the signs of the characteristic speeds f^′(_u±)$f^{\prime }\left(u_{\pm }\right)$. In the cases of 0≤f^′(_u−)<f^′(_u+)$0\le f^{\prime }\left(u_{-}\right)<f^{\prime }\left(u_{+}\right)$, we prove the global existence of solutions and asymptotic stability of the weak rarefaction waves when the initial disturbance is small. Also, we can get asymptotic stability of the strong rarefaction waves when f(u)$f\left(u\right)$ satisfies a certain growth condition.