Stable extendibility of vector bundles over real projective spaces

Let F$\mathbb{F}$ be either the real number field R$\mathbb{R}$ or the complex number field C$\mathbb{C}$ and RPⁿ${\mathbb{RP}}^n$ the real projective space of dimension n. Theorems A and C in Hemmi and Kobayashi (2008) [2] give necessary and sufficient conditions for a given F$\mathbb{F}$-vector bundle over RPⁿ${\mathbb{RP}}^n$ to be stably extendible to RP^m${\mathbb{RP}}^m$ for every m?n$m?n$. In this paper, we simplify the theorems and apply them to the tangent bundle of RPⁿ${\mathbb{RP}}^n$, its complexification, the normal bundle associated to an immersion of RPⁿ${\mathbb{RP}}^n$ in R^n+r${\mathbb{R}}^{n+r}$(r>0)$(r>0)$, and its complexification. Our result for the normal bundle is a generalization of Theorem A in Kobayashi et al. (2000) [8] and that for its complexification is a generalization of Theorem 1 in Kobayashi and Yoshida (2003) [5].     

The topological structure of fuzzy sets with sendograph metric

For a non-degenerate convex subset Y of the n  -dimensional Euclidean space Rⁿ${\mathbb{R}}^n$, let F(Y)$\mathbb{F}(Y)$ be the family of all fuzzy sets of Rⁿ${\mathbb{R}}^n$ which are upper semicontinuous, fuzzy convex and normal with compact supports contained in Y  . We show that the space F(Y)$\mathbb{F}(Y)$ with the topology of sendograph metric is homeomorphic to the separable Hilbert space ?²${?}^2$ if Y   is compact; and the space F(Rⁿ)$\mathbb{F}({\mathbb{R}}^n)$ is also homeomorphic to ?²${?}^2$.     

A unified proof on the partial regularity for suitable weak solutions of non-stationary and stationary Navier–Stokes equations

The partial regularity of the suitable weak solutions to the Navier–Stokes equations in Rⁿ${\mathbb{R}}^n$ with n=2,3,4$n=2,3,4$ and the stationary Navier–Stokes equations in Rⁿ${\mathbb{R}}^n$ for n=2,3,4,5,6$n=2,3,4,5,6$ are investigated in this paper. Using some elementary observation of these equations together with De Giorgi iteration method, we present a unified proof on the results of Caffarelli, Kohn and Nirenberg [1], Struwe [17], Dong and Du [5], and Dong and Strain [7]. Particularly, we obtain the partial regularity of the suitable weak solutions to the 4d non-stationary Navier–Stokes equations, which improves the previous result of [5], where Dong and Du studied the partial regularity of smooth solutions of the 4d Navier–Stokes equations at the first blow-up time.     

Energy decay for the linear Klein–Gordon equation and boundary control

In this work we study the asymptotic behavior of the solutions of the linear Klein–Gordon equation in R^N${\mathbb{R}}^N$, N?1$N?1$. We prove that local energy of solutions to the Cauchy problem decays polynomially. Afterwards, we use the local decay of energy to study exact boundary controllability for the linear Klein–Gordon equation in general bounded domains of R^N${\mathbb{R}}^N$, N?1$N?1$.     

On removable sets for convex functions

In the present article we provide a sufficient condition for a closed set F∈R^d$F\in {\mathbb{R}}^d$ to have the following property which we call c  -removability: Whenever a continuous function f:R^d→R$f:{\mathbb{R}}^d\to \mathbb{R}$ is locally convex on the complement of F  , it is convex on the whole R^d${\mathbb{R}}^d$. We also prove that no generalized rectangle of positive Lebesgue measure in R²${\mathbb{R}}^2$ is c-removable. Our results also answer the following question asked in an article by Jacek Tabor and Józef Tabor (2010) [5]: Assume the closed set F⊂R^d$F\subset {\mathbb{R}}^d$ is such that any locally convex function defined on R^d?F${\mathbb{R}}^d?F$ has a unique convex extension on R^d${\mathbb{R}}^d$. Is F   necessarily intervally thin (a notion of smallness of sets defined by their “essential transparency” in every direction)? We prove the answer is negative by finding a counterexample in R²${\mathbb{R}}^2$.     

An upper bound for a valence of a face in a parallelohedral tiling

Consider a face-to-face parallelohedral tiling of R^d${\mathbb{R}}^d$ and a (d−k)$(d-k)$-dimensional face F$F$ of the tiling. We prove that the valence of F$F$ (i.e. the number of tiles containing F$F$ as a face) is not greater than 2^k$2^k$. If the tiling is affinely equivalent to a Voronoi tiling for some lattice (the so called Voronoi case), this gives a well-known upper bound for the number of vertices of a Delaunay k$k$-cell. Yet we emphasize that such an affine equivalence is not assumed in the proof.     

Asymptotic average shadowing property on compact metric spaces

We prove that if for a continuous map f$f$ on a compact metric space X$X$, the chain recurrent set, R(f)$R\left(f\right)$ has more than one chain component, then f$f$ does not satisfy the asymptotic average shadowing property. We also show that if a continuous map f$f$ on a compact metric space X$X$ has the asymptotic average shadowing property and if A$A$ is an attractor for f$f$, then A$A$ is the single attractor for f$f$ and we have A=R(f)$A=R\left(f\right)$. We also study diffeomorphisms with asymptotic average shadowing property and prove that if M$M$ is a compact manifold which is not finite with dimM=2$dimM=2$, then the C¹$C^1$ interior of the set of all C¹$C^1$ diffeomorphisms with the asymptotic average shadowing property is characterized by the set of Ω$\Omega$-stable diffeomorphisms.     

Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation

We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical (α<1/2$\alpha <1/2$) dissipation α(−Δ)${(-\mathrm{\Delta })}^{\alpha }$: If a Leray–Hopf weak solution is Hölder continuous θ∈Cδ(R²)$\theta \in C^{\delta }({\mathbb{R}}^2)$ with δ>1−2α$\delta >1-2\alpha$ on the time interval [t₀,t]$[t_0,t]$, then it is actually a classical solution on (t₀,t]$(t_0,t]$.     

Harness processes and harmonic crystals

In the Hammersley harness processes the R$\mathbb{R}$-valued height at each site i∈Z^d$i\in {\mathbb{Z}}^d$ is updated at rate 1 to an average of the neighboring heights plus a centered random variable (the noise). We construct the process “a la Harris” simultaneously for all times and boxes contained in Z^d${\mathbb{Z}}^d$. With this representation we compute covariances and show L²$L^2$ and almost sure time and space convergence of the process. In particular, the process started from the flat configuration and viewed from the height at the origin converges to an invariant measure. In dimension three and higher, the process itself converges to an invariant measure in L²$L^2$ at speed t^1−d/2$t^{1-d/2}$ (this extends the convergence established by Hsiao). When the noise is Gaussian the limiting measures are Gaussian fields (harmonic crystals) and are also reversible for the process.     

Monotonicity of the reflected Bessel transition density on the diagonal

For α∈R$\alpha \in \mathbb{R}$, let p^R(t,x,x)$p^R(t,x,x)$ denote the diagonal of the transition density of the α$\alpha$-Bessel process in (0,1]$(0,1]$, killed at 0 and reflected at 1. As a function of x$x$, if either α≥3$\alpha \ge 3$ or α=1$\alpha =1$, then for t>0$t>0$, the diagonal is nondecreasing. This monotonicity property fails if 1≠α<3$1\ne \alpha <3$.     

Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations

We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical (α<1/2$\alpha <1/2$) dissipation α(−Δ)${(-\mathrm{\Delta })}^{\alpha }$. This study is motivated by a recent work of Caffarelli and Vasseur, in which they study the global regularity issue for the critical (α=1/2$\alpha =1/2$) QG equation [L. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, arXiv: math.AP/0608447, 2006]. Their approach successively increases the regularity levels of Leray–Hopf weak solutions: from L²$L^2$ to L^∞$L^{\infty }$, from L^∞$L^{\infty }$ to Hölder (Cδ$C^{\delta }$, δ>0$\delta >0$), and from Hölder to classical solutions. In the supercritical case, Leray–Hopf weak solutions can still be shown to be L^∞$L^{\infty }$, but it does not appear that their approach can be easily extended to establish the Hölder continuity of L^∞$L^{\infty }$ solutions. In order for their approach to work, we require the velocity to be in the Hölder space C1−2α$C^{1-2\alpha }$. Higher regularity starting from Cδ$C^{\delta }$ with δ>1−2α$\delta >1-2\alpha$ can be established through Besov space techniques and will be presented elsewhere [P. Constantin, J. Wu, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press].     

Linearly repetitive Delone sets are rectifiable

We show that every linearly repetitive Delone set in the Euclidean d  -space R^d${\mathbb{R}}^d$, with d?2$d?2$, is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice Z^d${\mathbb{Z}}^d$. In the particular case when the Delone set X   in R^d${\mathbb{R}}^d$ comes from a primitive substitution tiling of R^d${\mathbb{R}}^d$, we give a condition on the eigenvalues of the substitution matrix which ensures the existence of a homeomorphism with bounded displacement from X   to the lattice βZ^d$\beta {\mathbb{Z}}^d$ for some positive β. This condition includes primitive Pisot substitution tilings but also concerns a much broader set of substitution tilings.     

Hadwiger’s Theorem for definable functions

Hadwiger’s Theorem states that _En${\mathbb{E}}_n$-invariant convex-continuous valuations of definable sets in Rⁿ${\mathbb{R}}^n$ are linear combinations of intrinsic volumes. We lift this result from sets to data distributions over sets, specifically, to definable R$\mathbb{R}$-valued functions on Rⁿ${\mathbb{R}}^n$. This generalizes intrinsic volumes to (dual pairs of) non-linear valuations on functions and provides a dual pair of Hadwiger classification theorems.     

The Helmholtz equation in a non-smooth inclusion

In this paper, we study the Helmholtz equation in a non-smooth inclusion, i.e., in a doubly connected bounded domain B$B$ in R²$R^2$ with boundary ∂B$\partial B$ that consists of two disjoint closed curves Γ$\Gamma$ and _Γ0${\Gamma }_0$. The existence and uniqueness of a solution to the Helmholtz equation for mixed boundary conditions on Γ$\Gamma$ are obtained by using Riesz–Fredholm theory.     

Universal polynomials for Severi degrees of toric surfaces

The Severi variety parameterizes plane curves of degree d$d$ with δ$\delta$ nodes. Its degree is called the Severi degree. For large enough d$d$, the Severi degrees coincide with the Gromov–Witten invariants of CP²$\mathbb{C}{\mathbb{P}}^2$. Fomin and Mikhalkin (2010) [10] proved the 1995 conjecture that for fixed δ$\delta$, Severi degrees are eventually polynomial in d$d$.