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].     

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.     

On testing Hamiltonicity of graphs

Let us fix a function f(n)=o(nlnn)$f\left(n\right)=o(nlnn)$ and real numbers 0≤α<β≤1$0\le \alpha <\beta \le 1$. We present a polynomial time algorithm which, given a directed graph G$G$ with n$n$ vertices, decides either that one can add at most βn$\beta n$ new edges to G$G$ so that G$G$ acquires a Hamiltonian circuit or that one cannot add αn$\alpha n$ or fewer new edges to G$G$ so that G$G$ acquires at least e^−f(n)n!$e^{-f\left(n\right)}n!$ Hamiltonian circuits, or both.     

Constructing cubature formulae of degree 5 with few points

This paper is devoted to construct a family of fifth degree cubature formulae for n$n$-cube with symmetric measure and n$n$-dimensional spherically symmetrical region. The formula forn$n$-cube contains at most n²+5n+3$n^2+5n+3$ points and for n$n$-dimensional spherically symmetrical region contains only n²+3n+3$n^2+3n+3$ points. Moreover, the numbers can be reduced to n²+3n+1$n^2+3n+1$ and n²+n+1$n^2+n+1$ if n=7$n=7$ respectively, the latter of which is minimal.     

Tournaments associated with multigraphs and a theorem of Hakimi

A tournament of order n$n$ is usually considered as an orientation of the complete graph _Kn$K_n$. In this note, we consider a more general definition of a tournament that we call aC$C$-tournament, where C$C$ is the adjacency matrix of a multigraph G$G$, and a C$C$-tournament is an orientation of G$G$. The score vector of a C$C$-tournament is the vector of outdegrees of its vertices. In 1965 Hakimi obtained necessary and sufficient conditions for the existence of a C$C$-tournament with a prescribed score vector R$R$ and gave an algorithm to construct such a C$C$-tournament which required, however, some backtracking. We give a simpler and more transparent proof of Hakimi’s theorem, and then provide a direct construction of such a C$C$-tournament which works even for weighted graphs.     

Estimates for eigenvalues of the Paneitz operator

For an n  -dimensional compact submanifold Mⁿ$M^n$ in the Euclidean space R^N${\mathbf{R}}^N$, we study estimates for eigenvalues of the Paneitz operator on Mⁿ$M^n$. Our estimates for eigenvalues are sharp.     

On the length of an external branch in the Beta-coalescent

In this paper, we consider Beta(2−α,α)$(2-\alpha ,\alpha )$ (with 1<α<2$1<\alpha <2$) and related Λ$\Lambda$-coalescents. If T⁽ⁿ⁾$T^{\left(n\right)}$ denotes the length of a randomly chosen external branch of the n$n$-coalescent, we prove the convergence of n^α−1T⁽ⁿ⁾$n^{\alpha -1}{T}^{\left(n\right)}$ when n$n$ tends to ∞$\infty$, and give the limit. To this aim, we give asymptotics for the number σ⁽ⁿ⁾${\sigma }^{\left(n\right)}$ of collisions which occur in the n$n$-coalescent until the end of the chosen external branch, and for the block counting process associated with the n$n$-coalescent.     

Uniqueness of the Cheeger set of a convex body

We prove that if C⊂R^N$C\subset {\mathbb{R}}^N$ is an open bounded convex set, then there is only one Cheeger set inside C$C$ and it is convex. A Cheeger set of C$C$ is a set which minimizes the ratio perimeter over volume among all subsets of C$C$.     

Partial rigidity of degenerate CR embeddings into spheres

In this paper, we study degenerate CR embeddings f$f$ of a strictly pseudoconvex hypersurface M⊂Cⁿ⁺¹$M\subset {\mathbb{C}}^{n+1}$ into a sphere S$\mathbb{S}$ in a higher dimensional complex space C^N+1${\mathbb{C}}^{N+1}$. The degeneracy of the mapping f$f$ will be characterized in terms of the ranks of the CR second fundamental form and its covariant derivatives. In 2004, the author, together with X. Huang and D. Zaitsev, established a rigidity result for CR embeddings f$f$ into spheres in low codimensions. A key step in the proof of this result was to show that degenerate mappings are necessarily contained in a complex plane section of the target sphere (partial rigidity). In the 2004 paper, it was shown that if the total rank d$d$ of the second fundamental form and all of its covariant derivatives is <n$<n$ (here, n$n$ is the CR dimension of M$M$), then f(M)$f\left(M\right)$ is contained in a complex plane of dimension n+d+1$n+d+1$. The converse of this statement is also true, as is easy to see. When the total rank d$d$ exceeds n$n$, it is no longer true, in general, that f(M)$f\left(M\right)$ is contained in a complex plane of dimension n+d+1$n+d+1$, as can be seen by examples. In this paper, we carry out a systematic study of degenerate CR mappings into spheres. We show that when the ranks of the second fundamental form and its covariant derivatives exceed the CR dimension n$n$, then partial rigidity may still persist, but there is a “defect” k$k$ that arises from the ranks exceeding n$n$ such that f(M)$f\left(M\right)$ is only contained in a complex plane of dimension n+d+k+1$n+d+k+1$. Moreover, this defect occurs in general, as is illustrated by examples.     

Blow-up of periodic solutions to linearly degenerate quasilinear hyperbolic systems

An example is given to show that for the Cauchy problem of n×n$n\times n$ homogeneous linearly degenerate quasilinear hyperbolic systems with periodic initial data, when n>2$n>2$, C¹$C^1$ solutions may blow up in a finite time, no matter how small and smooth the initial data are.     

On the sum of two closed subspaces

If U,V$U,V$ are closed subspaces of a Fréchet space, then E$E$ is the direct sum of U$U$ and V$V$ if and only if E^′$E^{\prime }$ is the algebraic direct sum of the annihilators U^°$U^{°}$ and V^°$V^{°}$. We provide a simple proof of this (possibly well-known) result.     

Uniqueness properties of m-subharmonic functions in Cegrell classes

Let u,v$u,v$ be m-subharmonic functions defined on a domain Ω   in Cⁿ${\mathbb{C}}^n$. We are interested in giving sufficient conditions on u,v$u,v$ such that u=v$u=v$ on the whole domain Ω. Some applications to weak convergence of sequence of m-subharmonic functions are also discussed.     

Oscillation of harmonic functions for subordinate Brownian motion and its applications

In this paper, we establish an oscillation estimate of nonnegative harmonic functions for a pure-jump subordinate Brownian motion. The infinitesimal generator of such subordinate Brownian motion is an integro-differential operator. As an application, we give a probabilistic proof of the following form of relative Fatou theorem for such subordinate Brownian motion X$X$ in a bounded κ$\kappa$-fat open set; if u$u$ is a positive harmonic function with respect to X$X$ in a bounded κ$\kappa$-fat open set D$D$ and h$h$ is a positive harmonic function in D$D$ vanishing on D^c$D^c$, then the non-tangential limit of u/h$u/h$ exists almost everywhere with respect to the Martin-representing measure of h$h$.     

The Morse–Sard theorem for Clarke critical values

The Morse–Sard theorem states that the set of critical values of a C^k$C^k$ smooth function defined on a Euclidean space R^d${\mathbb{R}}^d$ has Lebesgue measure zero, provided k≥d$k\ge d$. This result is hereby extended for (generalized) critical values of continuous selections over a compactly indexed countable family of C^k$C^k$ functions: it is shown that these functions are Lipschitz continuous and the set of their Clarke critical values is null.     

Cone isomorphisms and almost surjective operators

Let E$E$ be a Banach lattice and F$F$ a Banach space. A bounded linear operator T:E→F$T:E\to F$ is an isomorphism on the positive cone of E$E$ if and only if T^∗$T^{\ast }$ is almost surjective. A dual version of this theorem holds also. A bounded linear operator T:F→E$T:F\to E$ is almost surjective if and only if T^∗$T^{\ast }$ is an isomorphism on the positive cone of F^∗$F^{\ast }$.     

Varieties of local integrability of analytic differential systems and their applications

In this paper we provide a characterization of local integrability for analytic or formal differential systems in Rⁿ${\mathbb{R}}^n$ or Cⁿ${\mathbb{C}}^n$ via the integrability varieties. Our result generalizes the classical one of Poincaré and Lyapunov on local integrability of planar analytic differential systems to any finitely dimensional analytic differential systems. As an application of our theory we study the integrability of a family of four-dimensional quadratic Hamiltonian systems.