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].     

Modules satisfying the weak Nakayama property

Let R$R$ be a commutative ring with identity. We will say that an R$R$-module M$M$ satisfies the weak Nakayama property, if IM=M$IM=M$, where I$I$ is an ideal of R$R$, implies that for any x∈M$x\in M$ there exists a∈I$a\in I$ such that (a−1)x=0$(a-1)x=0$. In this paper, we will study modules satisfying the weak Nakayama property. It is proved that if R$R$ is a local ring, then R$R$ is a Max ring if and only if J(R)$J\left(R\right)$, the Jacobson radical of R$R$, is T$T$-nilpotent if and only if every R$R$-module satisfies the weak Nakayama property.     

On the conjugacy classes in the orthogonal and symplectic groups over algebraically closed fields

Let F$\mathbb{F}$ be an algebraically closed field. Let V$\mathbb{V}$ be a vector space equipped with a non-degenerate symmetric or symplectic bilinear form B   over F$\mathbb{F}$. Suppose the characteristic of F$\mathbb{F}$ is sufficiently large  , i.e. either zero or greater than the dimension of V$\mathbb{V}$. Let I(V,B)$I(\mathbb{V},\mathbb{B})$ denote the group of isometries. Using the Jacobson–Morozov lemma we give a new and simple proof of the fact that two elements in I(V,B)$I(\mathbb{V},\mathbb{B})$ are conjugate if and only if they have the same elementary divisors.     

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${?}^2$.     

Multidimensional extension of the Morse–Hedlund theorem

A celebrated result of Morse and Hedlund, stated in 1938, asserts that a sequence x$x$ over a finite alphabet is ultimately periodic if and only if, for some n$n$, the number of different factors of length n$n$ appearing in x$x$ is less than n+1$n+1$. Attempts to extend this fundamental result, for example, to higher dimensions, have been considered during the last fifteen years. Let d≥2$d\ge 2$. A legitimate extension to a multidimensional setting of the notion of periodicity is to consider sets of Z^d${\mathbb{Z}}^d$ definable by a first order formula in the Presburger arithmetic 〈Z;<,+〉$〈\mathbb{Z};<,+〉$. With this latter notion and using a powerful criterion due to Muchnik, we exhibit a complete extension of the Morse–Hedlund theorem to an arbitrary dimension d$d$ and characterize sets of Z^d${\mathbb{Z}}^d$ definable in 〈Z;<,+〉$〈\mathbb{Z};<,+〉$ in terms of some functions counting recurrent blocks, that is, blocks occurring infinitely often.     

Analysis of symmetric matrix valued functions I

For any symmetric function f:Rⁿ?Rⁿ$f:{\mathbb{R}}^n?{\mathbb{R}}^n$, one can define a corresponding function on the space of n×n$n\times n$ real symmetric matrices by applying f$f$ to the eigenvalues of the spectral decomposition. We show that this matrix valued function inherits from f$f$ the properties of continuity, Lipschitz continuity, strict continuity, directional differentiability, Frechet differentiability, continuous differentiability.     

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$N?1$.     

Existence and boundary stabilization of solutions for the coupled semilinear system

We investigate the global existence of both strong and weak solutions for a semilinear coupled system with homogeneous feedback boundary conditions in bounded open domain Ω$\Omega$ in Rⁿ${\mathbb{R}}^n$ with n∈N$n\in \mathbb{N}$. We also prove the exponential decay of total energy associated with weak solutions.     

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${\mathbb{R}}^2$.     

Description of extended eigenvalues and extended eigenvectors of integration operators on the Wiener algebra

In the present paper we consider the Volterra integration operator V   on the Wiener algebra W(D)$W(\mathbb{D})$ of analytic functions on the unit disc D$\mathbb{D}$ of the complex plane C$\mathbb{C}$. A complex number λ$\lambda$ is called an extended eigenvalue of V if there exists a nonzero operator A   satisfying the equation AV=λVA$\mathit{AV}=\lambda \mathit{VA}$. We prove that the set of all extended eigenvalues of V   is precisely the set C?{0}$\mathbb{C}?\{0\}$, and describe in terms of Duhamel operators and composition operators the set of corresponding extended eigenvectors of V$V$. The similar result for some weighted shift operator on _?p${?}_p$ spaces is also obtained.     

Large deviations of infinite intersections of events in Gaussian processes

Consider events of the form {_Zs≥ζ(s),s∈S}$\{Z_s\ge \zeta \left(s\right),s\in S\}$, where Z$Z$ is a continuous Gaussian process with stationary increments, ζ$\zeta$ is a function that belongs to the reproducing kernel Hilbert space R$R$ of process Z$Z$, and S⊂R$S\subset \mathbb{R}$ is compact. The main problem considered in this paper is identifying the function β^∗∈R${\beta }^{\ast }\in R$ satisfying β^∗(s)≥ζ(s)${\beta }^{\ast }\left(s\right)\ge \zeta \left(s\right)$ on S$S$ and having minimal R$R$-norm. The smoothness (mean square differentiability) of Z$Z$ turns out to have a crucial impact on the structure of the solution. As examples, we obtain the explicit solutions when ζ(s)=s$\zeta \left(s\right)=s$ for s∈[0,1]$s\in [0,1]$ and Z$Z$ is either a fractional Brownian motion or an integrated Ornstein–Uhlenbeck process.     

A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation

We consider the semilinear parabolic equation _ut=Δu+u^p$u_t=\mathrm{\Delta }u+u^p$ on R^N${\mathbb{R}}^N$, where the power nonlinearity is subcritical. We first address the question of existence of entire solutions, that is, solutions defined for all x∈R^N$x\in {\mathbb{R}}^N$ and t∈R$t\in \mathbb{R}$. Our main result asserts that there are no positive radially symmetric bounded entire solutions. Then we consider radial solutions of the Cauchy problem. We show that if such a solution is global, that is, defined for all t?0$t?0$, then it necessarily converges to 0, as t→∞$t\to \infty$, uniformly with respect to x∈R^N$x\in {\mathbb{R}}^N$.     

Factorization properties of paratopological groups

In this article we continue the study of R$\mathbb{R}$-factorizability in paratopological groups. It is shown that: (1) all concepts of R$\mathbb{R}$-factorizability in paratopological groups coincide; (2) a Tychonoff paratopological group G   is R$\mathbb{R}$-factorizable if and only if it is totally ω  -narrow and has property ω-QU$\omega \text{-}\mathit{QU}$; (3) every subgroup of a _T1$T_1$ paratopological group G   is R$\mathbb{R}$-factorizable provided that the topological group G^?$G^{?}$ associated to G is a Lindelöf Σ-space, i.e., G is a totally Lindelöf Σ-space  ; (4) if Π=_∏i∈IGi$\Pi ={\prod }_{i\in I}{G}_i$ is a product of _T1$T_1$ paratopological groups which are totally Lindelöf Σ-spaces, then each dense subgroup of Π   is R$\mathbb{R}$-factorizable. These results answer in the affirmative several questions posed earlier by M. Sanchis and M. Tkachenko and by S. Lin and L.-H. Xie.     

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$1\ne \alpha <3$.