The adjoint representation inside the exterior algebra of a simple Lie algebra

For a simple complex Lie algebra g$\mathfrak{g}$ we study the space of invariants A=(?g^?⊗g^?)^g$A={(?{\mathfrak{g}}^{?}\otimes {\mathfrak{g}}^{?})}^{\mathfrak{g}}$, which describes the isotypic component of type g$\mathfrak{g}$ in ?g^?$?{\mathfrak{g}}^{?}$, as a module over the algebra of invariants (?g^?)^g${(?{\mathfrak{g}}^{?})}^{\mathfrak{g}}$. As main result we prove that A   is a free module, of rank twice the rank of g$\mathfrak{g}$, over the exterior algebra generated by all primitive invariants in (?g^?)^g${(?{\mathfrak{g}}^{?})}^{\mathfrak{g}}$, with the exception of the one of highest degree.     

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].     

Test ideals of non-principal ideals: Computations,jumping numbers,alterations and division theorems

Given an ideal a⊆R$\mathfrak{a}\subseteq R$ in a (log) Q$\mathbb{Q}$-Gorenstein F  -finite ring of characteristic p>0$p>0$, we study and provide a new perspective on the test ideal τ(R,a^t)$\tau (R,{\mathfrak{a}}^t)$ for a real number t>0$t>0$. Generalizing a number of known results from the principal case, we show how to effectively compute the test ideal and also describe τ(R,a^t)$\tau (R,{\mathfrak{a}}^t)$ using (regular) alterations with a formula analogous to that of multiplier ideals in characteristic zero. We further prove that the F  -jumping numbers of τ(R,a^t)$\tau (R,{\mathfrak{a}}^t)$ as t varies are rational and have no limit points, including the important case where R is a formal power series ring. Additionally, we obtain a global division theorem for test ideals related to results of Ein and Lazarsfeld from characteristic zero, and also recover a new proof of Skoda's theorem for test ideals which directly mimics the proof for multiplier ideals.     

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.     

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${?}^2$.     

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${\mathbb{R}}^2$.     

Measurability of semimartingale characteristics with respect to the probability law

Given a càdlàg process X$X$ on a filtered measurable space, we construct a version of its semimartingale characteristics which is measurable with respect to the underlying probability law. More precisely, let _Psem${\mathfrak{P}}_{sem}$ be the set of all probability measures P$P$ under which X$X$ is a semimartingale. We construct processes (B^P,C,ν^P)$(B^P,C,{\nu }^P)$ which are jointly measurable in time, space, and the probability law P$P$, and are versions of the semimartingale characteristics of X$X$ under P$P$ for each P∈_Psem$P\in {\mathfrak{P}}_{sem}$. This result gives a general and unifying answer to measurability questions that arise in the context of quasi-sure analysis and stochastic control under the weak formulation.