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${?}^2$.     

Maximum principles and isoperimetric inequalities for some Monge–Ampère-type problems

In this note we derive a maximum principle for an appropriate functional combination of u(x)$u(\mathbf{x})$ and |∇u|²${|\mathrm{\nabla }u|}^2$, where u(x)$u(\mathbf{x})$ is a strictly convex classical solution to a general class of Monge–Ampère equations. This maximum principle is then employed to establish some isoperimetric inequalities of interest in the theory of surfaces of constant Gauss curvature in R^N+1${\mathbb{R}}^{N+1}$.     

The spherical ergodic theorem revisited

In this paper, we give a new proof of a result of R. Jones showing almost everywhere convergence of spherical means of actions of R^d${\mathbb{R}}^d$ on L^p(X)$L^p(X)$-spaces are convergent for d?3$d?3$ and p>d/(d-1)$p>d/(d-1)$.     

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].     

Generic global rigidity of body–bar frameworks

A basic geometric question is to determine when a given framework G(p)$G(\mathbf{p})$ is globally rigid in Euclidean space R^d${\mathbb{R}}^d$, where G is a finite graph and p is a configuration of points corresponding to the vertices of G  . G(p)$G(\mathbf{p})$ is globally rigid in  R^d${\mathbb{R}}^d$ if for any other configuration q for G   such that the edge lengths of G(q)$G(\mathbf{q})$ are the same as the corresponding edge lengths of G(p)$G(\mathbf{p})$, then p is congruent to q. A framework G(p)$G(\mathbf{p})$ is redundantly rigid, if it is rigid and it remains rigid after the removal of any edge of G.     

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.     

Gaussian bounds for higher-order elliptic differential operators with Kato type potentials

Let P(D)$P(D)$ be a nonnegative homogeneous elliptic operator of order 2m   with real constant coefficients on Rⁿ${\mathbb{R}}^n$ and V   be a suitable real measurable function. In this paper, we are mainly devoted to establish the Gaussian upper bound for Schrödinger type semigroup e^−tH$e^{-tH}$ generated by H=P(D)+V$H=P(D)+V$ with Kato type perturbing potential V  , which naturally generalizes the classical result for Schrödinger semigroup e^−t(Δ+V)$e^{-t(\mathrm{\Delta }+V)}$ as V∈_K2(Rⁿ)$V\in K_2({\mathbb{R}}^n)$, the famous Kato potential class. Our proof significantly depends on the analyticity of the free semigroup e^−tP(D)$e^{-tP(D)}$ on L¹(Rⁿ)$L^1({\mathbb{R}}^n)$. As a consequence of the Gaussian upper bound, the L^p$L^p$-spectral independence of H is concluded.     

Laguerre functions on symmetric cones and recursion relations in the real case

In this article we derive differential recursion relations for the Laguerre functions on the cone Ω$\Omega$ of positive definite real matrices. The highest weight representations of the group Sp(n,R)$\mathrm{Sp}(n,\mathbb{R})$ play a fundamental role. Each such representation acts on a Hilbert space of holomorphic functions on the tube domain Ω+iSym(n,R)$\Omega +i\mathrm{Sym}(n,\mathbb{R})$. We then use the Laplace transform to carry the Lie algebra action over to L²(Ω,d_μν)$L^2(\Omega ,\mathrm{d}{\mu }_{\nu })$. The differential recursion relations result by restricting to a distinguished three-dimensional subalgebra, which is isomorphic to sl(2,R).$\mathfrak{sl}(2,\mathbb{R}).$