Annihilator-stability and unique generation

A ring R is said to be left uniquely generated if $Ra=Rb$ in R implies that $a=ub$ for some unit u in R. These rings have been of interest since Kaplansky introduced them in 1949 in his classic study of elementary divisors. Writing $\mathtt{l}(b)=\{r\in R|rb=0\}$, a theorem of Canfell asserts that R is left uniquely generated if and only if, whenever $Ra+\mathtt{l}(b)=R$ where $a,b\in R$, then $a?u\in \mathtt{l}(b)$ for some unit u in R. By analogy with the stable range 1 condition we call a ring with this property left annihilator-stable. In this paper we exploit this perspective on the left UG rings to construct new examples and derive new results. For example, writing J for the Jacobson radical, we show that a semiregular ring R is left annihilator-stable if and only if $R/J$ is unit-regular, an analogue of Bass' theorem that semilocal rings have stable range 1.     

First order logic without equality on relativized semantics

Let $\alpha \ge 2$ be any ordinal. We consider the class ${\mathsf{Drs}}_{\alpha }$ of relativized diagonal free set algebras of dimension α. With same technique, we prove several important results concerning this class. Among these results, we prove that almost all free algebras of ${\mathsf{Drs}}_{\alpha }$ are atomless, and none of these free algebras contains zero-dimensional elements other than zero and top element. The class ${\mathsf{Drs}}_{\alpha }$ corresponds to first order logic, without equality symbol, with α-many variables and on relativized semantics. Hence, in this variation of first order logic, there is no finitely axiomatizable, complete and consistent theory.     

The group of invariants of an inner function with finite spectrum

This paper determines the group of continuous invariants corresponding to an inner function Θ with finitely many singularities on the unit circle $\mathbb{T}$; that is, the continuous mappings $g:\mathbb{T}\to \mathbb{T}$ such that $\Theta °g=\Theta$ on $\mathbb{T}$. These mappings form a group under composition.     

Minimal unit vector fields with respect to Riemannian natural metrics

Let $(M,g)$ be a Riemannian manifold. We denote by $\stackrel{?}{G}$ an arbitrary Riemannian g-natural metric on the unit tangent sphere bundle $T_{1}M$, such metric depends on four real parameters satisfying some inequalities. The Sasaki metric, the Cheeger–Gromoll metric and the Kaluza–Klein metrics are special Riemannian g-natural metrics. In literature, minimal unit vector fields have been already investigated, considering $T_{1}M$ equipped with the Sasaki metric ${\stackrel{?}{G}}_S$ [12]. In this paper we extend such characterization to an arbitrary Riemannian g-natural metric $\stackrel{?}{G}$. In particular, the minimality condition with respect to the Sasaki metric ${\stackrel{?}{G}}_S$ is invariant under a two-parameters deformation of the Sasaki metric. Moreover, we show that a minimal unit vector field (with respect to $\stackrel{?}{G}$) corresponds to a minimal submanifold. Then, we give examples of minimal unit vector fields (with respect to $\stackrel{?}{G}$). In particular, we get that the Hopf vector fields of the unit sphere, the Reeb vector field of a K-contact manifold, and the Hopf vector field of a quasi-umbilical hypersurface with constant principal curvatures in a Kähler manifold, are minimal unit vector fields (with respect to $\stackrel{?}{G}$).     

A class of nonpositively curved Kähler manifolds biholomorphic to the unit ball in Cn

Let $(M,g)$ be a simply connected complete Kähler manifold with nonpositive sectional curvature. Assume that g has constant negative holomorphic sectional curvature outside a compact set. We prove that M is then biholomorphic to the unit ball in ${\mathbb{C}}^n$, where ${\dim }_{\mathbb{C}}M=n$. To cite this article: H. Seshadri, K. Verma, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Cohomology over Fiber products of local rings

Let S and T be local rings with common residue field k, let R be the fiber product $S{\times }_{k}T$, and let M be an S-module. The Poincaré series $P_M^R$ of M has been expressed in terms of $P_M^S$, $P_k^S$ and $P_k^T$ by Kostrikin and Shafarevich, and by Dress and Krämer. Here, an explicit minimal resolution, as well as theorems on the structure of ${\mathrm{Ext}}_R(k,k)$ and ${\mathrm{Ext}}_R(M,k)$ are given that illuminate these equalities. Structure theorems for the cohomology modules of fiber products of modules are also given. As an application of these results, we compute the depth of cohomology modules over a fiber product.     

Topological invariants for semigroups of holomorphic self-maps of the unit disk

Let $({\phi }_t)$, $({?}_t)$ be two one-parameter semigroups of holomorphic self-maps of the unit disk $\mathbb{D}?\mathbb{C}$. Let $f:\mathbb{D}\to \mathbb{D}$ be a homeomorphism. We prove that, if $f°{?}_t={\phi }_t°f$ for all $t\ge 0$, then f extends to a homeomorphism of $\stackrel{￣}{\mathbb{D}}$ outside exceptional maximal contact arcs (in particular, for elliptic semigroups, f extends to a homeomorphism of $\stackrel{￣}{\mathbb{D}}$). Using this result, we study topological invariants for one-parameter semigroups of holomorphic self-maps of the unit disk.     

Generic uniqueness of the bias vector of finite zero-sum stochastic games with perfect information

Mean-payoff zero-sum stochastic games can be studied by means of a nonlinear spectral problem. When the state space is finite, the latter consists in finding an eigenpair $(u,\lambda )$ solution of $T(u)=\lambda e+u$, where $T:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is the Shapley (or dynamic programming) operator, λ is a scalar, e is the unit vector, and $u\in {\mathbb{R}}^n$. The scalar λ yields the mean payoff per time unit, and the vector u, called bias, allows one to determine optimal stationary strategies in the mean-payoff game. The existence of the eigenpair $(u,\lambda )$ is generally related to ergodicity conditions. A basic issue is to understand for which classes of games the bias vector is unique (up to an additive constant). In this paper, we consider perfect-information zero-sum stochastic games with finite state and action spaces, thinking of the transition payments as variable parameters, transition probabilities being fixed. We show that the bias vector, thought of as a function of the transition payments, is generically unique (up to an additive constant). The proof uses techniques of nonlinear Perron–Frobenius theory. As an application of our results, we obtain an explicit perturbation scheme allowing one to solve degenerate instances of stochastic games by policy iteration.