On monoids over which all generators satisfy a flatness property

This paper addresses conditions under which all generators in the category of right S-acts (where S is a monoid) satisfy a flatness property. There are characterizations for monoids over which all generators satisfy a flatness property α where α can stand for freeness, projectivity, strong flatness, Condition (P), principal weak flatness and torsion freeness. To our knowledge, the problem has not been studied for other flatness properties such as weak flatness, Condition (E) and regularity. The present paper addresses this gap.     

Approximation and implementation of transformation based feedback laws for distributed parameter systems

A novel approach to the efficient numerical approximation and implementation of transformation based state feedback originating from backstepping or flatness based approaches is applied to a flatness based feedback law for the so called heavy rope system. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rugged modules: The opposite of flatness

Relative notions of flatness are introduced as a mean to gauge the extent of the flatness of any given module. Every module is thus endowed with a flatness domain and, for every ring, the collection of flatness domains of all of its modules is a lattice with respect to class inclusion. This lattice, the flatness profile of the ring, allows us, in particular, to focus on modules which have a smallest flatness domain (namely, one consisting of all regular modules.) We establish that such modules exist over arbitrary rings and we call them Rugged Modules. Rings all of whose (cyclic) modules are rugged are shown to be precisely the von Neumann regular rings. We consider rings without a flatness middle class (i.e., rings for which modules must be either flat or rugged.) We obtain that, over a right Noetherian ring every left module is rugged or flat if and only if every right module is poor or injective if and only if R = S×T, where S is semisimple Artinian and T is either Morita equivalent to a right PCI-domain, or T is right Artinian whose Jacobson radical properly contains no nonzero ideals. Character modules serve to bridge results about flatness and injectivity profiles; in particular, connections between rugged and poor modules are explored. If R is a ring whose regular left modules are semisimple, then a right module M is rugged if and only if its character left module M⁺ is poor. Rugged Abelian groups are fully characterized and shown to coincide precisely with injectively poor and projectively poor Abelian groups. Also, in order to get a feel for the class of rugged modules over an arbitrary ring, we consider the homological ubiquity of rugged modules in the category of all modules in terms of the feasibility of rugged precovers and covers for arbitrary modules.     

On Characterization of Monoids by Properties of Generators

Kilp and Knauer in(Comm. Algebra, 1992, 20(7), 1841–1856) gave characterizations of monoids when all generators in category of right S-acts(S is a monoid) satisfy properties such as freeness, projectivity, strong flatness, Condition(P), principal weak flatness, principal weak injectivity, weak injectivity, injectivity, divisibility, strong faithfulness and torsion freeness.Sedaghtjoo in(Semigroup Forum, 2013, 87: 653–662) characterized monoids by some other properties of generators including weak flatness, Condition(E) and regularity. To our knowledge,the problem has not been studied for properties mentioned above of(finitely generated, cyclic,monocyclic, Rees factor) right acts. In this article we answer the question corresponding to these properties and also f g-weak injectivity.     

On a Generalization of Weak Pullback Flatness

In 1998, Laan introduced weakly pullback flat acts and showed that weak pullback flatness is equivalent to the conjunction of Conditions (P) and (E′). In 2013, Golchin introduced Condition (P′), which is a generalization of Condition (P). In this article, we first define Condition (PF″), which lies strictly between weak pullback flatness and Condition (P′), and prove that Condition (PF″) coincides with the conjunction of Conditions (P′) and (E′). Furthermore, we give a classfication of monoids by Condition (PF″) of (cyclic, Rees factor) acts, and find a necessary and sufficient condition under which Condition (PF″) coincides with Condition (P′) (resp., weak pullback flatness, strong flatness) for Rees factor acts. Finally, we investigate Conditions (P′), (E′) and (PF″) using some new types of epimorphisms.     

Unimodular completion and direct flat representation in the context of differential flatness

Despite its significance in nonlinear controller design, the verification of the property of differential flatness and the identification of so-called flat outputs is a continuing research topic. However, the difficulty of both verification and identification heavily depends on the representation of the system. This contribution shows that flatness may be a far less complicated matter, depending on the choice of coordinates. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Constrained trace-optimization of polynomials in freely noncommuting variables

The study of matrix inequalities in a dimension-free setting is in the realm of free real algebraic geometry. In this paper we investigate constrained trace and eigenvalue optimization of noncommutative polynomials. We present Lasserre’s relaxation scheme for trace optimization based on semidefinite programming (SDP) and demonstrate its convergence properties. Finite convergence of this relaxation scheme is governed by flatness, i.e., a rank-preserving property for associated dual SDPs. If flatness is observed, then optimizers can be extracted using the Gelfand–Naimark–Segal construction and the Artin–Wedderburn theory verifying exactness of the relaxation. To enforce flatness we employ a noncommutative version of the randomization technique championed by Nie. The implementation of these procedures in our computer algebra system NCSOStoolsis presented and several examples are given to illustrate our results.     

混合优化RBF-BP网络的板形缺陷识别

针对传统板形模式识别方法存在精度低、鲁棒性弱的问题,提出了一种混合优化RBF-BP组合神经网络板形模式识别方法。首先利用自组织映射网络(SOM)对样本聚类,利用聚类后的网络拓扑结构确定RBF的中心,并计算RBF的宽度,克服了传统聚类算法随机选取中心导致聚类结果不稳定的问题。然后利用遗传算法(GA)良好的全局搜索能力优化整个网络的权值。RBF-BP组合神经网络是由一个RBF子网和一BP子网串联构成的,该网络同时具备BP神经网络能较好地预测未知样本的能力以及RBF神经网络的逼近速度快的优点。并以某900HC可逆冷轧机板形识别为应用背景,在MATLAB2010a环境下进行仿真实验,结果表明混合优化RBF-BP组合神经网络的板形模式识别方法能够识别出常见的板形缺陷,提高了板形缺陷识别精度并具有较好的鲁棒性,可以满足板带轧机高精度的板形控制要求。     

射影平坦Finsler度量的解析构造

利用Hamel关于射影平坦的基本方程,我们导出了Randers度量的λ形变保持射影平坦的充分条件.特别,对一类具有特殊旗曲率性质的Randers度量我们证明了这类度量一定存在保持射影平坦性的λ形变.     

A Tracking Controller for a Certain Class of Nonlinear Systems

Controller design methods based on flatness and passivity are well established. These methods are applied to PCHD systems such that flatness is used to achieve a high tracking performance and passivity ensures disturbance rejection and robustness. Since the error system, written in displacement coordinates, is not a PCHD system in general, we present conditions, which ensures this property. Therefore, one can apply methods from passivity control to stabilize the error dynamics. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Flatness,plateau limit and limit length of single-plateau functions

In this paper we consider iteration of single-plateau functions, an important class of continuous functions with infinitely many forts, and investigate changes of number and length of plateaux under iteration.We use the indices flatness, plateau limit and limit length to formulate those changes. Furthermore, we compute the flatness, plateau limit and limit length for all the nine types of single-plateau functions.     

On strongly flat and condition (P) <Emphasis Type="Italic">S</Emphasis>-posets

In this paper, we first present some homological classifications of pomonoids by using condition (P) and strongly flat properties. Unlike the case for acts, condition (P) and strongly flat coincide for cyclic right S-posets when all weakly right reversible convex subpomonoids of a pomonoid S are left collapsible. Thereby we characterize pomonoids over which strong flatness and condition (P) imply some other flatness properties. Furthermore, we characterize a pomonoid over which every right S-poset has a strongly flat (condition (P)) cover.     

Flatness of bisectors and the symmetry of distance

H. Busemann proved that the elementary spaces are characterized by the flatness of bisectors among all G-spaces with symmetric distance. In this paper we prove that in two dimensional straight G-spaces with a not necessarily symmetric distance the symmetry of distance is in fact implied by the flatness of bisectors. Thus Busemann's characterizations of the euclidean and hyperbolic planes are shown to hold among all straight two dimensional G-spaces with a not necessarily symmetric distance.     

Flatness and finiteness in the Mumford-Shah problem

We extend A. Bonnet's results about the solutions of the global Mumford-Shah problem (see Bonnet (1996)) by replacing his connected assumption by more natural flatness assumptions.     

Flatness in analytic mappings I. On an announcement of Hironaka

We present a stratification by “normal flatness” associated to an analytic mapping, analogous to Hironaka's classical result for analytic spaces. Our construction is based on a generic normal flatness theorem for mappings, proved using techniques concerning the variation of modules of meromorphically parametrized formal power series [1]. The existence of such a stratification was announced by Hironaka [13], but the other claims made in [13] are false. Counterexamples are also presented here.     

Left Kan extensions preserving finite products

We give conditions that ensure the preservation of finite products by left Kan extensions into cocomplete subcanonical sites. The conditions involve a suitable notion of flatness (interpreted in the internal logic of the site) of the functor that is extended and a good behavior of colimits used in the calculation of the left Kan extension. When the recipient category is a Grothendieck topos the good behavior of the colimits is granted and the flatness conditions turn out to be necessary as well as sufficient. We also show that the category of compactly generated Hausdorff spaces and that of small categories are equipped with suitable topologies that account for the well-known phenomenon of preservation of finite products by the geometric and categorical realization, respectively, of simplicial sets.     

Ricci flatness of asymptotically locally Euclidean metrics

In this article we study the metric property and the function theory of asymptotically locally Euclidean (ALE) Kähler manifolds. In particular, we prove the Ricci flatness under the assumption that the Ricci curvature of such manifolds is either nonnegative or nonpositive. The result provides a generalization of previous gap type theorems established by Greene and Wu, Mok, Siu and Yau, etc. It can also be thought of as a general positive mass type result. The method also proves the Liouville properties of plurisubharmonic functions on such manifolds. We also give a characterization of Ricci flatness of an ALE Kähler manifold with nonnegative Ricci curvature in terms of the structure of its cone at infinity.