Smooth Morphisms of Module Schemes

Let :A to B be a homomorphism of finite-dimensional algebrasover an algebraically closed field and the induced morphism of the associated module schemesfor any integer c 1. We prove that if the induced functor modB mod A is hom-controlled then the restriction of ^(c) to anyconnected component of is a composition of a smooth morphism followed by an immersion.Some new results on the types of singularities in the orbitclosures of module schemes are also proved. 2000 Mathematical Subject Classification: 14B05, 14L30, 16G10.     

Constructive Néron Desingularization of algebras with big smooth locus

An algorithmic proof of the General Néron Desingularization theorem and its uniform version is given for morphisms with big smooth locus. This generalizes the results for the one-dimensional case (cf. [10 Pfister, G., Popescu, D. (2017). Constructive General Neron Desingularization for one dimensional local rings. J. Symbolic Comput. 80:570–580.[Crossref], [Web of Science ®] , [Google Scholar]], [7 Khalid, A., Pfister, G., Popescu, D. (2018). A uniform General Neron Desingularization in dimension one. J. Algebra Appli. 16. arXiv:AC/1612.03416. [Google Scholar]]).     

Foliated Manifolds and Conformal Heat Morphisms

The objects under study, in this article, are Riemannian manifoldsfoliated by hypersurfaces. Looking at the transverse direction as time,we construct the generalised heat operator and, in the spirit of atime-space extension of harmonic morphisms, we introduce the conformal heat morphisms. Concrete examples of these concepts arepresented, as well as, a characterisation of conformal heat morphisms.Lastly, we calculate the heat Lie algebra of the generalised heatoperator.     

A Characterization of 1-, 2-, 3-, 4-Homomorphisms of Ordered Sets

We characterize totally ordered sets within the class of all ordered sets containing at least four-element chains. We use a simple relationship between their isotone transformations and the so called 1-endomorphism which is introduced in the paper. Later we describe 1-, 2-, 3-, 4-homomorphisms of ordered sets in the language of super strong mappings.     

On Aperiodic Relational Morphisms

We characterize aperiodic relational morphisms as those that are injective on regular $\H$-classes. This result is applied to obtain simple proofs and generalizations of McAlisters results on joins of aperiodic semigroups and groups. Also, we show that if $\pv H$ is a proper, non-trivial pseudovariety of groups, then \[\pv A\ast \pv H\subsetneq (\pv A\ast \pv G)\cap \ov {\pv H}.\] We provide coordinate-free formulations and proofs of Rhodess Presentation Lemma and generalizations. As an application, we give simpler proofs of Tilsons theorem on the complexity of semigroups with at most $2$ non-zero $\J$-classes and Rhodess theorem that complexity is not local.     

Infinitesimal non-crossing cumulants and free probability of type B

Free probabilistic considerations of type B first appeared in the paper of Biane, Goodman and Nica [P. Biane, F. Goodman, A. Nica, Non-crossing cumulants of type B, Trans. Amer. Math. Soc. 355 (2003) 2263-2303]. Recently, connections between type B and infinitesimal free probability were put into evidence by Belinschi and Shlyakhtenko [S.T. Belinschi, D. Shlyakhtenko, Free probability of type B: Analytic aspects and applications, preprint, 2009, available online at www.arxiv.org under reference arXiv:0903.2721]. The interplay between “type B” and “infinitesimal” is also the object of the present paper. We study infinitesimal freeness for a family of unital subalgebras A₁,…,Ak in an infinitesimal noncommutative probability space (A,φ,φ^′) and we introduce a concept of infinitesimal non-crossing cumulant functionals for (A,φ,φ^′), obtained by taking a formal derivative in the formula for usual non-crossing cumulants. We prove that the infinitesimal freeness of A₁,…,Ak is equivalent to a vanishing condition for mixed cumulants; this gives the infinitesimal counterpart for a theorem of Speicher from “usual” free probability. We show that the lattices NC(B)(n) of non-crossing partitions of type B appear in the combinatorial study of (A,φ,φ^′), in the formulas for infinitesimal cumulants and when describing alternating products of infinitesimally free random variables. As an application of alternating free products, we observe the infinitesimal analogue for the well-known fact that freeness is preserved under compression with a free projection. As another application, we observe the infinitesimal analogue for a well-known procedure used to construct free families of free Poisson elements. Finally, we discuss situations when the freeness of A₁,…,Ak in (A,φ) can be naturally upgraded to infinitesimal freeness in (A,φ,φ^′), for a suitable choice of a “companion functional” .     

A family of noetherian rings with their finite length modules under control

We investigate the category mod of finite length modules over the ring =A _k , where is a V-ring, i.e. a ring for which every simple module is injective, k a subfield of its centre and A an elementary k-algebra. Each simple module E _j gives rise to a quasiprogenerator P _j = A E _j . By a result of K. Fuller, P _j induces a category equivalence from which we deduce that mod _j mod EndP _j . As a consequence we can(1) construct for each elementary k-algebra A over a finite field k a nonartinian noetherian ring such that modA mod(2) find twisted versions of algebras of wild representation type such that itself is of finite or tame representation type (in mod)(3) describe for certain rings the minimal almost split morphisms in mod and observe that almost all of these maps are not almost split in Mod.     

Rigidity and flexibility analysis of a kind of surfaces of revolution and visualization

In this paper we consider a class of toroids generated by a meridian in a shape of quadrangle in an arbitrary position to the axis of revolution.We give geometrical meaning of the non-rigidity condition based on the position of the diagonals of meridian. Special classes of quadrangular meridians (trapezium, parallelogram, deltoid, … ) are discussed. The number of flexible toroids is considered. A tool SurfBend is developed for the rigidity and visualization.     

Riemannian foliations admitting transversal conformal fields

Let be a closed, connected Riemannian manifold with a foliation of codimension q and a bundle-like metric g _M . We study the relationship between several infinitesimal automorphisms. Moreover under the some curvature condition, if M admits a transversal conformal field, then is transversally isometric to the action of a finite subgroup of O(q) acting on the q-sphere of constant curvature.      

Harmonic morphisms from the Grassmannians and their non-compact duals

In this paper we give a unified framework for the construction of complex valued harmonic morphisms from the real, complex and quaternionic Grassmannians and their non-compact duals. This gives a positive answer to the corresponding open existence problem in the real and quaternionic cases. Mathematics Subject Classifications (2000): 58E20, 53C43, 53C12The author is a member of EDGE, Research Training Network HPRN-CT-2000-00101     

模代数的形变理论

本文构作了与模代数HA相适应的“形变复形”CH*(H,A),并利用形变复形的上同调群来刻画模代数HA的形变,证明了HA的无穷小形变的等价类是与形变复形的2阶上同调群HH2(H,A)是一一对应的,且当HH2(H,A)=0时,模代数HA为刚性的.     

On the existence of harmonic morphisms from symmetric spaces of rank one

Summary In this paper we give a unified framework for constructing harmonic morphisms from the irreducible Riemannian symmetric spaces ℍH ⁿ, ℂH ⁿ, ℝH ² _t+1, ℍP ⁿ, ℂP ⁿ and ℝP ²ⁿ⁺¹ of rank one. Using this we give a positive answer to the global existence problem for the non-compact hyperbolic cases. This work was supported by The Swedish Natural Science Research Council. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.