关于挠自由群分次环的注记

本文推广ＣＮａｓｔａｓｅｃｕ，Ｆ．ＶａｎＯｙｓｔａｅｙｅｎ的有关结果，并在一定条件下回答了Ｋ．Ｒ．Ｆｕｌｌｅｒ等人的公开问题．     

The Calabi Conjecture on Affine Maximal Surfaces

There are different versions of so called affine Bernstein conjectures, stated by Calabi and Chern, resp. In [T-W] the authors present a proof of Chern’s conjecture. In this paper we give an affirmative answer to Calabi’s conjecture.     

Homogeneous continua in 2-manifolds

In 1960 R.H. Bing [2] proved that every homogeneous plane continuum that contains an arc is a simple closed curve. At that time Bing [2, p. 228] asked if every 1-dimensional homogeneous continuum that contains an arc and lies on a 2-manifold is a simple closed curve. We prove that no 2-manifold contains uncountably many disjoint triods. We use this theorem and decomposition theorems of F.B. Jones [10] and H.C. Wiser [19] to answer Bing's question in the affirmative. We also prove that every homogeneous indecomposable continuum in a 2-manifold can be embedded in the plane. It follows from this result and another theorem of Wiser [20] that every homogeneous continuum that is properly contained in an orientable 2-manifold is planar.     

A Remark on a problem of Mrówka

Recently Mrówka [7] defined nearly-Lindelöf spaces (see Definition 2.1) and asked the following question: Is every nearly-Lindelof space a Lindelöf space? We will answer this question positively (see Corollary 2.5). However we will first prove some results concerning the related concepts of [a, b]-compactness (see Definitions 3.1 and 3.2) of topological spaces, from which the answer to Mrówka’s question will immediately follow.     

Une réponse négative au problème de Noether différentiel

According to a theorem first stated by Clifford, Noether [8] and Rosanes [12], which later received a complete proof by Castelnuovo [2], the Cremona group of the plane is generated by de Jonquières transformations. J.F. Ritt [10] asked whether such a result could be generalized to the differential plane. We give a negative answer, relying on a flat system of Rouchon.     

Regular Legendrian knots and the HOMFLY polynomial of immersed plane curves

We show that every unframed knot type in has a representative obtained by the Legendrian lifting of an immersed plane curve. This gives a positive answer to the question asked by V.I.Arnold in [3]. The Legendrian lifting lowers the framed version of the HOMFLY polynomial [20] to generic plane curves. We prove that the induced polynomial invariant can be completely defined in terms of plane curves only. Moreover it is a genuine, not Laurent, polynomial in the framing variable. This provides an estimate on the Bennequin-Tabachnikov number of a Legendrian knot. Received: 17 April 1996 / Revised: 12 May 1999 / Published online: 28 June 2000     

Archimedisierung absoluter Ebenen

In a nonarchimedean absolute plane let an origin o and a unit of distance [o, e] be given. By omitting those points which are infinitely distant to o relative to [o, e] and by identifying those of the resting points which are infinitely close together relative to [o, e], we get an archimedean absolute plane if certain two conditions are fulfilled. In the euclidean and in the hyperbolic case this geometric procedure corresponds to the well-known algebraic one in the underlying coordinate field.     

Piecewise polynomials with different smoothness degrees on polyhedral complexes

Abstract

For a given d-dimensional polyhedral complex Δ and a given degree k, we consider the vector space of piecewise polynomial functions on Δ of degree at most k with a different smoothness condition on each pair of adjacent d-faces of Δ. This is a finite dimensional vector space. The fundamental problem in Approximation Theory is to compute the dimension of this vector space. It is known that the dimension is given by a polynomial for sufficiently large k via commutative algebra. By using the technique of McDonald and Schenck [3] and extending their result to a plane polyhedral complex Δ with varying smoothness conditions, we determine this polynomial. This gives a complete answer for the dimension. At the end we discuss some examples through this technique.     

On the endomorphism ring of a module with relative chain conditions

A well-known result of Small states that if M is a noetherian left R-module having endomorphism ring S then any nil subring of S is nilpotent. Fisher [4] dualized this result and showed that if M is left artinian then any nil ideal of S is nilpotent. He gave a bound on the indices of nilpotency of nil subrings of the endomorphism rings of noetherian modules and raised the dual question of whether there are such bounds in the case of artinian modules. He gave an affirmative answer if the module is also assumed to be finitely-generated. Similar affirmative answers for modules with finite homogeneous length were given in [10] and [15]. On the other hand, the nilpotence of certain ideals of the endomorphism rings of modules noetherian relative to a torsion theory has been extensively studied. See [2,6,8,12,15,17]. Jirasko [11] dualized, in some sense, some of the results of [6] to torsion modules satisfying the descending chain conditions with respect to some radical.

In this paper we give a bound of indices of nilpotency on nil subrings of the endomorphism ring of a left R-module which is T-torsionfree with respect to some torsion theory T on R-mod. As a special case, we obtain an affirmative answer to Fisher's question. We also note that our results can be stated in an arbitrary Grothendieck category.     

Convergence to diffusions with regular boundaries

Earlier results on weak convergence to diffusion processes [8] are generalized to cases where the limiting diffusions may have regular boundaries. The boundaries may be adhesive or reflecting, and in each case we give two different sets of conditions for convergence. It is shown that these conditions are necessary and sufficient for convergence in the same sense as the conditions in [8]. We also extend our results to cases where the coefficients of the diffusions have simple discontinuities, in particular we thereby answer an open question by Keilson and Wellner [9]. Finally we formulate alternative sets of conditions for convergence, with these new sets being more convenient for instance when the sequence under investigation consists of pure jump Markov processes in continuous time.     

Existence of multiple positive periodic solutions for nonlinear functional difference equations

In this paper, we investigate the existence of multiple positive periodic solutions to a class of functional difference equations. We answer the open problems proposed by Y. Raffoul in [Electron. J. Differential Equations 55 (2002) 1-8] and the conditions obtained improve some recent results established there.     

关于3阶Carmichael数

朱文余和孙琦（见《数学进展》，2004，33（4）：505-507）提出了关于3阶Carmichael数的三个问题，我们（见《四川大学学报（自然科学版）》，2006，43（6）：1197-1201）肯定地回答了问题1．本文模仿Howe的寻找严格2阶Carmichael数（见Mathematics of Computation，2000，69（232）：1711—1719）的方法，提出寻找满足某种条件的3阶Carmichael数的方法，并用这种方法确实找到了几百个这样的数，因而完全肯定地回答了问题2．     

Coderivatives of normal cone mappings and Lipschitzian stability of parametric variational inequalities

In this paper, we provide a comprehensive study of coderivative formulas for normal cone mappings. This allows us to derive necessary and sufficient conditions for the Lipschitzian stability of parametric variational inequalities in reflexive Banach spaces. Our development not only gives an answer to the open questions raised in Yao and Yen (2009) [11], but also establishes generalizations and complements of the results given in Henrion et al. (2010) [4] and Yao and Yen (2009) [11] and [12].     

The classification of homotopy classes of bounded curvature paths

A bounded curvature path is a continuously differentiable piecewise C² path with bounded absolute curvature that connects two points in the tangent bundle of a surface. In this note we give necessary and sufficient conditions for two bounded curvature paths, defined in the Euclidean plane, to be in the same connected component while keeping the curvature bounded at every stage of the deformation. Following our work in [3], [2] and [4] this work finishes a program started by Lester Dubins in [6] in 1961.     

A note on the sample average approximation method for stochastic mathematical programs with complementarity constraints

Meng and Xu (2006) [3] proposed a sample average approximation (SAA) method for solving a class of stochastic mathematical programs with complementarity constraints (SMPCCs). After showing that under some moderate conditions, a sequence of weak stationary points of SAA problems converge to a weak stationary point of the original SMPCC with probability approaching one at exponential rate as the sample size tends to infinity, the authors proposed an open question, that is, whether similar results can be obtained under some relatively weaker conditions. In this paper, we try to answer the open question. Based on the reformulation of stationary condition of MPCCs and new stability results on generalized equations, we present a similar convergence theory without any information of second order derivative and strict complementarity conditions. Moreover, we carry out convergence analysis of the regularized SAA method proposed by Meng and Xu (2006) [3] where the convergence results have not been considered.     

RESTRICTING THE COMPARISON FUNCTOR OF AN ADJUNCTION TO PROJECTIVE OBJECTS

Abstract

This paper deals with projectives (in the sense of K.A.Hardie [5] relative to a right adjoint functor U: A → K. We answer the question, raised by R.-E. Hoffmann [6] p. 135, of knowing under what conditions there exists an equivalence between Proj u and Proj U^r, induced by the comparison functor Φ: A → K^T, where T denotes the monad induced by U. In the case, that U is an algebraic functor we also give necessary and sufficient conditions for the re gular projective objects to coincide with the U-projectives. Finally, we delineate how these results nay be applied in certain familiar situations.     

Cycle Covers of Planar 2-Edge-Connected Graphs

A graph with n vertices is said to have a small cycle cover provided its edges can be covered with at most (2n ? 1)/3 cycles. Bondy [2] has conjectured that every 2-connected graph has a small cycle cover. In [3] Lai and Lai prove Bondy’s conjecture for plane triangulations. In [1] the author extends this result to all planar 3-connected graphs, by proving that they can be covered by at most (n + 1)/2 cycles. In this paper we show that Bondy’s conjecture holds for all planar 2-connected graphs. We also show that all planar 2-edge-connected graphs can be covered by at most (3n ? 3)/4 cycles and we show an infinite family of graphs for which this bound is attained.     

Two questions on semi-clean group rings

It was proved in [4] that every group ring of a torsion abelian group over a commutative local ring is a semi-clean ring. It was asked in [4] whether every group ring of a torsion abelian group over a commutative clean ring is a semi-clean ring and whether every group ring of a torsion abelian group over a commutative semi-clean ring is a semi-clean ring. In this paper, we give a positive answer to question 1 and a negative answer to question 2.     

*-RINGS AND THEIR RADICALS

ABSTRACT

This paper is to investigate the class * of all prima rings without proper non-zero prima factors and to characterize the ‘minimal special radical’ containing the class * as well as the ‘maximal special radical’ the semisimple class of which contains the class *. This let us answer certain open questions put in [1] and [4].