Hereditary and Formally Smooth Coalgebras

We define formally smooth coalgebras and we study their relation with hereditary coalgebras. The main result of the paper establishes that a coalgebra with separable coradical is hereditary if and only if it is formally smooth if and only if it is the cotensor coalgebra , where C₀ is the coradical of C and N is a certain (C₀,C₀)–bicomodule. Presented by A. Verschoren Mathematics Subject Classifications (2000) primary: 16W30; secondary: 18G20. D. Ştefan: This author thanks the members of the Algebra Department of the University of Granada for their warm hospitality. He is especially grateful to Pascual Jara for the kind invitation to visit University of Granada. This research was partially supported by CERES, Contract 4-147.     

A local approach to functionals on

Let be a probability space and a Banach space. We show that the dual of can be ``locally' identified with

     

On a class of coupled Hamiltonian operators and their integrable hierarchies with two potentials

We discuss at first in this paper the Gauge equivalence among several u‐linear Hamiltonian operators and present explicitly the associated Gauge transformation of Bäcklund type among them. We then establish the sufficient and necessary conditions for the linear superposition of the discussed u‐linear operators and matrix differential operators with constant coefficients of arbitrary order to be Hamiltonian, which interestingly shows that the resulting Hamiltonian operators survive only up to the third differential order. Finally, we explore a few illustrative examples of integrable hierarchies from Hamiltonian pairs embedded in the resulting Hamiltonian operators.     

Algebraic reflexivity of the isometry group of some spaces of Lipschitz functions

We show that the isometry groups of Lip(X,d) and lip(X,dα) with α∈(0,1), for a compact metric space (X,d), are algebraically reflexive. We also prove that the sets of isometric reflections and generalized bi-circular projections on such spaces are algebraically reflexive. In order to achieve this, we characterize generalized bi-circular projections on these spaces.     

B(LF,ω～2 )-refinability of inverse limits

Let X be the limit of an inverse system {X α , παβ , Λ} and and let λ be the cardinal number of Λ. Assume that each projection πα : X→ X α is an open and onto map and X is λ-paracompact. We prove that if each X α is B(LF, ω 2 )-refinable (hereditarily B(LF, ω 2 )-refinable), then X is B(LF, ω 2 )-refinable (hereditarily B(LF, ω 2 )-refinable). Furthermore, we show that B(LF, ω 2 )-refinable spaces can be preserved inversely under closed maps.     

On String Graph Limits and the Structure of a Typical String Graph

We study limits of convergent sequences of string graphs, that is graphs with an intersection representation consisting of curves in the plane. We use these results to study the limiting behavior of a sequence of random string graphs. We also prove similar results for several related graph classes.     

Acyclic improper colouring of graphs with maximum degree 4

A k-colouring(not necessarily proper) of vertices of a graph is called acyclic, if for every pair of distinct colours i and j the subgraph induced by the edges whose endpoints have colours i and j is acyclic. We consider acyclic k-colourings such that each colour class induces a graph with a given(hereditary) property. In particular, we consider acyclic k-colourings in which each colour class induces a graph with maximum degree at most t, which are referred to as acyclic t-improper k-colourings. The acyclic t-improper chromatic number of a graph G is the smallest k for which there exists an acyclic t-improper k-colouring of G. We focus on acyclic colourings of graphs with maximum degree 4. We prove that 3 is an upper bound for the acyclic 3-improper chromatic number of this class of graphs. We also provide a non-trivial family of graphs with maximum degree4 whose acyclic 3-improper chromatic number is at most 2, namely, the graphs with maximum average degree at most 3. Finally, we prove that any graph G with Δ(G) 4 can be acyclically coloured with 4 colours in such a way that each colour class induces an acyclic graph with maximum degree at most 3.     

一致凸度量线性空间的自反性及其应用

为研究度量线性空间中凸集的逼近性质，Ｇ．Ｃ．Ａｈｕｊａ等引起了度量线性空间的严格凸性及一致凸性的定义。本文证明了完备的一致凸的度量线性空间是自反的。同时，作为应用，研究了最佳联合逼近元的存在性与唯一性问题。     

Transformation d'halphen

We give a modern version of the following result of Halphen: “degrees and classes of successive evolutes of a plane algebraic curve form two arithmetic progressions of the same ratio after a sufficient number of iterations”. The iteration of this transform provides a desingularisationprocess (branch by branch) of plane curves, different to the Nash transform and the Noether process. To prove these two points,we introduce systematically Puiseux's exponents, semi-groups of exponents and use elimination and duality formulae.     

Counting Special Families of Labelled Trees

A labelled tree rooted at its least labelled vertex is Least-Child-Being-Monk if it has the property that the least labelled child of 0 is a leaf. One of our main results is that the number of Least-Child-Being-Monk trees labelled on {0, 1, 2,... ,n + 1} is equal to nⁿ. More generally, let be the set of labelled trees on {0,1,2,..., n + 1}, such that the total number of descendants of the least labelled child of 0 is p. We prove that the cardinality of is equal to Furthermore, a labelled tree rooted at its least labelled vertex is Hereditarily-Least-Single if it has the property that every least child in this tree is a leaf. Let the number of Hereditarily-Least-Single trees with n vertices be h_n. We find a functional equation for the generating function of h(n) and derive a recurrence that will quickly compute h(n). Received November 13, 2004