Is there a fit between pedagogy and technology in online learning?

The study followed a group of online lecturers from different disciplines who were engaged in different levels of online teaching. The researchers' experiences with e-learning have indicated there are a variety of ways by which teaching staff approach e-learning. As new technologies provide a challenge to make learning an interactive and collaborative experience that is guided by a social constructivist approach to teaching and learning, some academic staff embrace the technology to enhance their pedagogy and others are reluctant to use the technology, although in the pedagogy they promote is a social constructivist learning approach. We conducted a qualitative research project in an attempt to answer the research questions of what pedagogies are used by teaching staff to facilitate e-learning, and how do teachers change their use and understanding of e-learning techniques. The study suggests that there is a continuum in the way the constructivist pedagogy had been implemented by the different university teachers and also a continuum in the way the technology had been embraced by them. From our observations, we categorised the university teachers in relation to their pedagogies (level of social constructivist approach) and to the level at which they used the technology, in order to explore how the relationship between these two elements changed. The study helps us understand how the technology enabled some of the teachers to develop their pedagogies and change their perspectives on social learning online. In addition, for others who used social features of the technology to an optimal level, the technology helped them accommodate and reinforce the notion of a social constructivist approach to teaching and learning. Finally, the interchange between the ability to use the technology and the adoption of social constructivist approach to teaching raised new questions in relation to implementation of online learning

Alexander-equivalent Zariski pairs of irreducible sextics

The existence of Alexander-equivalent Zariski pairs dealing with irreducible curves of degree 6 was proved by A. Degtyarev. However, up to now, no explicit example of such a pair was available (only the existence was known). In this paper, we construct the first concrete example.Comment: 21 pages, 19 figure

Nodal degenerations of plane curves and Galois covers

Globally irreducible nodes (i.e. nodes whose branches belong to the same irreducible component) have mild effects on the most common topological invariants of an algebraic curve. In other words, adding a globally irreducible node (simple nodal degeneration) to a curve should not change them a lot. In this paper we study the effect of nodal degeneration of curves on fundamental groups and show examples where simple nodal degenerations produce non-isomorphic fundamental groups and this can be detected in an algebraic way by means of Galois coverings.Comment: 16 pages, 3 figure

Configuration types and cubic surfaces

This paper is a sequel to the paper \cite{refGH}. We relate the matroid notion of a combinatorial geometry to a generalization which we call a configuration type. Configuration types arise when one classifies the Hilbert functions and graded Betti numbers for fat point subschemes supported at $n\le8$ essentially distinct points of the projective plane. Each type gives rise to a surface $X$ obtained by blowing up the points. We classify those types such that $n=6$ and $-K_X$ is nef. The surfaces obtained are precisely the desingularizations of the normal cubic surfaces. By classifying configuration types we recover in all characteristics the classification of normal cubic surfaces, which is well-known in characteristic 0 \cite{refBW}. As an application of our classification of configuration types, we obtain a numerical procedure for determining the Hilbert function and graded Betti numbers for the ideal of any fat point subscheme $Z=m_1p_1+...+m_6p_6$ such that the points $p_i$ are essentially distinct and $-K_X$ is nef, given only the configuration type of the points $p_1,...,p_6$ and the coefficients $m_i$.Comment: 14 pages, final versio

The New Old Lawyer: How Lawyers have Adapted to Mediation to Preserve their Power, Income, and Identity

This paper outlines the evolution of mediation in some common law jurisdictions from an idea most lawyers dismissed to a practice most now use. It highlights the attitudes and actions of lawyers as they have adjusted their practices to include mediation, and adapted mediation to suit their needs. In so doing perhaps it provides a glimpse into the future in those jurisdictions where mediation is still struggling for acceptance, and a caution about what price might have to be paid for such success

Long-range alternatives in Italian politics.

Caption titleAt head of title: Economic development. Italy. Political/8. R. Zariski. January 28, 1954463"--handwritten on leaf [1]. -- Series numbering handwritten on leaf [1]Includes bibliographical references (leaves [1]-5, 2nd group

Hilbert schemes of points on a locally planar curve and the Severi strata of its versal deformation

Let C be a locally planar curve. Its versal deformation admits a stratification by the genera of the fibres. The strata are singular; we show that their multiplicities at the central point are determined by the Euler numbers of the Hilbert schemes of points on C. These Euler numbers have made two prior appearances. First, in certain simple cases, they control the contribution of C to the Pandharipande-Thomas curve counting invariants of three-folds. In this context, our result identifies the strata multiplicities as the local contributions to the Gopakumar-Vafa BPS invariants. Second, when C is smooth away from a unique singular point, a special case of a conjecture of Oblomkov and Shende identifies the Euler numbers of the Hilbert schemes with the "U(infinity)" invariant of the link of the singularity. We make contact with combinatorial ideas of Jaeger, and suggest an approach to the conjecture.Comment: 16 page

On the Content of Polynomials Over Semirings and Its Applications

In this paper, we prove that Dedekind-Mertens lemma holds only for those semimodules whose subsemimodules are subtractive. We introduce Gaussian semirings and prove that bounded distributive lattices are Gaussian semirings. Then we introduce weak Gaussian semirings and prove that a semiring is weak Gaussian if and only if each prime ideal of this semiring is subtractive. We also define content semialgebras as a generalization of polynomial semirings and content algebras and show that in content extensions for semirings, minimal primes extend to minimal primes and discuss zero-divisors of a content semialgebra over a semiring who has Property (A) or whose set of zero-divisors is a finite union of prime ideals. We also discuss formal power series semirings and show that under suitable conditions, they are good examples of weak content semialgebras.Comment: Final version published at J. Algebra Appl., one reference added, three minor editorial change

Essential and inessential elements of a standard basis

In this paper we introduce the concept of inessential element of a standard basis of I, where I is any homogeneous ideal of a polynomial ring. An inessential element is, roughly speaking, a form of the basis whose omission produces an ideal having the same saturation of I; it becomes useless in any dehomogenization of I with respect to a linear form. We study the properties of the basis linked to the presence of inessential elements and give some examples.Comment: 15 page

Valuative analysis of planar plurisubharmonic functions

We show that valuations on the ring R of holomorphic germs in dimension 2 may be naturally evaluated on plurisubharmonic functions, giving rise to generalized Lelong numbers in the sense of Demailly. Any plurisubharmonic function thus defines a real-valued function on the set V of valuations on R and--by way of a natural Laplace operator defined in terms of the tree structure on V--a positive measure on V. This measure contains a great deal of information on the singularity at the origin. Under mild regularity assumptions, it yields an exact formula for the mixed Monge-Ampere mass of two plurisubharmonic functions. As a consequence, any generalized Lelong number can be interpreted as an average of valuations. Using our machinery we also show that the singularity of any positive closed (1,1) current T can be attenuated in the following sense: there exists a finite composition of blowups such that the pull-back of T decomposes into two parts, the first associated to a divisor with normal crossing support, the second having small Lelong numbers.Comment: Final version. To appear in Inventiones Math. 37 pages, 5 figure