“It is through others that we become ourselves.” A study of Vygotskian play in Russian and Irish schools.

50 years after publishing his seminal work on play and its role in child development, Vygotskian theory is still highly influential in education, and particularly in early years. This paper presents two examples of full integration of Vygotskian principles into schools in two very different settings. Both report improvements in learning and in well-being, and exemplify the theory-practice-theory cycle, highlighting the development of new theoretical constructs arising out of putting theory firmly into practice. In both settings, the positive results have come from years of effort, in which school personnel who may have been skeptical at first, have been inspired by the impact of adopting Vygotskian play on the children they teach. The Northern Ireland study shows that at least some of the Golden Key principles (mixed-age play and enhanced home-school links) translate perfectly into very different cultural-historical contexts.

Dynamic Job Scheduling on the Grid Environment using the Great Deluge Algorithm

The utilization of the computational Grid processor network has become a common method for researchers and scientists without access to local processor clusters to avail of the benefits of parallel processing for compute-intensive applications. As a result, this demand requires effective and efficient dynamic allocation of available resources. Although static scheduling and allocation techniques have proved effective, the dynamic nature of the Grid requires innovative techniques for reacting to change and maintaining stability for users. The dynamic scheduling process requires quite powerful optimization techniques, which can themselves lack the performance required in reaction time for achieving an effective schedule solution. Often there is a trade-off between solution quality and speed in achieving a solution. This paper presents an extension of a technique used in optimization and scheduling which can provide the means of achieving this balance and improves on similar approaches currently published.

On weak and strong Peano theorems

We say that the Peano theorem holds for a topological vector space $E$ if, for any continuous mapping $f : {\Bbb R}\times E \to E$ and any $(t(0), x(0))$ is an element of ${\Bbb R}\times E$, the Cauchy problem $\dot x(t) = f(t,x(t))$, $x(t(0)) = x(0)$, has a solution in some neighborhood of $t(0)$. We say that the weak version of Peano theorem holds for $E$ if, for any continuous map $f : {\Bbb R}\times E \to E$, the equation $\dot x(t) = f (t, x(t))$ has a solution on some interval. We construct an example (answering a question posed by S. G. Lobanov) of a Hausdorff locally convex topological vector space E for which the weak version of Peano theorem holds and the Peano theorem fails to hold. We also construct a Hausdorff locally convex topological vector space E for which the Peano theorem holds and any barrel in E is neither compact nor sequentially compact.

Time operators for one parameter semigroups

We develop two simple approaches to the construction of time operators for semigroups of continuous linear operators in Hilbert spaces provided that the generators of these semigroups are normal operators. The first approach enables us to give explicit formulas (in the spectral representations) both for the time operators and for their eigenfunctions. The other approach provides no explicit formula. However, it enables us to find necessary and sufficient conditions for the existence of time operators for semigroups of continuous linear operators in separable Hilbert spaces with normal generators. Time superoperators corresponding to unitary groups are also discussed.

On linear extension operators

We construct a countable-dimensional Hausdorff locally convex topological vector space $E$ and a stratifiable closed linear subspace $F$ subset of $E$ such that any linear extension operator from $C_b(F)$ to $C_b(E)$ is unbounded (here $C_b(X)$ stands for the Banach space of continuous bounded real-valued functions on $X$).

Three constructive examples of separable pre-Hilbert spaces nonisomorphic to their closed hyperplanes

An example of a sigma -compact infinite-dimensional pre-Hilbert space H is constructed such that any continuous linear operator T: H --> H is of the form T = lambdaI + F for some lambda is an element of R and for a finite-dimensional continuous linear operator F. A class of simple examples of pre-Hilbert spaces nonisomorphic to their closed hyperplanes is given. A sigma -compact pre-Hilbert space H isomorphic to H x R x R and nonisomorphic to H x R is also constructed.

Integral completeness of locally convex spaces

A locally convex space X is said to be integrally complete if each continuous mapping f: [0, 1] --> X is Riemann integrable. A criterion for integral completeness is established. Readily verifiable sufficient conditions of integral completeness are proved.

On stratifiable locally convex spaces

In the present paper we prove several results on the stratifiability of locally convex spaces. In particular, we show that a free locally convex sum of an arbitrary set of stratifiable LCS is a stratifiable LCS, and that all locally convex F'-spaces whose bounded subsets are metrizable are stratifiable. Moreover, we prove that a strict inductive limit of metrizable LCS is stratifiable and establish the stratifiability of many important general and specific spaces used in functional analysis. We also construct some examples that clarify the relationship between the stratifiability and other properties.