On the set of wild points of attracting surfaces in R3

Suppose that a closed surface S⊆R3is an attractor, notne-cessarily global, for a discrete dynamical system. Assuming that its set of wild points Wis totally disconnected, we prove that (up to an ambient homeomorphism) it has to be con-tained in a straight line. As a corollary we show that there exist uncountably many different 2-spheres in R3 none of which can be realized as an attractor for a homeomorphism. Our techniques hinge on a quantity r(K)that can be de-fined for any compact set K⊆R3and is related to “how wildly” it sits in R3. We establish the topological results that (i)r(W) ≤r(S)and (ii) any totally disconnected set having a finite rmust be contained in a straight line (up to an ambient homeomorphism). The main result follows from these and the fact that attractors have a finite r.The author is supported by the Spanish Ministerio de Economía y Competitividad (grant MTM 2015-63612-P)

Embedding of global attractors and their dynamics

Using shape theory and the concept of cellularity, we show that if $A$ is the global attractor associated with a dissipative partial differential equation in a real Hilbert space $H$ and the set $A-A$ has finite Assouad dimension $d$, then there is an ordinary differential equation in ${\mathbb R}^{m+1}$, with $m >d$, that has unique solutions and reproduces the dynamics on $A$. Moreover, the dynamical system generated by this new ordinary differential equation has a global attractor $X$ arbitrarily close to $LA$, where $L$ is a homeomorphism from $A$ into ${\mathbb R}^{m+1}$

A Manual for the Training of Short Term English Teachers as Required by Small-Scale Schools in Japan

This paper examines the business conditions facing small-scale English conversation schools in the increasingly competitive Japanese market. It looks at their staffing requirements and current training methods and then endeavors to develop and effective training programme that can be operated within the limited resources available to these schools. The intention is to provide new staff coming from non-teaching backgrounds a practical introduction to the realities of teaching through a two-week period of observation, team teaching, and supervised solo teaching followed by two months of directed study on classroom management and methodology. This allows for gradual assumption of responsibility for classes, latitude for trial and error in the learning process, and the passing on of accumulated knowledge. Training is supported by a manual, which recognizes the beginner\u27s lack of experience and attempts to introduce basic teaching techniques along with the rational for their use as simply and succinctly as possible. Emphasis is placed on understanding why things happen in the classroom so that trainees will learn to problem solve by themselves and recognize that reflection upon their successes and failures will enable them to continue to improve their standard of teaching. The ultimate aim thereby to produce thinking teachers who value self-development and see it as a necessary on-going process

A homotopical property of attractors

We construct a 2-dimensional torus T ⊆ R3 having the property that it cannot be an attractor for any homeomorphism of R3. To this end we show that the fundamental group of the complement of an attractor has certain finite generation property that the complement of T does not have