Every hierarchy of beliefs is a type

When modeling game situations of incomplete information one usually considers the players' hierarchies of beliefs, a source of all sorts of complications. Hars\'anyi (1967-68)'s idea henceforth referred to as the "Hars\'anyi program" is that hierarchies of beliefs can be replaced by "types". The types constitute the "type space". In the purely measurable framework Heifetz and Samet (1998) formalize the concept of type spaces and prove the existence and the uniqueness of a universal type space. Meier (2001) shows that the purely measurable universal type space is complete, i.e., it is a consistent object. With the aim of adding the finishing touch to these results, we will prove in this paper that in the purely measurable framework every hierarchy of beliefs can be represented by a unique element of the complete universal type space.Comment: 19 page

Greenhouse Effect in the Classroom: A Project- and Laboratory-Based Curriculum

This article describes a multifaceted curriculum for use in earth science classes from secondary school to undergraduate level. The centerpiece of the curriculum is a project-based investigation of greenhouse warming that can be conducted during one or more lab sessions using off-the-shelf materials. This process provides students with the opportunity to gain a deeper understanding of the climate system, the nature of scientific uncertainty, the burden of proof in ongoing research, and the difficulties of transferring scientific results to the public-policy realm. The main thrust of the lab demonstration and the curriculum itself is to help students distinguish the solid science in these issues from the array of questionable information they encounter in the press, on the internet, or in conversation. Educational levels: Primary elementary, Middle school, High school, Undergraduate lower division

Earthquake Magnitude and Intensity

In this exercise, students compare the amount of shaking caused by historic earthquakes, and use data from seismograms to determine Richter magnitude. They will also investigate moment magnitude, an alternative to Richter magnitude, and calculate a seismic moment. In the second portion of the exercise, students investigate earthquake intensity and prepare a map of intensity values from the 1994 Northridge, California earthquake, using actual reports of its effects. Introductory materials explain the difference between earthquake magnitude and intensity, point out the logarithmic nature of the Richter scale, and present criteria for assigning modified Mercalli intensity values to a particular location. The exercise includes instructions, maps, data, and study questions. A bibliography is also provided. Educational levels: High school, Undergraduate lower division

Coastal Terraces, Sea Level, and Active Tectonics

In this exercise, students investigate the use of coastal landforms from ancient shorelines in studying tectonic movements. Introductory materials explain how coastal landforms are classified on the basis of sediment supply and positions of the land relative to sea level, and describe the features of erosional coastal terraces. Using data on the coastal terraces of San Clemente Island, off the coast of Southern California, students will construct a topographic profile, measure shoreline angles, and calculate rates of uplift of the island relative to the sea. The exercise includes a map and stereo pair, data on sea level fluctuations and ages of coastal terraces, and a problem set with study questions. A bibliography is also provided. Educational levels: High school, Undergraduate lower division

Seismic Hazard Assessment: Conditional Probability

In this exercise, students investigate the use of conditional probability (the likelihood that a given event will occur within a specified time period) in assessing earthquake hazards. Introductory materials explain that conditional probability is based on the past history of earthquakes in a region and on how and when earthquakes recur; and discuss the different types of models that can be developed to predict recurrences. Using a table of probability values, students will calculate probabilities for earthquakes along the San Andreas and Wasatch Fault zones, and calculate probabilities that they will exceed a given acceleration (ground shaking) value. Example problems and a bibliography are provided. Educational levels: High school, Undergraduate lower division

Exploring the effectiveness of using big books in teaching primary English in Malaysian classrooms

There are many benefits of using stories for language and literacy development particularly in the areas of vocabulary(Cameron, 2001; Elley, 1989; Penno, Wilkinson & Moore 2002;Hargrave & Senechal, 2000) and reading comprehension. Yet, much is still unknown in second language (L2) contexts. This paper investigates the use of Big Books in primary English in Malaysia where English is taught as an official second language. It draws on transcript data from one teacher from one primary school to illustrate how students’ active participation and interest in language learning emerged as the teacher introduced Big Books instead of the textbook. Our data shows that practice with Big Books changes the interaction patterns and learning opportunities. Students were actively involved and they contributed to the discussions and made spontaneous comments about the stories. Most importantly,the teacher’s elicitation skill is essential in enhancing students’ learning. The Ministry of Education’s hopes to provide more active engagement and to increase students’ interests and motivation through stories were achieved

Numerical Dating

In this exercise, students investigate methods used by geologists studying active tectonics for determining ages in actual numbers of years. Introductory materials describe the three most-used techniques for dating material formed during the Quaternary Period (approximately the last 1.65 million years), discuss the concepts of radioactive decay and half-life, and explain how these may be used to determine the numerical age of an object or substance. The exercise includes a set of problems in which students calculate isotopic abundance, half-life, decay rate, and absolute age. Example problems and a bibliography are provided. Educational levels: High school, Undergraduate lower division