Uniqueness and non-uniqueness in percolation theory

This paper is an up-to-date introduction to the problem of uniqueness versus non-uniqueness of infinite clusters for percolation on ${\mathbb{Z}}^d$ and, more generally, on transitive graphs. For iid percolation on ${\mathbb{Z}}^d$, uniqueness of the infinite cluster is a classical result, while on certain other transitive graphs uniqueness may fail. Key properties of the graphs in this context turn out to be amenability and nonamenability. The same problem is considered for certain dependent percolation models -- most prominently the Fortuin--Kasteleyn random-cluster model -- and in situations where the standard connectivity notion is replaced by entanglement or rigidity. So-called simultaneous uniqueness in couplings of percolation processes is also considered. Some of the main results are proved in detail, while for others the proofs are merely sketched, and for yet others they are omitted. Several open problems are discussed.Comment: Published at http://dx.doi.org/10.1214/154957806000000096 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Rigorous computer analysis of the Chow-Robbins game

Flip a coin repeatedly, and stop whenever you want. Your payoff is the proportion of heads, and you wish to maximize this payoff in expectation. This so-called Chow-Robbins game is amenable to computer analysis, but while simple-minded number crunching can show that it is best to continue in a given position, establishing rigorously that stopping is optimal seems at first sight to require "backward induction from infinity". We establish a simple upper bound on the expected payoff in a given position, allowing efficient and rigorous computer analysis of positions early in the game. In particular we confirm that with 5 heads and 3 tails, stopping is optimal.Comment: 10 page

Teaching systems of linear equations in Sweden and China: What is made possible to learn?

A starting point for this study was an aim to understand better the relationbetween teaching and learning of mathematics. This interest was based onthe assumption that what is possible for students to learn about mathematicsmust be related to how they experience the mathematical content, which inturn must be related to how the content is handled during the mathematicslesson. Many factors that may have an impact on the teaching and learning ofmathematics are beyond the influence of teachers. However, the handling ofthe content what examples to use, what aspects of concepts and methods toemphasise, what exercises students should work on etc is something aboutwhich mathematics teachers must always make decisions. An intention withthis study was to produce results that could inform practice on the classroomlevel, as well as teacher education, in this respect. Another intention was tocontribute to the development of methods for analysing teaching that aresensitive to, and focus on, the specific content of instruction.Sixteen lessons from six classes in Sweden and China video recordedwithin the Learners Perspective Study were analysed and compared. Theanalysis was based on Variation theory and had its focus on differences in howthe same mathematical content was taught. The concept object of learning was used to denote what teachers try to teach and what students are supposedto learn. In this study the three objects of learning analysed were related tothe mathematics taught in the six classrooms systems of linear equations intwo unknowns, solutions to systems of linear equations in two unknowns and themethod of substitution. The analysis was done from the perspective of a studentin the classroom and aimed at describing what aspects of the objects of learningwere made possible to learn. An aspect was considered made possible toexperience if the corresponding dimension of variation was opened. The alternativewas that the aspect in question was taken for granted during teachingand kept invariant.The analytical approach employed made it possible to detect even subtle differencesin how the teachers handled the content. The teachers thereby madedifferent things possible to learn for the students. The description of these differencespoints out several aspects that could be so familiar to many teachersthat these aspects face the risk of being taken for granted in teaching. Further,findings regarding the use of systematic and deliberate variation in the Chineseclassrooms are discussed

Alternativ media och jämställdhet : En analys av artiklar på Avpixlat och dess kommentarer

Den här uppsatsen ämnar analysera diskursen om jämställdhet i en alternativ mediesite som upplyfter konservatism och politiska nyheter, Avpixlat. Uppsatsen fokuserar huvudsakligen på artiklar och kommentarer på delen ”jämställdhet” på deras hemsida, som är analyserad genom kvantitativ innehållsanalys och kritik diskursanalys. Den teoretiska utgångspunkten är ett intersektionellt angreppsätt och feministisk medieteori. Den huvudsaklige slutsatsen gällande jämställdhet på Avpixlat är att den diskuteras inom kontexten av kritik mot islam och invandring. Både artikelförfattaren och kommentarerna beskriver jämställdhetsdebatten som en kamp mellan de öppna progressiva Sverige och det fundamentalistiska elementen av islam och det förändringar som kan ske med det svenska samhället. Diskursen skiftar mellan att promota sexuell öppenhet (när de jämför den svenska livsstilen med islam), och att förhålla sig konservativt gentemot exempelvis gender mainstreaming och implementeringen av nya kvoteringslagar

Under the looking glass : Cinematographic differences between two film adaptations of Murder on the Orient Express

Det är ibland svårt att avgöra vilken betydelse kameraarbetet får i en film och exakt vad det förmedlar till tittaren. Genom att undersöka två filmer som baseras på samma ursprungsberättelse får vi möjlighet att utläsa kameraarbetets betydelse och skillnaden det gör för hur filmerna upplevs. Filmerna är Sidney Lumets Mordet på Orientexpressen (1974) och Kenneth Branaghs Mordet på Orientexpressen (2017). Det som undersöks i uppsatsen är hur olika kameratekniker används och hur det bidrar till berättandet. Det undersöks genom en metodbaserad på kompositionell analys och multimodalitet. Vår slutsats är att även om liknande sätt att arbeta med kameran förekommer i båda filmerna så har de skillnader som identifierats påverkat bland annat filmernas tema och hur deltagande tittaren blir i utredningen av mordet