Bi-algebras, generalised geometry and T-duality

A study of sigma models whose target space is a group G that admits a compatible Poisson structure is presented. The natural action of O(D,D;Z) on the generalised tangent bundle TG+T*G and a generalisation of the Courant bracket that appears are reviewed. This background provides a concrete example where the generalised geometry and doubled geometry descriptions are both well understood. Connections between the two formalisms are discussed and the world-sheet theory from Hamiltonian and Lagrangian perspectives is investigated. The comparisons between the approaches given by generalised geometry and doubled geometry suggest possible ways of generalising the analysis beyond the known examples.Comment: 43 page

Semi-Supervised Kernel PCA

We present three generalisations of Kernel Principal Components Analysis (KPCA) which incorporate knowledge of the class labels of a subset of the data points. The first, MV-KPCA, penalises within class variances similar to Fisher discriminant analysis. The second, LSKPCA is a hybrid of least squares regression and kernel PCA. The final LR-KPCA is an iteratively reweighted version of the previous which achieves a sigmoid loss function on the labeled points. We provide a theoretical risk bound as well as illustrative experiments on real and toy data sets

Mathematical retroreflectors

Retroreflectors are optical devices that reverse the direction of incident beams of light. Here we present a collection of billiard type retroreflectors consisting of four objects; three of them are asymptotically perfect retroreflectors, and the fourth one is a retroreflector which is very close to perfect. Three objects of the collection have recently been discovered and published or submitted for publication. The fourth object - notched angle - is a new one; a proof of its retroreflectivity is given.Comment: 32 pages, 19 figure

Mathematical Modeling Of Horizontal Displacement Of Above-ground Gas Pipelines

The modern geodetic equipment allows observations as soon as possible, providing high accuracy and productivity. Achieving high accuracy of measurement is impossible without taking into account external factors that create influence on an observation object. Therefore, in order to evaluate an influence of thermal displacement on the results of geodetic monitoring a mathematical model of horizontal displacement of above-ground pipelines was theoretically grounded and built. In this paper we used data of experimental studies on the existing pipelines "Soyuz" and "Urengoy - Pomary - Uzhgorod". Above-ground pipeline was considered as a dynamic system "building - environment". Based on the characteristics of dynamic systems the correlation between the factors of thermal influence and horizontal displacement of the pipeline axis was defined.Establishing patterns between input factors and output response of the object can be useful not only for geodetic control, but also for their consideration in the design of new objects. It was investigated that the greatest influence on the accuracy of geodetic observations can create dispersion of high-frequency oscillations caused by daily thermal displacement. The magnitude of displacement exceeds actual measurement error.The article presents the results of calculation of high-frequency oscillations of above-ground gas pipeline.The result made it possible to substantiate the accuracy and methodology of geodetic observations of the horizontal displacement of pipeline axes taking into account an influence of cyclical thermal displacement.Research results were recommended for use in practice for enterprises that serve the main gas pipelines and successfully tested by specialists of PJSC "Ukrtransgaz" (Kharkiv, Ukraine) during the technical state control of aerial pipeline crossing in Ukraine and also can be used to form the relevant regulations

Investigating mathematical investigations

There are good reasons why we may involve the students in doing mathematics investigations. Recent curricula encourage this sort of activity but we notice that its application in the classroom is not a simple matter. We discuss the issues that arise when students are presented with investigative tasks, with special interest in the dynamics of the classroom and in the role of the teacher. Our aim is to derive suggestions for classroom practice as well as for further research and teacher development

Mathematical Communication: What And How To Develop It In Mathematics Learning?

Mathematics is the language of symbols so that everyone who studied mathematics required having the ability to communicate using the language of these symbols. Mathematical communication skills will make a person could use mathematics for its own sake as well as others, so that will increase positive attitudes towards mathematics. Mathematical communication skills can support mathematical abilities, such as problem solving skills. With good communication skills then the problem will more quickly be represented correctly and this will support in solving problems. Students' mathematical communication skills can be developed in various ways, one with group discussions. Brenner (1998) found that the formation of small groups facilitate the development of mathematical communication skills. This paper describes the mathematical communication and how to develop the mathematical communication skills in learning mathematics. For further clarify the discussion, given also the example of learning that emphasizes the development of mathematical communication skills. Keywords: Mathematical Communication, Mathematics Learning

Mathematical Explanation: A Contextual Approach

PurposeIn this article, we aim to present and defend a contextual approach to mathematical explanation.MethodTo do this, we introduce an epistemic reading of mathematical explanation.ResultsThe epistemic reading not only clarifies the link between mathematical explanation and mathematical understanding, but also allows us to explicate some contextual factors governing explanation. We then show how several accounts of mathematical explanation can be read in this approach.ConclusionThe contextual approach defended here clears up the notion of explanation and pushes us towards a pluralist vision on mathematical explanation