Packing and covering with balls on Busemann surfaces

In this note we prove that for any compact subset $S$ of a Busemann surface $({\mathcal S},d)$ (in particular, for any simple polygon with geodesic metric) and any positive number $\delta$, the minimum number of closed balls of radius $\delta$ with centers at $\mathcal S$ and covering the set $S$ is at most 19 times the maximum number of disjoint closed balls of radius $\delta$ centered at points of $S$: $\nu(S) \le \rho(S) \le 19\nu(S)$, where $\rho(S)$ and $\nu(S)$ are the covering and the packing numbers of $S$ by ${\delta}$-balls.Comment: 27 page

The development of mathematical understanding and its application to Libyan secondary school mathematics

It is unfortunate that many students in Libyan schools and elsewhere dislike mathematics and are even afraid of studying it. This is partially related to the fact that mathematics is presented in an extensive fashion in both scope and depth of content and does not provide opportunities for the student population to understand it. The primary purpose of this study was to examine the problems of the method of presenting mathematics in Libyan secondary schools and to develop principles for understanding mathematics. There were four subordinate problems inherent in the main problems: 1) to study and analyze the characteristics of the new mathematics program in terms of content and method of presentation; 2) to study the problems of Libyan secondary school mathematics, specifically to determine weaknesses in the content and method of presentation; 3) to explore and develop criteria for understanding mathematics; and 4) to evaluate method of presenting mathematics in Libyan secondary schools in terms of the developed criteria

Correlation Functions in Two-Dimensional Critical Systems with Conformal Symmetry.

This thesis presents a study of certain conformal field theory (CFT) correlation functions that describe physical observables in conformally invariant two-dimensional critical systems. These are typically continuum limits of critical lattice models in a domain within the complex plane and with a boundary. Certain clusters, called boundary clusters, anchor to the boundary of the domain, and many of their features are governed by a conformally invariant probability measure. For example, percolation is an example of a critical lattice model, and when it is confined to a domain with a boundary, connected clusters of activated bonds that touch that boundary are the boundary clusters. This thesis is concerned with how the boundary clusters interact with each other according to that measure. One question that it considers are ``how likely are these clusters to repel each other or to connect with one another in a certain topological configuration?" Chapter one non-rigorously derives an already well-known elliptic system of differential equations closely tied to this matter by using standard techniques of CFT, chapters two and three rigorously infer certain properties concerning the solution space of this system, and chapter four uses some of those results to predict an answer to this question. This thesis also considers local variations of this question such as ``what regions of the domain do the perimeters of the boundary clusters explore," and ``how often will several boundary clusters connect at just a single, specified point in the domain?" Chapter five predicts precise answers to these questions. All of these answers are quantitative predictions that we verify via high-precision computer simulation. Chapters four and five also present these simulation results. Further material that supplements chapter one is included in two appendices.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/94010/1/smflores_1.pd

Scale and abstraction : the sensitivity of fire regime simulation to nuisance parameters

Fire plays a key role in ecosystem dynamics and its impact on environmental, social and economic assets is increasingly a critical area of research. Fire regime simulation models are one of many approaches that provide insights into the relative importance of factors driving the dynamics of fire-vegetation systems. Fire propagates as a contagious process and simulation is an approach that captures this behaviour explicitly, integrating spatial and temporal data to produce auto-correlated patterns of fire regimes. However, when formulating these models, time and many aspects of space must be made discrete. These parameters are 'nuisance parameters': parameters necessary for the model formulation but not otherwise of interest. Fire growth simulations are therefore discrete approximations of continuous non-linear systems, and it might be expected that the values chosen for these nuisance parameters will be important. While it is well known that discrete geometries have consequences for the shape and area of simulated fires, no research has investigated the consequence this may have for estimates of the relative importance of the various drivers of fire regimes. I argue that nuisance parameters can be demonstrated to be unimportant for this class of model. I use the idea of 'importance' to underline the need for context with such an assertion. With sufficient replication, any parameter can be found statistically significant. A parameter is important, on the other hand, if different values produce qualitatively different outcomes. Models are commonly either re-parameterised to account for changes in resolution or scaling-up methods applied if such exist. I will further argue that such differences as there are in model outputs due to spatial resolution, cannot be accounted for by either re-parameterising or using a common approach that allows resolution to vary over the spatial extent. A set of experiments were devised using a published fire regime simulation model, modified, verified and validated, to isolate just those aspects of the model's sensitivity to resolution and discrete geometries that are unavoidable or intrinsic to these choices. This new model was used to test the above hypotheses, using peer-reviewed treatments that stand as yardsticks by which formal estimates of the importance of nuisance parameters can be made. As estimated by the model, neither spatio-temporal resolution nor any of the various choices available for discrete geometries, altered the model predictions. As expected, it is spatial resolution that has the greatest impact on running times for the model but this study finds that neither calibration, nor taking an approach that allows resolution to vary over the spatial extent, can account for differences in model outputs that arise from running simulations at coarser resolutions. All models are abstractions and a good model should ideally hold over levels of abstraction. This is rarely the case, but this study shows that results obtained through simulation in estimating the drivers of fire frequency in large landscapes, are robust with regard to these aspects of abstraction. This adds considerable confidence to a significant body of work that has used this approach over the last two decades

Metastability for the Ising model on the hexagonal lattice

We consider the Ising model on the hexagonal lattice evolving according to Metropolis dynamics. We study its metastable behavior in the limit of vanishing temperature when the system is immersed in a small external magnetic field. We determine the asymptotic properties of the transition time from the metastable to the stable state and study the relaxation time and the spectral gap of the Markov process. We give a geometrical description of the critical configurations and show how not only their size but their shape varies depending on the thermodynamical parameters. Finally we provide some results concerning polyiamonds of maximal area and minimal perimeter

Proceedings of the 6th International Workshop of the EARSeL Special Interest Group on Forest Fires Advances in Remote Sensing and GIS Applications in Forest Fire Management Towards an Operational Use of Remote Sensing in Forest Fire Management

During the last two decades, interest in forest fire research has grown steadily, as more and more local and global impacts of burning are being identified. The definition of fire regimes as well as the identification of factors explaining spatial and temporal variations in these fire characteristics are recently hot fields of research. Changes in these fire regimes have important social and ecological implications. Whether these changes are mainly caused by land use or climate warming, greater efforts are demanded to manage forest fires at different temporal and spatial scales. The European Association of Remote Sensing Laboratories (EARSeL)’s Special Interest Group (SIG) on Forest Fires was created in 1995, following the initiative of several researchers studying Mediterranean fires in Europe. It has promoted five technical meetings and several specialised publications since then, and represents one of the most active groups within the EARSeL. The SIG has tried to foster interaction among scientists and managers who are interested in using remote sensing data and techniques to improve the traditional methods of fire risk estimation and the assessment of fire effect. The aim of the 6th international workshop is to analyze the operational use of remote sensing in forest fire management, bringing together scientists and fire managers to promote the development of methods that may better serve the operational community. This idea clearly links with international programmes of a similar scope, such as the Global Monitoring for Environment and Security (GMES) and the Global Observation of Forest Cover/Land Dynamics (GOFC-GOLD) who, together with the Joint Research Center of the European Union sponsor this event. Finally, I would like to thank the local organisers for the considerable lengths they have gone to in order to put this material together, and take care of all the details that the organization of this event requires.JRC.H.3-Global environement monitorin