Mathematical Modelling: Concepts and Case Studies

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Glioblastoma is an aggressive form of brain tumour, which is characterised by life expectancies of less than 2 years from diagnosis and currently has no cure. The only intervention available to a patient is having the infected area of their brain cut away as soon as the tumour cells are observed. Unfortunately, these technologies may have undesirable consequences for the electricity networks supplying our homes and businesses.

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The possible plethora of low carbon technologies, like electric vehicles, heat pumps and photovoltaics, will lead to increased pressure on the local electricity networks from larger and less predictable demands. The motion of weights attached to a chain or string moving on a frictionless pulley is a classic problem of introductory physics used to understand the relationship between force and acceleration.

In their recently published paper Oxford Mathematicians Dominic Vella and Alain Goriely and colleagues looked at the dynamics of the chain when one of the weights is removed and thus one end is pulled with constant acceleration. The use of mathematical models to describe the motion of a variety of biological organisms has been the subject of much research interest for several decades.

Case Studies and Films

This picture shows the "Z" machine at Sandia Labs in New Mexico producing, for a tiny fraction of a second, TW of power - about times the average electricity consumption of the entire planet. How should such extreme behaviour be described mathematically? The International Congresses of Mathematicians ICMs take place every four years at different locations around the globe, and are the largest regular gatherings of mathematicians from all nations.

However, as much as the assembled mathematicians may like to pretend that these gatherings transcend politics, they have always been coloured by world events: the congresses prior to the Second World War saw friction between French and German mathematicians, for example, whilst Cold War political tensions likewise shaped the conduct of later congresses. The concept of equilibrium is one of the most central ideas in economics. It is one of the core assumptions in the vast majority of economic models, including models used by policymakers on issues ranging from monetary policy to climate change, trade policy and the minimum wage.

But is it a good assumption? For example, let us consider the case of a particle moving on the real line. Homogenization theory aims to understand the properties of materials with complicated microstructures, such as those arising from flaws in a manufacturing process or from randomly deposited impurities. The goal is to identify an effective model that provides an accurate approximation of the original material.

Oxford Mathematician Benjamin Fehrman discusses his research. The discomfort experienced when a kidney stone passes through the ureter is often compared to the pain of childbirth. Severe pain can indicate that the stone is too large to naturally dislodge, and surgical intervention may be required. A ureteroscope is inserted into the ureter passing first through the urethra and the bladder in a procedure called ureteroscopy.

Fusion energy may hold the key to a sustainable future of electricity production. However some technical stumbling blocks remain to be overcome. Even the sturdiest solid solutions suffer damage over time, which could be avoided by adding a thin liquid coating. In many natural systems, such as the climate, the flow of fluids, but also in the motion of certain celestial objects, we observe complicated, irregular, seemingly random behaviours.

These are often created by simple deterministic rules, and not by some vast complexity of the system or its inherent randomness. A typical feature of such chaotic systems is the high sensitivity of trajectories to the initial condition, which is also known in popular culture as the butterfly effect. Oxford Mathematician Riccardo W. Its study is at the interface of probability, number theory, analysis, and geometry.

The applications to physics include the study of ocean waves, earthquakes, sound and other types of waves. At the beginning of the twentieth century, some minor algebraic investigations grabbed the interest of a small group of American mathematicians. The problems they worked on had little impact at the time, but they may nevertheless have had a subtle effect on the way in which mathematics has been taught over the past century.

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In a seminal paper, Alan Turing mathematically demonstrated that two reacting chemicals in a spatially uniform mixture could give rise to patterns due to molecular movement, or diffusion. This is a particularly striking result, as diffusion is considered to be a stabilizing mechanism, driving systems towards uniformity think of a drop of dye spreading in water.

What does boiling water have in common with magnets and the horizon of black holes? They are all described by conformal field theories CFTs! We are used to physical systems that are invariant under translations and rotations. Imagine a system which is also invariant under scale transformations. Such a system is described by a conformal field theory.

Remarkably, many physical systems admit such a description and conformal field theory is ubiquitous in our current theoretical understanding of nature. Oxford Mathematician Tom Oliver talks about his research in to the rich mine of mathematical information that are L-functions. Americans drink an average of 3. When carrying a liquid, common sense says walk slowly and refrain from overfilling the container. But easier said than followed.

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Cue sloshing. They produce this cold light using a light-emitting molecule, the luciferin, and a complementary enzyme, luciferase. This process is known as bioluminescence. Over the last five decades, software and computation has grown to become integral to the scientific process, for both theory and experimentation. Here he discusses his latest work. It turns out that the situation is not as simple as it looks and these two areas might be connected. It is probably well-known that number theory is the source of some of the oldest and most accessible questions in mathematics:.

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Governments around the world are seeking to address the economic and humanitarian consequences of climate change. One of the most graphic indications of warming temperatures is the melting of the large ice caps in Greenland and Antarctica. This is a litmus test for climate change, since ice loss may contribute more than a metre to sea-level rise over the next century, and the fresh water that is dumped into the ocean will most likely affect the ocean circulation that regulates our temperature.

To be useful, there must be a group of interested and informed users, who have both the tools and the skills to analyse the data to uncover misconduct, and then lobby governments and donors to listen to and act on the findings.

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Michael Bonsall , Professor of Mathematical Biology at Oxford University's Department of Zoology, discusses his research in population biology, what it tells us about species evolution and, in particular, why grandmothering is important to humans. From studying the rhythmic movements, researchers at the Universities of York, Birmingham, Oxford and Kyoto University, Japan, have developed a mathematical formula which makes it easier to understand and predict how sperm make the journey to fertilise an egg.

This knowledge will help scientists to gauge why some sperm are successful in fertilisation and others are not. The classic picture of how spheres deform e. This deformation is also very cheap in terms of the elastic energy and so it has long been assumed that this is what a physical shell e. Cancer is a complex and resilient set of diseases and the search for a cure requires a multi-strategic approach. In an interview with Rolling Stone Magazine in , Bob Dylan was pushed to define himself: Do you think of yourself primarily as a singer or a poet?

Case Studies and Films | Mathematical Institute

As part of our series of research articles deliberately focusing on the rigour and intricacies of mathematics and its problems, Oxford Mathematician Eduardo Casali discusses his work. Think of a mathematician and you might imagine an isolated individual fueled by coffee whose immaculate if incomprehensible papers may, in the fullness of time, via a decades-long dry chain of citations, be made use of by an industrialist via one or two other dedicated mathematicians.

As part of our series of research articles deliberately focusing on the rigour and complexity of mathematics and its problems, Oxford Mathematician Gui-Qiang G Chen discusses his work on the Mathematics of Shock Reflection-Diffraction. X-ray imaging is an important technique for a variety of applications including medical imaging, industrial inspection and airport security. An X-ray image shows a two-dimensional projection of a three-dimensional body. The original 3D information can be recovered if multiple images are given of the same object from different viewpoints.

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Across the physical and biological sciences, mathematical models are formulated to capture experimental observations. Often, multiple models are developed to explore alternate hypotheses.

Mathematical Modelling: Concepts and Case Studies Mathematical Modelling: Concepts and Case Studies
Mathematical Modelling: Concepts and Case Studies Mathematical Modelling: Concepts and Case Studies
Mathematical Modelling: Concepts and Case Studies Mathematical Modelling: Concepts and Case Studies
Mathematical Modelling: Concepts and Case Studies Mathematical Modelling: Concepts and Case Studies
Mathematical Modelling: Concepts and Case Studies Mathematical Modelling: Concepts and Case Studies

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