Hollis Williams

The University of Warwick · School of Engineering

‘Perpetual motion’ is a hypothetical type of motion which continues forever without any external energy input contributing to this system. Students should know that this is generally impossible because of energy losses due to friction or other non-conservative forces, or because some assumption has been made which violates the first or second law o...

A classic magic trick involves filling a bucket with liquid and then turning it upside down so that the liquid stays in the bucket. This does not work under normal circumstances because of the Rayleigh-Taylor instability. Although the atmospheric pressure pushing at the free surface from below is theoretically enough to support the surface, in real...

Popular article about the rule-based game Baba is You.

Mirror symmetry is an exciting and challenging area of current research which sits at the intersection point between algebraic geometry and string theory. It is concerned with a class of geometric objects known as Calabi-Yau manifolds, which first entered theoretical physics as a way to deal with the large number of extra spatial dimensions which t...

The Riemann-Roch theorem is a classical result which forms a beautiful algebraic connection between complex analysis on a compact Riemann surface and a global topological property of that surface (the genus). We present a survey of the theorem and its many variants and generalisations. We also provide an alternative elementary proof of the Riemann-...

We suggest that it might be possible to resolve the vacuum energy problem by assuming the reality of a many worlds interpretation of quantum mechanics. The suggested resolution is that the enormous theoretical prediction for the vacuum energy density is actually the value distributed across all the parallel universes in a superposition. It is assum...

We discuss the two main perspectives from which a mathematician might view a maze: that of graph theory and that of problem-solving.

The ungula is a geometric solid which has been studied since antiquity. In particular, Archimedes demonstrated using geometric means that the volume of an ungula is one-sixth of the volume of an enclosing cube. We will find the volume by performing a triple integral in cylindrical coordinates: this computation is somewhat simpler and more transpare...

Recent experimental work has found that cavity formation during the impact of a wettable sphere on a free liquid surface can be suppressed by decreasing the density of the ambient gas \cite{williams}. It was suggested that this phenomenon can be attributed to the gas slowing sealing of the thin crown sheet which forms behind the sphere during splas...

Titanomagnetite sand (known more popularly as ironsand) is a type of granular material which contains high amounts of iron in the form of magnetite, as well as other metal impurities such as calcium and titanium. The surface morphology and constituent elements of these sands has been studied in detail and X-ray diffraction has been used to characte...

Mixtures of granular matter and liquid (known as granular suspensions) are ubiquitous in nature, with perhaps the most famous example being quicksand. The theoretical study of these mixtures is complicated due to the lack of constitutive relations for general flow conditions [1]. Other studies have characterised the Rayleigh-Taylor instabilities wh...

We study a version of the two-degree-of-freedom double pendulum in which the two point masses are replaced by rigid bodies of irregular shape and nonconservative forces are permitted. We derive the equations of motion by analysing the forces involved in the framework of screw theory. This distinguishes the work from similar studies in the literatur...

A key concept in current fluid dynamics and its applications to biology and technology is a phenomenon known as wetting. Wetting is familiar from everyday life and is simply the ability of a liquid to stay in contact with a solid surface. The wettability depends on the properties of the liquid and the solid and can be characterized by the static eq...

Formation of a splash crown and a cavity following the impact of a sphere on a body of liquid is a classical problem. In the related problem of droplet splashing on a flat surface, it has been established that the properties of the surrounding gas can influence the threshold speed at which splashing occurs. At lower impact speeds, this is due mainl...

In this article, we construct a very simple double pendulum (the concept of a pendulum should be familiar to all beginning students of classical mechanics). Since a double pendulum has two degrees of freedom, we suggest that this pendulum can be used in the classroom environment to illustrate the concept of normal modes and we use video software to...

In daily life we are familiar with the fact that a glass or a ceramic mug which is knocked from a table shatters when it hits the ground. However, the phenomenology behind this event is perhaps less familiar due to the speed with which it happens. Examination of the recovered fragments shows that the mug generally breaks up into several large piece...

We present a number of exact solutions to the linearised Grad equations for non-equilibrium rarefied gas flows and heat flows. The solutions include the flow and pressure fields associated to a point force placed in a rarefied gas flow close to a no-slip boundary and the temperature field for a point heat source placed in a heat flow close to a tem...

Formation of a splash crown and a cavity following the impact of a sphere on a body of liquid is a classical problem. In the related problem of a droplet splashing on a flat surface, it has been established that the properties of the surrounding gas can influence the splashing threshold. At lower impact speeds, this is due mainly to the influence o...

We revisit the connection between the Ricci flow and isoperimetric inequalities on surfaces which are diffeomorphic to the 2-sphere. The Ricci flow is used to prove a version of the isoperimetric inequality on the round 2-sphere. We prove that the Cheeger isoperimetric constant is non-decreasing under Ricci flow on topological 2-spheres. A topologi...

The classical Penrose inequality relates the mass of an asymptotically flat spacetime to the total area of its black holes. We present a PDE-based proof of the Penrose inequality for a special class of perturbations of Schwarzschild initial data. The proof is based around linearising solutions to a version of the Jang equation which is modified to...

It is well-known that Newton’s work on mechanics depended in a crucial way on the previous observations of Galileo. The key insight of Galileo was that one can analyse the motion of bodies using experiments and mathematical equations. One experimental observation which roughly emerges from this work in modern terms is that two objects of different...

In industrial applications, it is crucial to be able to predict and model formation of cavitation bubbles. CFD solvers based on a mesh-free method known as Smoothed Particle Hydrodynamics (SPH) have recently become popular for modelling of complicated free surface flows. In this article, we demonstrate that the presence of cavitation bubbles can be...

Formation of a splash crown and a cavity following the impact of a sphere on a body of liquid is a classical problem. In the related problem of a droplet splashing on a flat surface, it has been established that the properties of the surrounding gas can influence the splashing threshold. At lower impact speeds, this is due mainly to the influence o...

Este documento es una traducción de 'Como aprendo árabe', una charla de Steve Kaufmann.

Bird wings are frequently modelled in the aerodynamics literature under the assumption that they are static aerofoils. Air flows over real bird wings can violate this assumption, both because of the topology of the wing and the fact that real wings typically undergo flapping and twisting motions during flight. There are many computational and numer...

Granular flows appear frequently in the natural world and in civil engineering applications. These flows can exhibit features which are surprising and counter-intuitive and are often used to test the limits of the classical continuum approximation for modelling of fluid flows. An important sub-class of the granular flows are the gravity-driven gran...

The notion of a boundary condition is typically considered to be somewhat advanced and not suitable to be introduced in high school level physics. In this article, we give a simple visual demonstration of the difference between Dirichlet and Neumann boundary conditions for a string which oscillates according to the one-dimensional wave equation. Th...

Increased intensity of extreme climatic events and natural hazards, combined with sea level rise due to global warming, has increased the vulnerability of nearshore and coastal regions to extreme flooding and erosion. The existing hard-engineered infrastructures for flood protection are mainly built from concrete with very high carbon emissions thr...

Este documento es una traducción de 'Alquimia musical', un diálogo entre Robert Bosnak y Brian Carroll (más conocido por su nombre artístico Buckethead). Las opiniones expresadas en este diálogo son responsabilidad exclusiva de Robert Bosnak y Brian Carroll y no representan la posición del traductor.

We demonstrate that the usual optical methods can be applied in a simple way to quickly estimate thickness of liquid sheets on sub-millimetre scales. It is shown that an approximate linear relation can be obtained between gap thickness and several of the usual RBG and HSV variables (red and green pixel values in the RBG case and saturation in the H...

A common trick used by hostel occupants where a kettle is unavailable is to heat water using shaving blades [1]. To make the device, one first strips and ties the ends of insulated copper wires to a pair of stainless steel razor blades. The blades are then joined by some insulating material (by embedding the blades into a pair of matchsticks at top...

In certain quantum gravity theories known as emergent gravity theories, it is proposed that spacetime is divided up into a discrete structure at the Planck scale. Such theories are likely ruled out by the failure to observe violations of Lorentz invariance at such length scales. Another reason to doubt these theories is the fact that one would expe...

We suggest that it might be possible to resolve the vacuum energy problem by assuming the reality of a many worlds interpretation of quantum mechanics. The suggested resolution is that the enormous theoretical prediction for the vacuum energy density is actually the value distributed across all the parallel universes in a superposition. It is assum...

We give an outline of thermonuclear fusion and explain some of the major engineering and applied mathematical problems which would need to be overcome in order to achieve controlled fusion on Earth as a method of power generation.

‘Perpetual motion’ is a hypothetical type of motion which continues forever without any external energy input contributing to this system. Students should know that this is generally impossible because of energy losses due to friction or other non-conservative forces, or because some assumption has been made which violates the first or second law o...

The long jump is a track and field event in which athletes attempt to jump as far as possible from a take-off point into a sandpit. In the 1970s, an athlete called Tuariki Delamere tried to introduce a ‘front flip’ technique where one goes into a front tuck instead of using a more regular technique like the hitch kick and argued that it would allow...

Mirror symmetry is an exciting and challenging area of current research which sits at the intersection point between algebraic geometry and string theory. It is concerned with a class of geometric objects known as Calabi-Yau manifolds, which first entered theoretical physics as a way to deal with the large number of extra spatial dimensions which t...

A topic which causes some confusion in the literature on rolling motion is the rolling friction experienced by a sphere as it rolls down a groove in an inclined slope. This set-up is often considered in the calculus of variations literature in relation to the brachistochrone and tautochrone problems [1 – 4]. In this article, we propose to simplify...

Increased intensity of extreme climatic events and natural hazards, combined with sea level rise due to global warming, has increased the vulnerability of nearshore and coastal regions to extreme flooding and erosion. The existing hard-engineered infrastructures for flood protection are mainly built from concrete with very high carbon emissions thr...

It is well-known that eigenvalues in quantum mechanics cannot be assigned to physical properties independently of the measurement context. We argue that it might be possible to relate the contexts of the Mermin-Peres magic square using causality relations in a way which makes explanations using hidden variable models unnecessary and unappealing.

A simple optical method is proposed for estimating thickness of liquid sheets on sub-millimetre scales. It is shown that an approximate linear relation can be obtained between gap thickness and several of the usual RBG and HSV variables (red and green pixel values in the RBG case and saturation in the HSV case). The method is viable as long as the...

A classic magic trick involves filling a bucket with liquid and then turning it upside down so that the liquid stays in the bucket (apparently due to surface tension). This does not work under normal circumstances because of the Rayleigh-Taylor instability. Although the atmospheric pressure pushing at the free surface from below is theoretically en...

Popular article about the rule-based game Baba is You.

The classical Penrose inequality relates the mass of an asymptotically flat spacetime to the total area of its black holes. We present a PDE-based proof of the Penrose inequality for certain maximal perturbations of Schwarzschild spacetime. The proof is based around linearising solutions to a version of the Jang equation which is modified to deal w...

The Crookes radiometer (also known as a light mill) is a fascinating sunlight-powered device, in which a set of vanes is placed inside a glass bulb within which a partial vacuum has been pulled. The vanes then rotate when sunlight shines on the bulb. The reason for the turning of the vanes was subject to intense debate and many students still have...

We use a tensile testing machine to create stress-strain plots and determine Young’s modulus for some ductile and plastic materials. Supplementary videos are also provided.

Popular article in which we count the number of elements in the symmetry group of the tetrahedron and mention connections with chemistry via the methane molecule (a molecule with perfect tetrahedral symmetry).

We outline the basic theory behind white light interferometry and the workings of a typical light interferometer microscope. We study WLI images obtained for rough and smooth chrome steel spheres to illustrate the principle that curved rough surfaces can be imaged with such a device as long as the surface roughness is kept within certain limits.

A key problem which must be introduced in engineering and materials science education is the concept of flexural rigidity of a beam. Teaching this concept also requires introduction of Young’s modulus and (in some cases) the Euler-Bernoulli equation. The intuition behind this equation is typically limited and first principles derivations involve ad...

Fluid dynamics is a classical subject with its roots in antiquity. It studies all types of fluid motion, where a fluid can be a liquid or a gas. It has evolved throughout the 20th century and formed connections with other branches of science (for example, chemistry and reaction kinetics via fluid dynamics of combustion, and superconductivity via th...

We show that solutions exist to a simplified version of the system of equations obtained by coupling Bray’s conformal flow of metrics and the generalised Jang equation. This would establish the Penrose inequality for a class of conformally flat perturbations of Schwarzschild spacetime, provided that one can additionally prove that existence of a su...

In their most uncomplicated form mathematical models are essentially just mathematical descriptions of real-world systems. Stringing together variables and parameters into an equation we can attempt to describe complex behaviours that change with respect to time. Today mathematical models are used for everything, from predicting exam grades, to the...

Smocked spaces are a special class of metric spaces which were introduced as a generalisation of pulled thread spaces. We investigate convergence of these spaces, showing that the smocked space obtained from the Hausdorff limit of a sequence of smocking sets is equivalent to the Gromov-Hausdorff limit of the corresponding smocked spaces and also ob...

The physical problem of a body of water in a tank that drains through a hole in the base is a classical problem which has been studied since at least the time of Torricelli. To fixate this in a student’s mind, one could ask them to visualize a bathtub that is being drained through the plughole or a bottle being drained through a tap. This problem i...

A uniform magnetic field is one which has the same strength and direction at all points, where the study of uniform magnetic fields coupled to constant electric currents is known as magnetostatics [1]. In practice, uniform magnetic fields can be difficult to create. One of the most common ways of producing a magnetic field which is uniform over a l...

The notion of a boundary condition is typically considered to be somewhat advanced and not suitable to be introduced in high school level physics. In this article, we give a simple visual demonstration of the difference between Dirichlet and Neumann boundary conditions for a string which oscillates according to the one-dimensional wave equation. Th...

The Young modulus is a key concept in elasticity theory and mechanics of solids. Measurements of Young’s modulus in a classroom setting have been studied many times in the literature. These studies typically focus on simple or inexpensive ways of measuring the Young modulus, or they describe alternative indirect ways to measure the modulus which do...

We study a version of the two-degree-of-freedom double pendulum in which the two point masses are replaced by identical rigid bodies of irregular shape and nonconservative forces are permitted. We derive the equations of motion by considering the forces in the system in the framework of screw theory. This distinguishes the work from similar studies...

Wavelet analysis allows us to analyse classes of function not susceptible to Fourier analysis by using a series representation of square-integrable functions instead of sinusoidal functions. One of the most recent developments in this area is the wavelet transform. Wavelet analysis is becoming especially important from the practical viewpoint of da...

It is well-known that eigenvalues in quantum mechanics cannot be assigned to physical properties independently of the measurement context. We consider dependence of physical properties in the Mermin-Peres magic square on changes to the context, arguing that values for properties can switch when the context is changed in a way which is implied by th...

We outline the basic theory behind white light interferometry and the workings of a typical light interferometer microscope. We study WLI images obtained for rough and smooth chrome steel spheres to illustrate the principle that curved rough surfaces can be imaged with such a device as long as the surface roughness is kept within certain limits.

Water waves on the surface of the ocean are a complicated and interesting physical phenomenon. The classic example of a surface water wave would be a single-component sinusoidal wave, but realistic water waves do not take such a simplistic form. The next step up in complexity which one might consider consider is that of a spectrum-based wave. Besid...

Granular flows appear frequently in the natural world and in civil engineering applications. These flows can exhibit features which are surprising and counter-intuitive and are often used to test the limits of the classical continuum approximation for modelling of fluid flows. An important sub-class of the granular flows are the gravity-driven gran...

The Crookes radiometer was invented by William Crookes in the nineteenth century (he originally observed the effect upon which the device is based during unrelated experimental work to determine the atomic mass of thallium) [1]. In the device (shown in Figure 1), one mounts a set of vanes inside a sealed glass sphere such that the vanes are able to...

The hourglass is a time-keeping device which dates back to the Middle Ages. It is similar to a water clock but uses a powdered granular material such as sand as the substance which flows rather than a liquid [1]. Although the sand flows from top to bottom in an hourglass, this is clearly a different type of flow to the fluid flows which students ar...

We outline the physics involved in the use of scanning electron microscopy (SEM) and modernise the usual textbook treatment by explaining how one can now obtain high-quality images with non-conductive specimens. As a concrete example of an application in biology, we provide several magnification series of a flea obtained using SEM.

We study the age-old question: 'How long does it take to empty a bathtub?' More specifically, we use some physical assumptions and high-school calculus to determine how much longer it takes for the second half of the bath to empty in comparison to the first half, and find that the answer is perhaps not what one would naively expect. No knowledge of...

A key concept in current fluid dynamics and its applications to biology and technology is a phenomenon known as wetting. Wetting is familiar from everyday life and is simply the ability of a liquid to stay in contact with a solid surface. The wettability depends on the properties of the liquid and the solid and can be characterized by the static eq...

Bird wings are frequently modelled in the aerodynamics literature under the assumption that they are static aerofoils. Air flows over real bird wings can violate this assumption, both because of the topology of the wing and the fact that real wings typically undergo flapping and twisting motions during flight. There are many computational and numer...

Formation of a splash crown and a cavity following the impact of a sphere on a body of liquidis a classical problem. In the related problem of a droplet splashing on a flat surface, it has beenestablished that the properties of the surrounding gas can influence the splashing threshold. Atlower impact speeds, this is due mainly to the influence of g...

A topic which causes some confusion in the literature on rolling motion is the rolling friction experienced by a sphere as it rolls down a groove in an inclined slope. This set-up is often considered in the calculus of variations literature in relation to the brachistochrone and tautochrone problems [1 – 4]. In this article, we propose to simplify...

A common way to help babies to exercise in a safe way is to use a toy called a baby bouncer. The baby typically sit in a harness which is attached to a frame. The cord and spring which attach the harness to the frame can be considered as a model spring, hence the baby can move around and try to maintain balance as the spring oscillates. This is the...

We investigate the extent to which spectral triples from the theory of C *-algebras appear in theoretical physics. (Talk given at the Warwick Mathematics Institute Postgraduate Seminar 06/10/2021).

Traffic calming uses specific measures to reduce and control vehicle speeds. One of the most common of these is the 'yellow bar marking', also known as an 'optical strip'. Optical strips are raised lines painted across the road to alert drivers travelling at high speed of approaching hazards as a perceptual cue and through vibrations (hence the alt...

We present a number of exact solutions to the linearised Grad equations for non-equilibrium rarefied gas flows and heat flows. The solutions include the flow and pressure fields associated to a point force placed in a rarefied gas flow close to a no-slip boundary and the temperature field for a point heat source placed in a heat flow close to a tem...

We show that solutions exist to a simplified version of the system of equations obtained by coupling Bray’s conformal flow of metrics and the generalised Jang equation. This would establish the Penrose inequality for a class of conformally flat perturbations of Schwarzschild spacetime, provided that one can additionally prove that existence of a su...

Popular article in which we count the number of elements in the symmetry group of the tetrahedron and mention connections with chemistry via the methane molecule (a molecule with perfect tetrahedral symmetry).

The method of fundamental solutions (MFS) is a numerical method which enables one to model a flow using fundamental solutions (Green functions) for the equations of motion. We proceed to carry out some simple numerical computations which confirm the effectiveness of the method (at least for very simple physical scenarios).

Archimedes' Stomachion is a puzzle which has received the attention of several leading mathematicians. We describe the Stomachion from a mathematical viewpoint.

We summarise Edward Witten's 1989 paper “Quantum Field Theory and the Jones Polynomial” and compare Witten's approach to topological quantum field theories via Feynman integrals and Wilson lines with the mathematical interpretation via modular tensor categories. (Expanded version of a talk at Warwick Mathematics Postgraduate Seminar, June 2021).

This is a record of some notes on Feynman diagrams and the Feynman rules which were written as additional material to supplement a reading module on QFT at the University of Warwick taught by Minhyong Kim (the attendees were a mix of mathematicians and physicists). As this is a first course on QFT, the exposition follows the user-friendly version o...

A mathematical model is an idealisation of a problem which can be used to obtain results, observations or insights about that problem. Note that the problem does not necessarily need to involve a physical system and we will give two important examples from the biological sciences: the SIR model for the spread of infectious diseases and the ‘resonan...

The method of fundamental solutions (MFS) is a numerical method which enables one to model a flow using fundamental solutions (Green functions) for the equations of motion. We proceed to carry out some simple numerical computations which confirm the effectiveness of the method (at least for very simple physical scenarios).

We go through examples of some computations with gamma matrices. Although these can be tedious, it is important to have facility with these when doing QFT.

In pure mathematics, a group is an abstraction of a set of symmetries of a mathematical object. Despite the simplicity of the formal definition, the study and classification of groups is a vast subject and the various sub-branches of group theory have been the subject of intensive study for most of the twentieth century and beyond (geometric group...

We go through some of the the tree-level QED calculation of the total Compton scattering cross section in detail: this should be of use for students taking a first course on QFT when used in conjunction with the references.

In this article, we construct a very simple double pendulum (the concept of a pendulum should be familiar to all beginning students of classical mechanics). Since a double pendulum has two degrees of freedom, we suggest that this pendulum can be used in the classroom environment to illustrate the concept of normal modes and we use video software to...

As suggested by the name, homotopy theory first originated in the setting of algebraic topology but has now split off into a separate technical discipline which has been applied to algebraic geometry, homological algebra and category theory. Although there are many accounts of category theory in the mathematical literature, this book attempts to br...

We solve the unrestrained brachistochrone problem for sliding objects in the presence of sliding friction only. This is a well-posed extremal problem which cannot be solved using the methods of Euler and Lagrange due to the nature of the constraints.

The scanning electron microscope (SEM) was a key development in the history of biology: this device used the new discoveries of subatomic physics to create images with much higher resolutions at larger magnifications than those available to conventional microscopes. Such devices are now also commonly used in materials science and electronics. In th...

We study the acid-catalysed reaction of iodine with propanone, a classic demonstration chemistry experiment at high school level. Although the reaction is complicated, we show that studying the speed of the reaction reduces down to a simple differential equation which can be studied with calculus techniques, forming a prototype for more complicated...

Categories are abstract structures which nonetheless have extremely wide-ranging applications in many areas of mathematics. Roughly speaking, if one thinks of an abelian category M as a collection of objects linked together by arrows, the localisation of the category whose objects are complexes of the objects in M is the derived category (this turn...

Solutions are presented to all of Nomura's mazes.

We discuss the significance of superpositions of unitary operators in the formalism of quantum mechanics. We show that with this viewpoint, it can be demonstrated that one can observe a measurement with zero Ozawa uncertainty in a physically realisable feedback set-up which uses polarised photons coupled to spin. We derive a set of conditions under...

It is well-known that Newton’s work on mechanics depended in a crucial way on the previous observations of Galileo. The key insight of Galileo was that one can analyse the motion of bodies using experiments and mathematical equations. One experimental observation which roughly emerges from this work in modern terms is that two objects of different...

We consider the decay of the Higgs boson to W+W− at a proposed Large Hadron Electron Collider and determine the likelihood of detecting a signal for the Higgs mass from its decay product W-jets by imposing cuts to select candidate jet pairs and optimizing the value of the angular separation ∆R. It was found that at the LHeC experiment (CM energy √s...

It is well-known that Archimedes calculated the volume of the geometric solid known as the ungula to be one-sixth of the volume of an enclosing cube. Since then, several mathematicians have confirmed this result using calculus and trigonometry. We perform a version of the computation in cylindrical coordinates which is somewhat simpler than some of...

We give an outline of thermonuclear fusion and explain some of the major engineering and applied mathematical problems which would need to be overcome in order to achieve controlled fusion on Earth as a method of power generation.

The physical problem of a body of water in a tank which drains through a hole in the base is a classical problem which has been studied since at least the time of Torricelli [7]. To fixate this in a student's mind, one could ask them to visualise a bathtub which is being drained through the plughole or a bottle being drained through a tap. This pro...