Strict inequality in the box-counting dimension product formulas

We supplement the well known upper and lower box-counting product inequalities to give a new product formula for subsets of metric spaces. We develop a procedure for constructing sets so that the upper and lower box-counting dimensions of these sets and their product can take arbitrary values satisfying the above product formula. In particular we illustrate how badly behaved both the lower and upper box-counting dimensions can be on taking products

Dimension prints and the avoidance of sets for flow solutions of non-autonomous ordinary differential equations

We provide a criterion for a generalised flow solution of a non-autonomous ordinary differential equation to avoid a subset of the phase space. This improves on that established by Aizenman for the autonomous case, where avoidance is guaranteed if the underlying vector field is sufficiently regular and the subset has sufficiently small box-counting dimension. We define the r-codimension print of a subset $S\subset \R^{n}\times [0,T]$, which is a subset of $(0,\infty]^{2}$ that encodes the dimension of S in a way that distinguishes spatial and temporal detail. We prove that the subset S is avoided by a generalised flow solution with underlying vector field in $L^{p}(0, T; L^{q}(R^{n}))$ if the Holder conjugates (q^{*}; p^{*}) are in the r-codimension print of S

On solutions of the transport equation in the presence of singularities

We consider the transport equation on $[0,T]\times \R^n$ in the situation where the vector field is $BV$ off a set $S\subset [0,T]\times \R^n$. We demonstrate that solutions exist and are unique provided that the set of singularities has a sufficiently small anisotropic fractal dimension and the normal component of the vector field is sufficiently integrable near the singularities. This result improves upon recent results of Ambrosio who requires the vector field to be of bounded variation everywhere. In addition, we demonstrate that under these conditions almost every trajectory of the associated regular Lagrangian flow does not intersect the set $S$ of singularities. Finally, we consider the particular case of an initial set of singularities that evolve in time so the singularities consists of curves in the phase space, which is typical in applications such as vortex dynamics. We demonstrate that solutions of the transport equation exist and are unique provided that the box-counting dimension of the singularities is bounded in terms of the H\"older exponent of the curves

On the regularity of Lagrangian trajectories corresponding to suitable weak solutions of the Navier-Stokes equations

The putative singular set S in space-time of a suitable weak solution u of the 3D Navier–Stokes equations has box-counting dimension no greater than 5/3. This allows one to prove that almost all trajectories avoid S. Moreover, for each point x that does not belong to S, one can find a neighbourhood U of x such that the function u is continuous on U and space derivatives of u are bounded on every compact subset of U. It follows that almost all Lagrangian trajectories corresponding to u are C^{1} functions of time (Robinson & Sadowski, Nonlinearity 2009). We recall the main idea of the proof, give examples that clarify in what sense the uniqueness of trajectories is considered, and make some comments on how this result might be improved

Generalised Cantor sets and the dimension of products

In this paper we consider the relationship between the Assouad and box-counting dimension and how both behave under the operation of taking products. We introduce the notion of ‘equi-homogeneity’ of a set, which requires a uniformity in the cardinality of local covers at all length-scales and at all points, and we show that a large class of homogeneous Moran sets have this property. We prove that the Assouad and box-counting dimensions coincide for sets that have equal upper and lower box-counting dimensions provided that the set ‘attains’ these dimensions (analogous to ‘s-sets’ when considering the Hausdorff dimension), and the set is equi-homogeneous. Using this fact we show that for any α ∈ (0, 1) and any β, γ ∈ (0, 1) such that β + γ ≥ 1 we can construct two generalised Cantor sets C and D such that dimBC = αβ, dimBD = α γ, and dimAC = dimAD = dimA (C × D) = dimB (C × D) = α

Engaging with maths online - teaching mathematics collaboratively and inclusively through a pandemic and beyond

This case study details several concrete approaches to integrating the use of student-loaned iPads in the teaching of mathematics in Higher Education. Although there is a scarcity of rigorous studies into the efficacy of tablet devices for improved educational outcomes, previous case studies have argued that tablet devices, if used, should be integrated into the whole learning experience. The mathematics teaching team at Middlesex University have developed an inclusive digital pedagogy over the last five years that enabled us to effectively respond to the remote teaching imposed by the Covid-19 pandemic by loaning iPads to all students on specialist mathematics programmes. As we begin the return to campus we continue to integrate these devices into our teaching to address the observed “digital divide” in Generation-Z students which is characterised not by access to smart devices but by the digital skills to use them as effective learning tools. This is particularly relevant at Middlesex University which is disproportionately affected by digital poverty amongst its student population. We discuss the use of virtual whiteboard apps, the necessity of handwritten mathematics, the rich integration of multimedia content, persistent collaborative “problem solving spaces”, and how a common hardware platform allows for varied and equitable inclusive assessment. We also report the results of students’ surveys of iPad use during the remote-only 2020-21 academic year

Design and experimental evaluation of a test rig for ultrasonic monitoring of sccelerated corrosion

The design of a test rig for ultrasonic monitoring of corroding steel samples and results of its experimental evaluation are presented. In order to accelerate and quantify corrosion, the monitored sample acts as an anode in a typical constant current electroplating setup. An Arduino-powered microcontroller is used to log the relevant voltage and current values on an SD card, and a high accuracy ultrasonic waveform acquisition instrument is employed to record the ultrasonic waveforms reflected form the corroding surface. Experimental assessment of the designed rig confirmed that it met the design objectives and could be used for experimental studies of accelerated corrosion in steel samples

Survival extrapolation using the poly-Weibull model.

Recent studies of (cost-) effectiveness in cardiothoracic transplantation have required estimation of mean survival over the lifetime of the recipients. In order to calculate mean survival, the complete survivor curve is required but is often not fully observed, so that survival extrapolation is necessary. After transplantation, the hazard function is bathtub-shaped, reflecting latent competing risks which operate additively in overlapping time periods. The poly-Weibull distribution is a flexible parametric model that may be used to extrapolate survival and has a natural competing risks interpretation. In addition, treatment effects and subgroups can be modelled separately for each component of risk. We describe the model and develop inference procedures using freely available software. The methods are applied to two problems from cardiothoracic transplantation

Revising the role of the history of mathematics in post-pandemic world

In this short philosophical and discursive paper, the main objective is to reassess a new emergent role of the history of mathematics in order to bring about greater diversity and engagement in the mathematical sciences. The discussion is based around the project undertaken at a North London university and their partner pre-university college, which piloted the larger national project in the UK in the local context. The success of the project, it is further suggested, would greatly benefit from a framework in which the history of mathematics as a humanistic discipline is closely related to viewing mathematics as a virtuous practice. We also include a short summary about the lives and careers of two Serbian mathematicians, Judita Cofman, and Milica Ilić-Dajović, to showcase how learning about the ways in which marginalisation takes place can help students position themselves and contextualise their priorities as they enter the professional mathematics landscape

Impacts of climate change on dissolved oxygen concentration relevant to the coastal and marine environment around the UK

The decline in dissolved oxygen and onset of oxygen deficiency and hypoxia are naturally occurring phenomenon in aquatic environments, typically occurring on seasonal timescales. Over decadal timescales, there has been a measurable decline in dissolved oxygen concentrations in the global ocean due to warming caused by anthropogenic activity. Approximately 15% of the global decline in oxygen has been attributed to reduced solubility in response to ocean warming, with the remaining 85% due to intensified stratification. The relative contribution of these factors in coastal and shelf-sea waters is currently unknown. In UK waters, sustained observations in the North Sea reveal the recent onset of oxygen deficiency in late summer, partially due to ocean warming. Models designed to represent coastal and shelf sea processes suggest there are large parts of the Celtic Sea, English Channel and Irish Sea that are prone to oxygen deficiency, but data is too sparse in time and space to support these findings. In addition, the ability of models to accurately represent oxygen dynamics is still under debate due to correct representation of physical and biological processes within models. Physical processes play a key role in the development of oxygen deficient regions and thus understanding how oxygen concentrations will respond to climate change requires a coupled physical and biogeochemical approach