An archive of Dent's papers, that relate to her life and work in the 1920s in the physics department at the University of Bristol, is held in the Special Collections of the University of Bristol Arts and Social Sciences Library, in Tyndall Avenue, Bristol. Included in that archive is a series of s, written in the 1930s by members of the Clifton Hill House Old Students' Association, that include news and photographs of Dent, her family, and friends. A
blue plaque was unveiled in Dent's honour on 9 August 2025 at StAntony's Heritage Centre, Trafford Park, Manchester.
Atomic force microscopy In 1928, LennardJones and Dent published two papers, "" and "", that for the first time, outlined a calculation of the
potential of the electric field in a vacuum, produced by a thin sodium chloride crystal surface. They gave an expression for the electric potential produced by a system of
point charges in vacuum (although not a real cubic sodium chloride ionic lattice). The expression for the potential in vacuum, \varphi_{0}\left(r\right), at the point r = {x, y, z},
near the cubic lattice of point ions with different signs, the charge e_{k}, and the period
a (a crystalline solid is distinguished by the fact that the atoms making up the crystal are arranged in a periodic fashion), can be represented in the form: {{NumBlk|:| \varphi_{0}\left(r\right)= \frac{2\pi}{a^{2}} \sum_{l,m}\sum_{s} \frac{\left(-1\right)^{s} \exp \left[ -\left(\frac{2\pi}{a}\right) \sqrt{l^{2}+m^{2}} \left(z+z_{s}\right) \right]} {\left|k_{l,m}\right|} \times \sum_{k}e_{k}\cos \left[k_{l,m}\left(r_{\parallel}-r_{k}\right)\right] :r_{\parallel}=\left\{x,y\right\} is the lateral vector that fixes the observation point coordinates in the sample plane. :k_{l,m} is the
reciprocal lattice vector. :
s is the number of planes to be calculated inside the crystal;
s set to zero would calculate the surface plane. The expression sums the set of potential static charges for the surface and lower planes of the crystal lattice. LennardJones and Dent showed that this expression forms a rapidly convergent
Fourier series. Harold Eugene Buckley, a crystallographic researcher at the University of Manchester until his death in 1959, had suggested that their results should be treated with caution. For example, the contraction a crystal plane would suffer under the conditions prescribed would not be the same as that of a similar plane with a solid mass of crystal behind it. Another difficulty arises because calculation of crystal surface field force fields are so great that simplifying assumptions have to be made to render them capable of a solution. Michael Jaycock and Geoffrey Parfitt, then respectively senior lecturer in
surface and colloid chemistry at
Loughborough University of Technology and professor of
chemical engineering at
Carnegie Mellon University, concurred with Buckley, noting that "an ideal crystal, in which the ionic positions at the surface were identical to those achieved in the bulk crystal... is obviously extremely improbable." However, they acknowledged that the LennardJones and Dent model was singularly elegant, and like most researchers working before the advent of modern computers, they were limited in what could be attempted computationally. Nonetheless, LennardJones and Dent demonstrated that the force exerted on a single ion, by a surface with evenly distributed positive and negative ions, decreases very rapidly with increasing distance. Later work by Jason Cleveland, Manfred Radmacher, and
Paul Hansma, has shown that this result has direct application to
atomic force microscopy by predicting that noncontact imaging is possible only at small tipsample separations.
Reduced major axis regression attempts to model the relationship between two variables by fitting a linear equation (straight line) to observed data The theoretical underpinnings of standard
least squares regression analysis are based on the assumption that the independent variable (often labelled as
x) is measured without error as a design variable. The dependent variable (labeled
y) is modeled as having uncertainty or error. Both independent and dependent measurements may have multiple sources of error. Therefore, the underlying least squares regression assumptions can be violated. Reduced major axis (RMA) regression is specifically formulated to handle errors in both the
x and
y variables. If the estimate of the ratio of the error variance of
y to the error variance of
x is denoted by '
, then the reduced major axis method assumes that ' can be approximated by the ratio of the total variances of
x and
y. RMA minimizes both vertical and horizontal distances of the data points from the predicted line (by summing areas) rather than the least squares sum of squared vertical (
yaxis) distances. In Dent's 1935 paper on linear regression, entitled "", she admitted that when the variances in the
x and
y variables are unknown, "we cannot hope to find the true positions of the observed points, but only their most probable positions." However, by treating the probability of the errors in terms of
Gaussian error functions, she contended that this expression may be regarded as "a function of the unknown quantities", or the
likelihood function of the data distribution. Furthermore, she argued that
maximising this function to obtain the
maximum likelihood estimation, subject to the condition that the points are
collinear, will give the parameters for the line of best fit. She then deduced formulae for the errors in estimating the
centroid and the line inclination when the data consists of a single (unrepeated) observation.
Maurice Kendall and Alan Stuart showed that the maximum likelihood estimator of a likelihood function, depending on a parameter \theta, satisfies the following
quadratic equation: {{NumBlk|:| \theta^{2}x^{T}y+\left[\lambda x^{T}x-y^{T}y\right]\theta-\lambda x^{T}y=0 :where x and y are the \mathbf{X} and \mathbf{Y}
vectors in a
covariance matrix giving the
covariance between each pair of
x and
y variables. The
superscript T indicates the
transpose of the matrix. Using the
quadratic formula to solve for the positive
root (or zero) of (): {{NumBlk|:| \theta_{ML}\equiv\frac{y^{T}y-\lambda x^{T}x+\sqrt{(y^{T}y-\lambda x^{T}x)^{2} + 4\lambda (x^{T}y)^{2}}}{2x^{T}y} Inspection of () shows that as '''' tends to plus infinity, the positive root tends to: {{NumBlk|:| \theta_{x}, equal to \frac{x^{T}y}{x^{T}x} Correspondingly, as '''' tends to zero, the root tends to: {{NumBlk|:| \theta_{y}, equal to \frac{y^{T}y}{x^{T}y} Dent had solved the maximum likelihood estimator in the case where the covariance matrix is not known. Dent's maximum likelihood estimator is the
geometric mean of \theta_{x} and \theta_{y}, equivalent to: {{NumBlk|:| \theta_{ML}\equiv\sqrt{\frac{x^{T}y}{x^{T}x}\times{\frac{y^{T}y}{x^{T}y}}}\equiv\sqrt{\frac{{y^{T}y}}{{x^{T}x}}}, where x^{T}y is positive.
Dennis Lindley repeated Dent's analysis and stated that Dent's geometric mean estimator is not a consistent estimator for the likelihood function, and that the gradient of the estimate will have a bias, and this remains true even if the number of observations tends to infinity. Subsequently,
Theodore Anderson pointed out that the likelihood function has no maximum in this case, and therefore, there is no maximum likelihood estimator. Kenneth Alva Norton, a former consulting engineer with the then
National Bureau of Standards, responded to Lindley, stating Lindley's own methods and assumptions lead to a biased prediction. Furthermore,
Albert Madansky, late H. G. B. Alexander professor of business administration at
University of Chicago Booth School of Business, noted that Lindley took the wrong root for the quadratic in () for the case where x^{T}y is negative.
Richard J. Smith has stated that Dent was the first to develop a RMA regression method for line fitting that built on the work of Robert Adcock in "" (1878) and Charles Kummell in "" (1879). It is now believed that she was the first to propose what is often called the geometric mean functional relationship estimator of slope, and that her essential arguments can be generalised to any number of variables. Moreover, although her solution has its theoretical limitations, it is of practical importance, as it likely represents the best
a priori estimate if nothing is known about the true error distribution in the model. It is generally much less reasonable to assume that all the error, or residual scatter, is attributable to one of the variables.
Electrical design using digital computers In the 1950s, British electrical engineers would rarely use a digital computer, and if they did, it would be to solve some complicated equation outside the scope of
analogue computers. To a certain extent, engineers were deterred by the difficulty and the time taken to program a particular problem. Furthermore, the varied and often unique problems that arise in electrical design practice, together with the degree of uncertainty of the numerical data of many problems, accentuated this tendency. On 10 April 1956, Dent and Brian Birtwistle presented their paper, "", to the Convention on Digital Computer Techniques at the
Institution of Electrical Engineers. The paper was intended to show, by describing three relatively simple applications, that the digital computer could be a useful aid to the electrical design engineer. The three example problems were: The Ferranti Mark 1 computer at the University of Manchester was used for the calculations in the three problems. Dent was allowed to use the university's
library of
subroutines, from which the following were taken and incorporated into the programs: •
Exponential. •
Sine and cosine. •
Square root. • Solution of
simultaneous equations. •
Inversion of matrices. • Integration of differential equations by
Runge–Kutta methods. The first problem of calculating the impulse voltage distribution on transformer windings took about five hours of machine time. Conversely, a hand calculation, using a method described by Thomas John Lewis in "" (1954), took around three months. The use of a computer in the second problem allowed for a more accurate solution as it was possible to include
nonlinear magnetic characteristics in the calculation. In the last problem, the torque and speed curves for the synchronous motors were calculated in around fifteen minutes. Their paper was one of the first to recognise that highspeed digital computers could provide considerable assistance to the electrical design engineer by carrying out automatically the optimum design of products. Significant research had been devoted to determining a transformer's internal
transient voltage distribution. Early attempts were hampered by computational limitations encountered when solving large numbers of coupled differential equations with analogue computers. It was not until Dent, with Hartill and Miles, in "" (1958), recognised the limitations of the analogue models and developed a digital computer model, and associated program, where nonuniformity in the
transformer windings could be introduced and any input voltage applied. == Publications ==