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We present and analyse a novel manifestation of the revival phenomenon for linear spatially periodic evolution equations, in the concrete case of three nonlocal equations that arise in water wave theory and are defined by convolution kernels. Revival in these cases is manifested in the form of dispersively quantised cusped solutions at rational times. We give an analytic description of this phenomenon, and present illustrative numerical simulations.

We investigate, both analytically and numerically, dispersive fractalisation and quantisation of solutions to periodic linear and nonlinear Fermi–Pasta–Ulam–Tsingou systems. When subject to periodic boundary conditions and discontinuous initial conditions, e.g., a step function, both the linearised and nonlinear continuum models for FPUT exhibit fractal solution profiles at irrational times (as determined by the coefficients and the length of the interval) and quantised profiles (piecewise constant or perturbations thereof) at rational times. We observe a similar effect in the linearised FPUT chain at timestwhere these models have validity, namelyt= O(h−2), wherehis proportional to the intermass spacing or, equivalently, the reciprocal of the number of masses. For nonlinear periodic FPUT systems, our numerical results suggest a somewhat similar behaviour in the presence of small nonlinearities, which disappears as the nonlinear force increases in magnitude. However, these phenomena are manifested on very long time intervals, posing a severe challenge for numerical integration as the number of masses increases. Even with the high-order splitting methods used here, our numerical investigations are limited to nonlinear FPUT chains with a smaller number of masses than would be needed to resolve this question unambiguously.

We establish Liouville correspondences for the integrable two-component Camassa–Holm hierarchy, the two-component Novikov $$($$ Geng–Xue $$)$$ hierarchy, and the two-component dual dispersive water wave hierarchy using the related Liouville transformations. This extends previous results for scalar Camassa–Holm and KdV hierarchies and Novikov and Sawada–Kotera hierarchies to the multicomponent case.

С помощью преобразований Лиувилля установлены соответствия для интегрируемой двухкомпонентной иерархии Камассы-Холма, двухкомпонентной иерархии Новикова (иерархии Гэна-Сюэ) и двухкомпонентной иерархии уравнений волн на воде с двойной дисперсией. Это расширяет на многокомпонентный случай более ранние результаты, посвященные скалярным иерархиям Камассы-Холма и Кортевега-де Фриза, а также иерархиям Новикова и Савады-Котеры.

We use moving frames to construct and classify the joint invariants and joint differential invariants of binary and ternary forms. In particular, we prove that the differential invariant algebra of ternary forms is generated by a single third order differential invariant. To connect our results with earlier analysis of Kogan, we develop a general method for relating differential invariants associated with different choices of cross-section.

In this study we challenge the paradigm of using the Boyle van ’t Hoff (BvH) relation to relate cell size as a linear function of inverse extracellular osmotic pressure for short time periods (~5 to 30 mins). We present alternative models that account for mechanical resistance (turgor model) and ion-osmolyte leakage (leak model), which is not accounted for by the BvH relation. To test the BvH relation and the alternative models, we conducted a meta-analysis of published BvH datasets, as well as new experiments using a HepG2 cell line. Our meta-analysis showed that the BvH relation may be assumed of the hypertonic region but cannot be assumed a priori over the hyper- and hypotonic region. Both alternative models perform better than the BvH relation but are nearly indistinguishable when plotted. The return to isotonic conditions plot indicated neither alternative model accurate predicts return volumes for HepG2 cells. However, a combined turgor-leak model accurately predicts both the BvH plot and the return to isotonic conditions plot. Moreover, this turgor-leak model provides a facile method to estimate the membrane-cortex Young’s modulus and the cell membrane permeability to intracellular ions/osmolytes during periods of osmotic challenge, and predicts a novel passive method of volume regulation without the need for ion pumps.

Image feature points are detected as pixels which locally maximise a detector function, two commonly used examples of which are the (Euclidean) image gradient and the Harris–Stephens corner detector. A major limitation of these feature detectors is that they are only Euclidean-invariant. In this work, we demonstrate the application of a 2D equi-affine-invariant image feature point detector based on differential invariants as derived through the equivariant method of moving frames. The fundamental equi-affine differential invariants for 3D image volumes are also computed.