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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.

We show how to compute the circular area invariant of planar curves, and the spherical volume invariant of surfaces, in terms of line and surface integrals, respectively. We use the Divergence Theorem to express the area and volume integrals as line and surface integrals, respectively, against particular kernels; our results also extend to higher dimensional hypersurfaces. The resulting surface integrals are computable analytically on a triangulated mesh. This gives a simple computational algorithm for computing the spherical volume invariant for triangulated surfaces that does not involve discretizing the ambient space. We discuss potential applications to feature detection on broken bone fragments of interest in anthropology.

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.

Machine learning (ML), being now widely accessible to the research community at large, has fostered a proliferation of new and striking applications of these emergent mathematical techniques across a wide range of disciplines. In this paper, we will focus on a particular case study: the field of paleoanthropology, which seeks to understand the evolution of the human species based on biological and cultural evidence. As we will show, the easy availability of ML algorithms and lack of expertise on their proper use among the anthropological research community has led to foundational misapplications that have appeared throughout the literature. The resulting unreliable results not only undermine efforts to legitimately incorporate ML into anthropological research, but produce potentially faulty understandings about our human evolutionary and behavioral past. The aim of this paper is to provide a brief introduction to some of the ways in which ML has been applied within paleoanthropology; we also include a survey of some basic ML algorithms for those who are not fully conversant with the field, which remains under active development. We discuss a series of missteps, errors, and violations of correct protocols of ML methods that appear disconcertingly often within the accumulating body of anthropological literature. These mistakes include use of outdated algorithms and practices; inappropriate train/test splits, sample composition, and textual explanations; as well as an absence of transparency due to the lack of data/code sharing, and the subsequent limitations imposed on independent replication. We assert that expanding samples, sharing data and code, re-evaluating approaches to peer review, and, most importantly, developing interdisciplinary teams that include experts in ML are all necessary for progress in future research incorporating ML within anthropology.

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.

In this paper, we study the differential invariants and the invariant heat flow in centro-affine geometry, proving that the latter is equivalent to the inviscid Burgers' equation. Furthermore, we apply the centro-affine invariants to develop an invariant algorithm to match features of objects appearing in images. We show that the resulting algorithm compares favorably with the widely applied scale-invariant feature transform (SIFT), speeded up robust features (SURF), and affine-SIFT (ASIFT) methods.

Given $n$ distinct points $\mathbf{x}_1, \ldots, \mathbf{x}_n$ in $\mathbb{R}^d$, let $K$ denote their convex hull, which we assume to be $d$-dimensional, and $B = \partial K $ its $(d-1)$-dimensional boundary. We construct an explicit one-parameter family of continuous maps $\mathbf{f}_{\varepsilon} \colon \mathbb{S}^{d-1} \to K$ which, for $\varepsilon > 0$, are defined on the $(d-1)$-dimensional sphere and have the property that the images $\mathbf{f}_{\varepsilon}(\mathbb{S}^{d-1})$ are codimension $1$ submanifolds contained in the interior of $K$. Moreover, as the parameter $\varepsilon$ goes to $0^+$, the images $\mathbf{f}_{\varepsilon}(\mathbb{S}^{d-1})$ converge, as sets, to the boundary $B$ of the convex hull. We prove this theorem using techniques from convex geometry of (spherical) polytopes and set-valued homology. We further establish an interesting relationship with the Gauss map of the polytope $B$, appropriately defined. Several computer plots illustrating our results will be presented., 26 pages

We prove that an effective, analytic action of a connected Lie group G on an analytic manifold M becomes free on a comeager subset of an open subset of M when prolonged to a frame bundle of sufficiently high order. We further prove that the action of G becomes free on a comeager subset of an open subset of a submanifold jet bundle over M of sufficiently high order, thereby establishing a general result that underlies Lie's theory of symmetry groups of differential equations and the equivariant method of moving frames.

In general, the ubiquitous χ^{(2)} nonlinear directional coupler, where nonlinearity and evanescent coupling are intertwined, is nonintegrable. We rigorously demonstrate that matching excitation to the even or odd fundamental supermodes yields dynamical analytical solutions for any phase matching in a symmetric coupler. We analyze second harmonic generation and optical parametric amplification regimes and study the influence of fundamental field parity and power on the operation of the device. These fundamental solutions are useful to develop applications in classical and quantum fields such as all-optical modulation of light and quantum-states engineering.

In general, the ubiquitous χ^{(2)} nonlinear directional coupler, where nonlinearity and evanescent coupling are intertwined, is nonintegrable. We rigorously demonstrate that matching excitation to the even or odd fundamental supermodes yields dynamical analytical solutions for any phase matching in a symmetric coupler. We analyze second harmonic generation and optical parametric amplification regimes and study the influence of fundamental field parity and power on the operation of the device. These fundamental solutions are useful to develop applications in classical and quantum fields such as all-optical modulation of light and quantum-states engineering.

The structure group and the involutive differential system that characterize the pseudo-group of contact transformations on a jet space are determined.

This paper introduces a new, fully recursive algorithm for computing moving frames and differential invariants of Lie pseudo-group actions. The recursive method avoids unwieldy symbolic expressions that complicate the treatment of large scale applications of the equivariant moving frame method. The development leads to novel results on partial moving frames, structure equations, and new differential operators underlying the moving frame construction. In particular, our methods produce a streamlined computational algorithm for determining moving frames and differential invariants of finite-dimensional Lie group actions.

Because Nature seems to strive for efficiency, many systems arising in physical applications are founded on a minimization principle. In a mechanical system, the stable equilibrium configurations minimize the potential energy. In an electrical circuit, the current adjusts itself to minimize the power. In optics and relativity, light rays follow the paths of minimal distance — the geodesics on the curved space-time manifold. Solutions to most of the boundary value problems arising in applications to continuum mechanics are also characterized by a minimization principle, which is then employed to design finite element numerical approximations to their solutions, [61, 81]. Optimization — finding minima or maxima — is ubiquitous throughout mathematical modeling, physics, engineering, economics, and data science, including the calculus of variations, differential geometry, control theory, design and manufacturing, linear programming, machine learning, and beyond.

The geometry of Euclidean space is founded on the familiar properties of length and angle. The abstract concept of a norm on a vector space formalizes the geometrical notion of the length of a vector. In Euclidean geometry, the angle between two vectors is specified by their dot product, which is itself formalized by the abstract concept of an inner product. Inner products and norms lie at the heart of linear (and nonlinear) analysis, in both finite-dimensional vector spaces and infinite-dimensional function spaces. A vector space equipped with an inner product and its associated norm is known as an inner product space. It is impossible to overemphasize their importance for theoretical developments, practical applications, and the design of numerical solution algorithms.

We describe computational algorithms for constructing the explicit power series expansions for normal forms of submanifolds under transformation groups. The procedure used to derive the coefficients relies on the recurrence formulae for differential invariants provided by the method of equivariant moving frames.

Linear algebra is the core of modern applied mathematics. Its humble origins are to be found in the need to solve “elementary” systems of linear algebraic equations. But its ultimate scope is vast, impinging on all of mathematics, both pure and applied, as well as numerical analysis, statistics, data science, physics, engineering, mathematical biology, financial mathematics, and every other discipline in which mathematical methods are required. A thorough grounding in the methods and theory of linear algebra is an essential prerequisite for understanding and harnessing the power of mathematics throughout its multifaceted applications

Vector spaces and their ancillary structures provide the common language of linear algebra, and, as such, are an essential prerequisite for understanding contemporary applied (and pure) mathematics. The key concepts of vector space, subspace, linear independence, span, and basis will appear, not only in linear systems of algebraic equations and the geometry of n-dimensional Euclidean space, but also in the analysis of linear differential equations, linear boundary value problems, Fourier analysis, signal processing, numerical methods, and many, many other fields. Therefore, in order to master modern linear algebra and its applications, the first order of business is to acquire a firm understanding of fundamental vector space constructions.

So far, our physical applications of linear algebra have concentrated on statics: unchanging equilibrium configurations of mass–spring chains, circuits, and structures, all modeled by linear systems of algebraic equations. It is now time to set the universe in motion. In general, a (continuous) dynamical system refers to the (differential) equations governing the time-varying motion of a physical system, be it mechanical, electrical, chemical, fluid, thermodynamical, biological, financial, . . . . Our immediate goal is to solve the simplest class of continuous dynamical models, which are first order autonomous linear systems of ordinary differential equations.

Motivated by the dynamics of neuronal responses, we analyze the dynamics of the Fitzhugh-Nagumo slow-fast system with delayed self-coupling. This system provides a canonical example of a canard explosion for sufficiently small delays. Beyond this regime, delays significantly enrich the dynamics, leading to mixed-mode oscillations, bursting and chaos. These behaviors emerge from a delay-induced subcritical Bogdanov-Takens instability arising at the fold points of the S-shaped critical manifold. Underlying the transition from canard-induced to delay-induced dynamics is an abrupt switch in the nature of the Hopf bifurcation.

We examine the relationships between the differential invariants of objects and of their images under a surjective map. We analyze both the case when the underlying transformation group is projectable and hence induces an action on the image, and the case when only a proper subgroup of the entire group acts projectably. In the former case, we establish a constructible isomorphism between the algebra of differential invariants of the images and the algebra of fiber-wise constant (gauge) differential invariants of the objects. In the latter case, we describe residual effects of the full transformation group on the image invariants. Our motivation comes from the problem of reconstruction of an object from multiple-view images, with central and parallel projections of curves from three-dimensional space to the two-dimensional plane serving as our main examples., This paper includes corrections and additions to the published version

In this paper, we study explicit correspondences between the integrable Novikov and Sawada-Kotera hierarchies, and between the Degasperis-Procesi and Kaup-Kupershmidt hierarchies. We show how a pair of Liouville transformations between the isospectral problems of the Novikov and Sawada-Kotera equations, and the isospectral problems of the Degasperis-Procesi and Kaup-Kupershmidt equations relate the corresponding hierarchies, in both positive and negative directions, as well as their associated conservation laws. Combining these results with the Miura transformation relating the Sawada-Kotera and Kaup-Kupershmidt equations, we further construct an implicit relationship which associates the Novikov and Degasperis-Procesi equations.

The orbital stability of the peaked solitary-wave solutions for a generalization of the modified Camassa–Holm equation with both cubic and quadratic nonlinearities is investigated. The equation is a model of asymptotic shallow-water wave approximations to the incompressible Euler equations. It is also formally integrable in the sense of the existence of a Lax formulation and bi-Hamiltonian structure. It is demonstrated that, when the Camassa–Holm energy counteracts the effect of the modified Camassa–Holm energy, the peakon and periodic peakon solutions are orbitally stable under small perturbations in the energy space.

I trace the main steps of the first fifty-five years of my career as an applied mathematician, pausing from time to time to describe problems that arose in asymptotics and numerical analysis and had far-reaching effects on this career. Lecture delivered at Asymptotics and Applied Analysis, Conference in Honor of Frank W. J. Olver's 75th Birthday, January 10–14, 2000, San Diego State University, San Diego, California. Editors' Note: Frank W. J. Olver died on April 23, 2013. The following text was typed by his son, Peter J. Olver, from handwritten notes found among his papers. At times the writing is unpolished, including incomplete sentences, but the editors have decided to leave it essentially the way it was written. However, for clarity, some abbreviations have been written out in full. A couple of handwritten words could not be deciphered, and a guess for what was intended is enclosed in brackets: […]. Endnotes have been made into footnotes within the body of the article. References were mostly not included in the handwritten text, but rather listed in order at the end. Citations to references have been included at the appropriate point in the text.

We derive conditions on the initial data, including cases where the initial momentum density is not of one sign, that produce blow-up of the induced solution to the modified integrable Camassa–Holm equation with cubic nonlinearity. The blow-up conditions are formulated in terms of the initial momentum density and the average initial energy.

The use of extracorporeal membrane oxygenation (ECMO) for reversible pulmonary failure in critically ill patients has increased over the last few decades. Nosocomial infections are a major complication of ECMO and fungi have been found to be a common cause. Herein, we describe a case of invasive pulmonary aspergillosis following ECMO, which was successfully treated with combination antifungal therapy and interferon-gamma.

When posed on a periodic domain in one space variable, linear dispersive evolution equations with integral polynomial dispersion relations exhibit strikingly different behaviors depending upon whether the time is rational or irrational relative to the length of the interval, thus producing the Talbot effect of dispersive quantization and fractalization. The goal here is to show that these remarkable phenomena extend to nonlinear dispersive evolution equations. We will present numerical simulations, based on operator splitting methods, of the nonlinear Schrodinger and Korteweg--deVries equations with step function initial data and periodic boundary conditions. For the integrable nonlinear Schrodinger equation, our observations have been rigorously confirmed in a recent paper of Erdoǧan and Tzirakis, [10].