A new recursive implementation of the equivariant moving frame method for infinite-dimensional Lie pseudo-group actions is presented. It allows pseudo-group normalizations to be performed before the prolonged action is computed, thereby avoiding the introduction of unnecessarily complicated intermediate expressions. Several examples are provided to illustrate the algorithm.

In this paper, we investigate the revivals of the one-dimensional periodic Schrödinger equation with a piecewise C^2 potential function. As has been observed through numerical simulations of the equation with various initial data and potential functions, the solution, while remaining fractalized at irrational times, exhibits a form of revival at rational times. The goal is to prove that the solution at these rational times is given by a finite linear combination of translations and dilations of the initial datum, plus an additional continuous term, which we call "continuous revival". In pursuit of this result, we present a review of relevant properties of the periodic Schrödinger equation as an eigenvalue problem, including asymptotic results on both the eigenvalues and eigenfunctions.

In this paper, the dispersive revival and fractalization phenomena for bidirectional dispersive equations on a bounded interval subject to periodic boundary conditions and discontinuous initial profiles are investigated. Firstly, we study the periodic initial-boundary value problem of the linear beam equation with step function initial data, and analyze the manifestation of the revival phenomenon for the corresponding solution at rational times. Next, we extend the investigation to periodic initial-boundary value problems of more general bidirectional dispersive equations. We prove that, if the initial functions are of bounded variation, the dynamical evolution of such periodic problems depend essentially upon the large wave number asymptotics of the associated dispersion relations. Integral polynomial or asymptotically integral polynomial dispersion relations produce dispersive revival/fractalization rational/irrational dichotomies, whereas those with non-polynomial growth result in fractal profiles at all times. Finally, numerical experiments, in the concrete case of the nonlinear beam equation, are used to demonstrate how such effects persist into the nonlinear regime.

We prove the convergence of normal form power series for suitably nonsingular analytic submanifolds under a broad class of infinite-dimensional Lie pseudo-group actions. Our theorem is illustrated by a number of examples, and includes, as a particular case, Chern and Moser's celebrated convergence theorem for normal forms of real hypersurfaces. The construction of normal forms relies on the equivariant moving frame method, while the convergence proof is based on the realization that the normal form can be recovered as part of the solution to an initial value problem for an involutive system of differential equations, whose analyticity is guaranteed by the Cartan-Kähler Theorem.

The equivariant method of moving frames is applied to formulate a systematic method for explicitly determining general equivariant maps and thereby establishing a nonlinear generalization of a formula attributed to Malgrange.

Within anthropology, the use of three-dimensional (3D) imaging has become increasingly standard and widespread since it broadens the available avenues for addressing a wide range of key issues. The ease with which 3D models can be shared has had major impacts for research, cultural heritage, education, science communication, and public engagement, as well as contributing to the preservation of the physical specimens and archiving collections in widely accessible data bases. Current scanning protocols have the ability to create the required research quality 3D models; however, they tend to be time and labor intensive and not practical when working with large collections. Here we describe a streamlined, Batch Artifact Scanning Protocol we have developed to rapidly create 3D models using a medical CT scanner. Though this method can be used on a variety of material types, we use a large collection of experimentally broken ungulate limb bones. Using the Batch Artifact Scanning Protocol, we were able to efficiently create 3D models of 2,474 bone fragments at a rate of less than 3 minutes per specimen, as opposed to an average of 50 minutes per specimen using structured light scanning.

Distinguishing agents of bone modification at paleoanthropological sites is an important means of understanding early hominin evolution. Fracture pattern analysis is used to help determine site formation processes, including whether hominins were hunting or scavenging for animal food resources. Determination of how these behaviors manifested in ancient human sites has major implications for our biological and behavioral evolution, including social and cognitive abilities, dietary impacts of having access to in-bone nutrients like marrow, and cultural variation in butchering and food processing practices. Nevertheless, previous analyses remain inconclusive, often suffering from lack of replicability, misuse of mathematical methods, and/or failure to overcome equifinality. In this paper, we present a new approach aimed at distinguishing bone fragments resulting from hominin and carnivore breakage. Our analysis is founded on a large collection of scanned three-dimensional models of fragmentary bone broken by known agents, to which we apply state of the art machine learning algorithms. Our classification of fragments achieves an average mean accuracy of 77% across tests, thus demonstrating notable, but not overwhelming, success for distinguishing the agent of breakage. We note that, while previous research applying such algorithms has claimed higher success rates, fundamental errors in the application of machine learning protocols suggest that the reported accuracies are unjustified and unreliable. The systematic, fully documented, and proper application of machine learning algorithms leads to an inherent reproducibility of our study, and therefore our methods hold great potential for deciphering when and where hominins first began exploiting marrow and meat, and clarifying their importance and influence on human evolution.

Modern archaeological methods increasingly utilize 3D virtual representations of objects, computationally intensive analyses, high resolution scanning, large datasets, and machine learning. With higher resolution scans, challenges surrounding computational power, memory, and file storage quickly arise. Processing and analyzing high resolution scans often requires memory-intensive workflows, which are infeasible for most computers and increasingly necessitate the use of super-computers or innovative methods for processing on standard computers. Here we introduce a novel protocol for en-masse micro-CT scanning of small objects with a mostly-automated processing workflow that functions in memory-limited settings. We scanned 1,112 animal bone fragments using just 10 micro-CT scans, which were post-processed into individual PLY files. Notably, our methods can be applied to any object (with discernible density from the packaging material) making this method applicable to a variety of inquiries and fields including paleontology, geology, electrical engineering, and materials science. Further, our methods may immediately be adopted by scanning institutes to pool customer orders together and offer more affordable scanning. The work presented herein is part of a larger program facilitated by the international and multi-disciplinary research consortium known as Anthropological and Mathematical Analysis of Archaeological and Zooarchaeological Evidence (AMAAZE). AMAAZE unites experts in anthropology, mathematics, and computer science to develop new methods for mass-scale virtual archaeological research. Overall, our new scanning method and processing workflows lay the groundwork and set the standard for future mass-scale, high resolution scanning studies.

We use the method of equivariant moving frames to revisit the problem of normal forms and equivalence of nondegenerate real hypersurfaces M ⊂ℂ^2 under the pseudo-group action of holomorphic transformations. The moving frame recurrence formulae allows us to systematically and algorithmically recover the results of Chern and Moser for hypersurfaces that are either non-umbilic at a point p∈ M or umbilic in an open neighborhood of it. In the former case, the coefficients of the normal form expansion, when expressed as functions of the jet of the hypersurface at the point, provide a complete system of functionally independent differential invariants that can be used to solve the equivalence problem. We prove that under a suitable genericity condition, the entire algebra of differential invariants for such hypersurfaces can be generated, through the operators of invariant differentiation, by a single-real differential invariant of order 7. We then apply the method of moving frames to construct new convergent normal forms for the intermediate but overlooked case of nondegenerate real hypersurfaces at singularly umbilic points, namely those umbilic points where the hypersurface is not identically umbilic in a neighborhood thereof.

The method of equivariant moving frames is employed to construct and completely classify the differential invariants for the action of the projective group on functions defined on the two-dimensional projective plane. While there are four independent differential invariants of order $\leq 3$ , it is proved that the algebra of differential invariants is generated by just two of them through invariant differentiation. The projective differential invariants are, in particular, of importance in image processing applications.

The structure of algebras of differential invariants, particularly their generators, is investigated using the symbolic invariant calculus provided by the method of equivariant moving frames. We develop a computational algorithm that will, in many cases, determine whether a given set of differential invariants is generating. As an example, we establish a new result that the Gaussian curvature generates all the differential invariants for Euclidean surfaces in three-dimensional space.

A new, elementary algorithm for constructing complete, minimal sets of generating invariants for finite or, more generally, discrete group actions, both linear and nonlinear, is proposed. The resulting fundamental invariants are piecewise analytic and endowed with a rewrite rule that enables one to immediately express any other invariant (polynomial, rational, smooth, analytic, etc.) as a function thereof. The construction is inspired by the method of equivariant moving frames for Lie group actions.

Historically, the mathematical study of physical phenomena has proceeded in three phases. The introductory phase relies on linear models, and is a necessary prerequisite to the deeper understanding of the actual nonlinear regime. Since different wave numbers propagate at different speeds, localized initial data will disperse in time, albeit with conservation of energy. Weakly nonlinear models cannot describe the fully nonlinear, real world phenomena such as wave breaking, shocks, waves of maximal height, large amplitude disturbances, and so on. The classical, analytic traveling wave solutions are given by classical analytic orbits to the dynamical system. The admission of nonanalytic traveling wave solutions indicates that classical dynamical systems analysis must be significantly broadened to explain how a classical analytic orbit can give rise to a nonanalytic solution.

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 article, 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 (e.g., bones, genetics) and cultural (e.g., stone tools) 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 the 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 article 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 the use of outdated algorithms and practices; inappropriate testing/training 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 the progress in future research incorporating ML within anthropology and beyond.

We explicitly characterize boundary conditions that are compatible with low order variational principles. The freedom afforded by adding in a null Lagrangian without altering the Euler–Lagrange equation significantly expands the range of variationally admissible boundary conditions, although not all possibilities are permitted. Applications to several fundamental problems arising in elastostatics, including bars, beams, and plates, are presented.

The most fundamental attribute of motion in the macroscopic physical universe is that all objects move continuously. We demonstrate how this basic fact can be used to motivate the modern topological definition of continuity of functions. This relatively simple definition can then be used to establish a new, rigorous foundation for calculus that does not rely on limits, and thus forms a compelling route into the mathematics that is used to model and understand our world. In this note, I will revisit the continuity of motion from a new perspective. The aim is to understand both concepts at a more basic level — one that does not rely on the standard, but unintuitive and technically intricate epsilon-delta definition of limit that was introduced two centuries ago by Bolzano, Cauchy, and Weierstrass, [2], and has, ever since, formed the almost universally accepted foundation for rigorous calculus. Instead, in [14], calculus can be based on the topological definition of a continuous function as one December 1, 2020 † English translation from [5].