arXiv subjects

Peter J. Olver

Publications and source records attributed to Peter J. Olver.

At least 19 recordsLinked to original sources

arXiv2026

Continuous Revival of the Periodic Schrödinger Equation with Piecewise $C^2$ Potentials

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.

math.AP↗

arXiv2026

Moving Frames for Lie Pseudo-groups: A Recursive Implementation

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.

math-ph↗

arXiv2025

Convergence of Normal Form Power Series for Infinite-Dimensional Lie Pseudo-Group Actions

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.

math-ph↗

arXiv2025

New Revival Phenomena for Bidirectional Dispersive Hyperbolic Equations

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.

math.AP↗

arXiv2025

The Batch Artifact Scanning Protocol: A new method using computed tomography (CT) to rapidly create three-dimensional models of objects from large collections en masse

Within anthropology, the use of three-dimensional (3D) imaging has become increasingly common and widespread since it broadens the available avenues for addressing a wide range of key anthropological issues. The ease with which 3D models can be generated and shared has major impact on 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 to rapidly create 3D models using a medical CT scanner. While this method can be used on a variety of material types, we have, for specificity, applied our protocol to a large collection of experimentally broken ungulate limb bones. By employing the Batch Artifact Scanning Protocol, we were able to efficiently create 3D models of 2,474 bone fragments at a rate of less than 4 minutes per specimen.

cs.CV↗

arXiv2024

En masse scanning and automated surfacing of small objects using Micro-CT

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 {\em 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.

cs.CV↗

arXiv2022

Use and Misuse of Machine Learning in Anthropology

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.

cs.LG↗

arXiv2022

Using machine learning on new feature sets extracted from 3D models of broken animal bones to classify fragments according to break agent

Distinguishing agents of bone modification at paleoanthropological sites is at the root of much of the research directed at understanding early hominin exploitation of large animal resources and the effects those subsistence behaviors had on early hominin evolution. However, current methods, particularly in the area of fracture pattern analysis as a signal of marrow exploitation, have failed to overcome equifinality. Furthermore, researchers debate the replicability and validity of current and emerging methods for analyzing bone modifications. Here we present a new approach to fracture pattern analysis aimed at distinguishing bone fragments resulting from hominin bone breakage and those produced by carnivores. This new method uses 3D models of fragmentary bone to extract a much richer dataset that is more transparent and replicable than feature sets previously used in fracture pattern analysis. Supervised machine learning algorithms are properly used to classify bone fragments according to agent of breakage with average mean accuracy of 77% across tests.

cs.CV↗

arXiv2022

Normal forms, moving frames, and differential invariants for nondegenerate hypersurfaces in C^2

We use the method of equivariant moving frames to revisit the problem of normal forms and equivalence of nondegenerate real hypersurfaces M \subset C^2 under the pseudo-group action of holomorphic transformations. The moving frame recurrence formulae allow us to systematically and algorithmically recover the results of Chern and Moser for hypersurfaces that are either non-umbilic at a point p \in 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 moving frames to construct new convergent normal forms for nondegenerate real hypersurfaces at singularly umbilic points, namely those umbilic points where the hypersurface is not identically umbilic around them.

math.DG↗

arXiv2020

Dispersive Fractalization in Linear and Nonlinear Fermi-Pasta-Ulam-Tsingou Lattices

We investigate, both analytically and numerically, dispersive fractalization and quantization 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 linearized 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 quantized profiles (piecewise constant or perturbations thereof) at rational times. We observe a similar effect in the linearized FPUT chain at times $t$ where these models have validity, namely $t = \mathrm{O}(h^{-2})$, where $h$ is 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 behavior 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.

nlin.PS↗

arXiv2020

The Virtual Goniometer: A new method for measuring angles on 3D models of fragmentary bone and lithics

The contact goniometer is a commonly used tool in lithic and zooarchaeological analysis, despite suffering from a number of shortcomings due to the physical interaction between the measuring implement, the object being measured, and the individual taking the measurements. However, lacking a simple and efficient alternative, researchers in a variety of fields continue to use the contact goniometer to this day. In this paper, we present a new goniometric method that we call the virtual goniometer, which takes angle measurements virtually on a 3D model of an object. The virtual goniometer allows for rapid data collection, and for the measurement of many angles that cannot be physically accessed by a manual goniometer. We compare the intra-observer variability of the manual and virtual goniometers, and find that the virtual goniometer is far more consistent and reliable. Furthermore, the virtual goniometer allows for precise replication of angle measurements, even among multiple users, which is important for reproducibility of goniometric-based research. The virtual goniometer is available as a plug-in in the open source mesh processing packages Meshlab and Blender, making it easily accessible to researchers exploring the potential for goniometry to improve archaeological methods and address anthropological questions.

math.NA↗

arXiv2020

New Revival Phenomena for Linear Integro-Differential Equations

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.

math.AP↗

arXiv2020

Feature Matching and Heat Flow in Centro-Affine Geometry

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.

math.DG↗

arXiv2020

Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

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.

math.MG↗

arXiv2019

Computation of Circular Area and Spherical Volume Invariants via Boundary Integrals

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.

math.NA↗

arXiv2019

The $n$ body matrix and its determinant

The primary purpose of this note is to prove two recent conjectures concerning the $n$ body matrix that arose in recent papers of Escobar-Ruiz, Miller, and Turbiner on the classical and quantum $n$ body problem in $d$-dimensional space. First, whenever the positions of the masses are in a nonsingular configuration, meaning that they do not lie on an affine subspace of dimension $\leq n-2$, the $n$ body matrix is positive definite and, hence, defines a Riemannian metric on the space coordinatized by their interpoint distances. Second, its determinant can be factored into the product of the order $n$ Cayley--Menger determinant and a mass-dependent factor that is also of one sign on all nonsingular mass configurations. The factorization of the $n$ body determinant is shown to be a special case of an intriguing general result proving the factorization of determinants of a certain form.

math-ph↗

arXiv2019

Symmetry-based analytical solutions to the χ^{(2)} nonlinear directional coupler

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 fields 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.

physics.optics↗

arXiv2018

Points of constancy of the periodic linearized Korteweg--deVries equation

We investigate the points of constancy in the piecewise constant solution profiles of the periodic linearized Korteweg--deVries equation with step function initial data at rational times. The solution formulas are given by certain Weyl sums, and we employ number theoretic techniques, including Kummer sums, in our analysis. These results constitute an initial attempt to understand the phenomenon of "fractalization" observed at irrational times.

math.NT↗