Peter J. Olver
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View article: Using Moving Frames to Construct Equivariant Maps
Using Moving Frames to Construct Equivariant Maps Open
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.
View article: Convergence of Normal Form Power Series for Infinite-Dimensional Lie Pseudo-Group Actions
Convergence of Normal Form Power Series for Infinite-Dimensional Lie Pseudo-Group Actions Open
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 …
View article: Continuous Revival of the Periodic Schrödinger Equation with Piecewise $C^2$ Potential
Continuous Revival of the Periodic Schrödinger Equation with Piecewise $C^2$ Potential Open
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 pot…
View article: En masse scanning and automated surfacing of small objects using Micro-CT
En masse scanning and automated surfacing of small objects using Micro-CT Open
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 surround…
View article: New Revival Phenomena for Bidirectional Dispersive Hyperbolic Equations
New Revival Phenomena for Bidirectional Dispersive Hyperbolic Equations Open
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 …
View article: Projective invariants of images
Projective invariants of images Open
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 …
View article: Use and Misuse of Machine Learning in Anthropology
Use and Misuse of Machine Learning in Anthropology Open
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 pape…
View article: Using machine learning on new feature sets extracted from 3D models of broken animal bones to classify fragments according to break agent
Using machine learning on new feature sets extracted from 3D models of broken animal bones to classify fragments according to break agent Open
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 …
View article: 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
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 Open
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 c…
View article: Normal forms, moving frames, and differential invariants for nondegenerate hypersurfaces in C^2
Normal forms, moving frames, and differential invariants for nondegenerate hypersurfaces in C^2 Open
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 recurren…
View article: Non-Analytic Solutions of Nonlinear Wave Models
Non-Analytic Solutions of Nonlinear Wave Models Open
This paper surveys recent work of the coauthors on nonanalytic solutions to nonlinear wave models. We demonstrate the connection between nonlinear dispersion and the existence of a remarkable variety of nonclassical solutions, including pe…
View article: New revival phenomena for linear integro–differential equations
New revival phenomena for linear integro–differential equations Open
We present and analyze 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 kern…
View article: Dispersive fractalisation in linear and nonlinear Fermi–Pasta–Ulam–Tsingou lattices
Dispersive fractalisation in linear and nonlinear Fermi–Pasta–Ulam–Tsingou lattices Open
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 in…
View article: Continuous Maps from Spheres Converging to Boundaries of Convex Hulls
Continuous Maps from Spheres Converging to Boundaries of Convex Hulls Open
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, e…
View article: The Virtual Goniometer: A new method for measuring angles on 3D models of fragmentary bone and lithics
The Virtual Goniometer: A new method for measuring angles on 3D models of fragmentary bone and lithics Open
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 in…
View article: Feature Matching and Heat Flow in Centro-Affine Geometry
Feature Matching and Heat Flow in Centro-Affine Geometry Open
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 devel…
View article: Dispersive Fractalization in Linear and Nonlinear Fermi-Pasta-Ulam-Tsingou Lattices
Dispersive Fractalization in Linear and Nonlinear Fermi-Pasta-Ulam-Tsingou Lattices Open
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 in…
View article: Computation of Circular Area and Spherical Volume Invariants via Boundary Integrals
Computation of Circular Area and Spherical Volume Invariants via Boundary Integrals Open
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 integra…
View article: Revivals and fractalisation in the linear free space Schrödinger equation
Revivals and fractalisation in the linear free space Schrödinger equation Open
We consider the one-dimensional linear free space Schrödinger equation on a bounded interval subject to homogeneous linear boundary conditions. We prove that, in the case of pseudoperiodic boundary conditions, the solution of the initial-b…
View article: Dispersive Lamb systems
Dispersive Lamb systems Open
Under periodic boundary conditions, a one-dimensional dispersive medium driven by a Lamb oscillator exhibits a smooth response when the dispersion relation is asymptotically linear or superlinear at large wave numbers, but unusual fractal …
View article: The $n$ Body Matrix and Its Determinant
The $n$ Body Matrix and Its Determinant Open
The primary purpose of this note is to prove two recent conjectures\nconcerning the $n$ body matrix that arose in recent papers of Escobar-Ruiz,\nMiller, and Turbiner on the classical and quantum $n$ body problem in\n$d$-dimensional space.…
View article: Revivals and Fractalisation in the Linear Free Space Schr\\"odinger\n Equation
Revivals and Fractalisation in the Linear Free Space Schr\\"odinger\n Equation Open
We consider the one-dimensional linear free space Schr\\"odinger equation on a\nbounded interval subject to homogeneous linear boundary conditions. We prove\nthat, in the case of pseudoperiodic boundary conditions, the solution of the\nini…
View article: Points of constancy of the periodic linearized Korteweg–deVries equation
Points of constancy of the periodic linearized Korteweg–deVries equation Open
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 formulae are given by certain Weyl sums…
View article: Liouville correspondences between multi-component integrable hierarchies
Liouville correspondences between multi-component integrable hierarchies Open
In this paper, 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 by means of the relat…
View article: Affine Differential Invariants for Invariant Feature Point Detection
Affine Differential Invariants for Invariant Feature Point Detection Open
Image feature points are detected as pixels which locally maximize 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 det…
View article: On the commutator of ${\mathcal{C}^{\infty}}$ -symmetries and the reduction of Euler–Lagrange equations
On the commutator of ${\mathcal{C}^{\infty}}$ -symmetries and the reduction of Euler–Lagrange equations Open
Se presenta un nuevo procedimiento para reducir en cuatro unidades el orden de las ecuaciones de Euler-Lagrange asociadas a problemas variacionales de orden n que involucran integrales de una sola variable dependiente e independiente. En p…
View article: Liouville Correspondences between Integrable Hierarchies
Liouville Correspondences between Integrable Hierarchies Open
In this paper, we study explicit correspondences between the integrable\nNovikov and Sawada-Kotera hierarchies, and between the Degasperis-Procesi and\nKaup-Kupershmidt hierarchies. We show how a pair of Liouville transformations\nbetween …
View article: Bäcklund transformations for tri-Hamiltonian dual structures of multi-component integrable systems
Bäcklund transformations for tri-Hamiltonian dual structures of multi-component integrable systems Open
In this article, the Bäcklund transformation based-approach is explored to obtain Hamiltonian operators of multi-component integrable systems which are governed by compatible tri-Hamiltonian dual structures. The resulting Hamiltonian opera…