arXiv research

We discuss how the shape of a special Cosserat rod can be represented as a path in the special Euclidean algebra. By shape we mean all those geometric features that are invariant under isometries of the three-dimensional ambient space. The representation of the shape as a path in the special Euclidean algebra is intrin…

This paper develops a new method for constructing splines on Lie groups using Poisson equation solutions.

In this paper we will discuss some new developments in the design of numerical methods for optimal control problems of Lagrangian systems on Lie groups. We will construct these geometric integrators using discrete variational calculus on Lie groups, deriving a discrete version of the second-order Euler-Lagrange equatio…

Closed-form relations and approximations for SE(3) derivatives for robust numerical simulations.

The paper introduces new uniformity and homogeneity concepts for Cosserat media.

New equations for Cosserat media motions derived from bundle automorphisms.

Upper and lower bounds for hyperbolic rod complements in 3-torus volumes.

Complete classification of rod complements in 3-torus using topology.

Characterizes null Lagrangians in Cosserat elasticity.

We combine and extend the work of Alexander & Antman \cite{alexander.82} and Fuller \cite{fuller.71,fuller.78} to give a framework within which precise definitions can be given of topological and geometrical quantities characterising the contortion of open rods undergoing large deformations under end loading. We use th…

Rod flow models Adam's behavior at the edge of stability.

Study of rod packings in 3-torus using 3-manifold geometry.

Study toric gravitational instantons using rod structures and inequalities.

There are many industrial situations where rods are used to stir a fluid, or where rods repeatedly stretch a material such as bread dough or taffy. The goal in these applications is to stretch either material lines (in a fluid) or the material itself (for dough or taffy) as rapidly as possible. The growth rate of mater…

Geometrically reformulates Cosserat solid mechanics using differential geometry.

Study evaluates profitability of Islamic banks in Bangladesh using ROA, ROE, and ROD.

The energy minimization problem associated to uniform, isotropic, linearly elastic rods leads to a geometric variational problem for the rod centerline, whose solutions include closed, knotted curves. We give a complete description of the space of closed and quasiperiodic solutions. The quasiperiodic curves are paramet…

A new ODE model explains gradient descent dynamics near edge of stability.

A Lie groupoid, called \textit{second-order non-holonomic material Lie groupoid}, is associated in a natural way to any Cosserat media. This groupoid is used to give a new definition of homogeneity which does not depend on a reference crystal. The corresponding Lie algebroid, called \textit{second-order non-holonomic m…

We discuss several issues regarding material homogeneity and strain compatibility for materially uniform thin elastic shells from the viewpoint of a 3-dimensional theory, with small thickness, as well as a 2-dimensional Cosserat theory. A relationship between inhomogeneity and incompatibility measures under the two des…

Bicycle paths form geodesics in 3D subspaces, related to Kirchhoff rods.

New method stabilizes tensegrity structures suitable for engineering.

Kirchhoff energy is a classical functional on the space of arclength-parameterized framed curves whose critical points approximate configurations of springy elastic rods. We introduce a generalized functional on the space of framed curves of arbitrary parameterization, which model rods with axial stretch or cross-secti…

We derive a dimensionally-reduced limit theory for an $n$-dimensional nonlinear elastic body that is slender along $k$ dimensions. The starting point is to view an elastic body as an $n$-dimensional Riemannian manifold together with a not necessarily isometric $W^{1,2}$-immersion in $n$-dimensional Euclidean space. The…

Proves uniqueness and existence of toric gravitational instantons.

Study p-Willmore disks with boundary energies, finding equilibrium configurations.

In-plane drill rotations are impossible for smooth shells.

Weaved helices form mechanically stable 3D structures.

We suggest an alternative mathematical model for the massless neutrino. Consider an elastic continuum in 3-dimensional Euclidean space and assume that points of this continuum can experience no displacements, only rotations. This framework is a special case of the so-called Cosserat theory of elasticity. Rotations of p…

The correspondence of stationary, axisymmetric, asymptotically flat space-times and bundles over a reduced twistor space has been established in four dimensions. The main impediment for an application of this correspondence to examples in higher dimensions is the lack of a higher-dimensional equivalent of the Ernst pot…

In 1870, R. Clausius found the virial theorem which amounts to introduce the trace of the stress tensor when studying the foundations of thermodynamics, as a way to relate the absolute temperature of an ideal gas to the mean kinetic energy of its molecules. In 1901, H. Poincar{é} introduced a duality principle in analy…

Paper introduces untangling number to quantify 3-periodic tangle complexity.

The purpose of this short notice is to present an elementary summary of a few recent results obtained through the application of the formal theory of systems of partial differential equations and Lie pseudo groups to engineering (elasticity theory, electromagnetism, coupling phenomena) and mathematical (gauge theory, g…

The paper deals with the Weyl equation which is the massless Dirac equation. We study the Weyl equation in the stationary setting, i.e. when the spinor field oscillates harmonically in time. We suggest a new geometric interpretation of the stationary Weyl equation, one which does not require the use of spinors, Pauli m…

The paper deals with the Weyl equation which is the massless Dirac equation. We study the Weyl equation in the stationary setting, i.e. when the the spinor field oscillates harmonically in time. We suggest a new geometric interpretation of the stationary Weyl equation, one which does not require the use of spinors, Pau…

This review discusses solutions to Einstein's equations using twistor theory.

The paper generalizes the second Pappus-Guldin theorem for calculating volumes of bodies.

What is the longest rope on the unit sphere? Intuition tells us that the answer to this packing problem depends on the rope's thickness. For a countably infinite number of prescribed thickness values we construct and classify all solution curves. The simplest ones are similar to the seamlines of a tennis ball, others e…

The Backlund transformation for pseudospherical surfaces, which is equivalent to that of the sine-Gordon equation, can be restricted to give a transformation on space curves that preserves constant torsion. We study its effects on closed curves (in particular, elastic rods) that generate multiphase solutions for the vo…

A linkage is a finite graph with lengths assigned to each edge. A planar realization is a map to the plane which preserves edge lengths. It can be thought of as a mechanical device formed from stiff rods and rotating joints. We look at the configuration space of all planar realizations of a linkage (following work of K…

We study the maximal entropy per unit generator of push-point mapping classes on the punctured disk. Our work is motivated by fluid mixing by rods in a planar domain. If a single rod moves among N-fixed obstacles, the resulting fluid diffeomorphism is in the push-point mapping class associated with the loop in π_1(D^2 …

We derive variational formulas for the total Q-prime curvature under the deformation of strictly pseudoconvex domains in a complex manifold. We also show that the total Q-prime curvature agrees with the renormalized volume of such domains with respect to the complete Einstein-Kähler metric. In the appendix, by Rod Gove…

Existence proved for specific types of gravitational instantons.

We suggest an alternative mathematical model for the electron in which the dynamical variables are a coframe (field of orthonormal bases) and a density. The electron mass and external electromagnetic field are incorporated into our model by means of a Kaluza-Klein extension. Our Lagrangian density is proportional to ax…

A mechanical linkage is a mechanism made of rigid rods linked together by flexible joints, in which some vertices are fixed and others may move. The partial configuration space of a linkage is the set of all the possible positions of a subset of the vertices. We characterize the possible partial configuration spaces of…

Study Bernstein-Gelfand-Gelfand complexes on Lipschitz domains, computing cohomology and applying to elasticity models.

A mechanical linkage is a mechanism made of rigid rods linked together by flexible joints, in which some vertices are fixed and others may move. The partial configuration space of a linkage is the set of all the possible positions of a subset of the vertices. We characterize the possible partial configuration spaces of…

We prove a relation between the scaling $h^β$ of the elastic energies of shrinking non-Euclidean bodies $S_h$ of thickness $h\to 0$, and the curvature along their mid-surface $S$. This extends and generalizes similar results for plates [BLS16, LRR] to any dimension and co-dimension. In particular, it proves that the na…