arXiv research

Examining orbits ending in binary collisions for three equal masses under an inverse cube force.

Consider the equal mass planar $4$-body problem with a potential corresponding to an inverse \textit{cube} force. The Jacobi-Maupertuis principle reparametrizes the dynamics as geodesics of a certain metric. We examine the curvature of this geodesic flow in the reduced space on the collinear and parallelogram invariant…

We construct new examples of normal (metric) currents using inverse systems of cube complexes. For any $N\ge 2$ we provide examples of $N$-dimensional normal currents whose associated vector fields are simple, and whose supports are purely $2$-unrectifiable and have Nagata dimension $N$. We show that in $l^\infty$ norm…

We discuss two generalizations of the inverse problem of the calculus of variations, one in which a given mechanical system can be brought into the form of Lagrangian equations with non-conservative forces of a generalized Rayleigh dissipation type, the other leading to Lagrangian equations with so-called gyroscopic fo…

Improved weather forecasting using deep CNN on cubed-sphere grid.

Smooth flows for physical systems with smooth energies and forces.

Paper uses black-box inference to estimate non-linear latent force models.

Darboux inverses explain Kepler orbits on curved surfaces.

Simulation-based inference tackles complex inverse problems in science.

In this paper, we prove the short-time existence of hyperbolic inverse (mean) curvature flow (with or without the specified forcing term) under the assumption that the initial compact smooth hypersurface of $\mathbb{R}^{n+1}$ ($n\geqslant2$) is mean convex and star-shaped. Several interesting examples and some hyperbol…

Paper proves Gromov's cube inequality in all dimensions.

A toric cube is a subset of the standard cube defined by binomial inequalities. These basic semialgebraic sets are precisely the images of standard cubes under monomial maps. We study toric cubes from the perspective of topological combinatorics. Explicit decompositions as CW-complexes are constructed. Their open cells…

Simplified and extended a method for rearranging infinite configurations of cubes.

Neural network enhances seismic imaging in salt-prone areas.

CAMEL embeds data into a manifold using curvature-augmented forces.

Generalizes Leighton's theorem to cube complexes.

In this paper we introduce a representation of knots and links called a cube diagram. We show that a property of a cube diagram is a link invariant if and only if the property is invariant under two types of cube diagram operations. A knot homology is constructed from cube diagrams and shown to be equivalent to knot Fl…

Counterexample disproves completeness of model space forcing regularity in infinite-dimensional Lie groups.

Given a triangulation of a closed topological cube, we show that (under some technical condition) there is an essentially unique tiling of a rectangular parallelepiped by cubes, indexed by the vertices of the triangulation. Moreover, i - the combinatorics is preserved, and ii- the boundary is preserved: vertices corres…

New geometric spine for Artin groups defined by cube complexes.

Study mapping class groups on CAT(0) cube complexes.

For a knot K the cube number is a knot invariant defined to be the smallest n for which there is a cube diagram of size n for K. We will show that the cube number detects chirality in all cases computed thus far, and distinguishes certain legendrian knots.

We give a characterization of alternating link exteriors in terms of cubed complexes. To this end, we introduce the concept of a "signed BW cubed-complex", and give a characterization for a signed BW cubed-complex to have the underlying space which is homeomorphic to an alternating link exterior.

We define an integer-valued invariant of special cube complexes called the genus, and prove that having genus one characterizes special cube complexes with abelian fundamental group. Using the genus, we obtain a new proof that the fundamental group of a special cube complex is either free abelian or surjects onto a non…

For a knot $K$ the cube number is a knot invariant defined to be the smallest $n$ for which there is a cube diagram of size $n$ for $K$. There is also a Legendrian version of this invariant called the \emph{Legendrian cube number}. We will show that the Legendrian cube number distinguishes the Legendrian left hand toru…

Groups act on cube complexes with varying dimensions.

Finite stature proven for cube complexes with cyclonormal edges.

Cube edges curves minimize systole length.

We demonstrate the homogeneity of the Hilbert Cube. In particular, we construct explicit self-homeomorphisms of the Hilbert cube so that given any two points, a homeomorphism moving one to the other may be realized.

Near isospectrality forces full isospectrality for compact quotients of symmetric spaces.

Beside simplices, $n$-cubes form an important class of simple polyhedra. Unlike hyperbolic Coxeter simplices, hyperbolic Coxeter $n$-cubes are not classified. We show that there is no hyperbolic Coxeter $n$-cube for $n\geq~6$, and provide a full classification for $n\leq 5$. Our methods, which are essentially of combin…

In this short note we highlight some of the differences between cube diagrams and grid diagrams. We also list examples of small cube diagrams for all knots up to 7 crossings and give some examples of links.

Lean 4 formalizes Stokes' theorem for smooth singular cubes.

Groups on CAT(0) cube complexes grow exponentially uniformly.

CUBE explains models by balanced experiments and contrasts.

We prove that smooth cube manifolds have normal smooth structures.

We show that groups satisfying Kazhdan's property (T) have no unbounded actions on finite dimensional CAT(0) cube complexes, and deduce that there is a locally CAT(-1) Riemannian manifold which is not homotopy equivalent to any finite dimensional, locally CAT(0) cube complex.

Solve arc diagrams on surfaces via branched covers.

Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.

New group acts on complex but not in lower dimensions.

We count orientable small covers over cubes. We also get estimates for $O_n/R_n$, where $O_n$ is the number of orientable small covers and $R_n$ is the number of all small covers over an $n$-cube up to the Davis-Januszkiewicz equivalence.

Lecture notes on CAT(0) cube complexes for beginners.

Extends folding techniques to study subgroups of CAT(0) cube complexes.

The study finds conditions for groups acting on CAT(0) cube complexes to have infinite girth.

We introduce a class of spaces, called real cubings, and study the stucture of groups acting nicely on these spaces. Just as cubings are a natural generalisation of simplicial trees, real cubings can be regarded as a natural generalisation of real trees. Our main result states that a finitely generated group $G$ acts n…

We combine ideas of Scott and Swarup on good position for almost invariant subsets of a group with ideas of Sageev on constructing cubings from such sets. We construct cubings which are more canonical than in Sageev's original construction. We also show that almost invariant sets can be chosen to be in very good positi…

The paper extends ternary algebra concepts using cube roots of unity.

Develops theory of relatively geometric actions on CAT(0) cube complexes.