arXiv research

The paper examines the geometry of a curve's centre symmetry set.

Affine $λ$-equidistants of convex polygons with parallel opposite sides have applications to isoperimetric inequalities.

In this paper the generic bifurcations of the Minkowski symmetry set for 1-parameter families of plane curves are classified and the necessary and sufficient geometric criteria for each type are given. The Minkowski symmetry set is an analogue of the standard Euclidean symmetry set, and is defined to be the locus of ce…

This paper is centred on solving differential equations by symmetry groups for first order ODEs and is in response to Starrett (2007). It also explores the possibility of averting the assumptions by Olver (2000) that, in practice finding the solutions of the linearized symmetry condition is usually a much more difficul…

Study of symmetries in 2D Yang-Mills theory, including orbifolds and higher forms.

Constructs an asymptotic metric for moduli space of centred hyperbolic monopoles.

Develops methods to learn centre groupings from summary statistics in multi-centre studies.

We discuss $SU(2)$ Bogomolny monopoles of arbitrary charge $k$ invariant under various symmetry groups. The analysis is largely in terms of the spectral curves, the rational maps, and the Nahm equations associated with monopoles. We consider monopoles invariant under inversion in a plane, monopoles with cyclic symmetry…

We discuss the spectral curves and rational maps associated with $SU(2)$ Bogomolny monopoles of arbitrary charge $k$. We describe the effect on the rational maps of inverting monopoles in the plane with respect to which the rational maps are defined, and discuss the monopoles invariant under such inversion. We define t…

Expected centre of mass for random embeddings is constant.

New method constructs axial vector fields and defines quasi-local spin-angular momentum.

Monotonicity formulae play a crucial role for many geometric PDEs, especially for their regularity theories. For minimal submanifolds in a Euclidean ball, the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball, then its area is at least that of the equatorial disk. R…

Study on Hardy-Littlewood maximal operators on manifolds with bounded geometry.

We study the existence of cocompact lattices in Lie groups with bi-invariant metric of signature $(2,n-2)$. We assume in addition that the Lie groups under consideration are simply-connected, indecomposable and solvable. Then their centre is one- or two-dimensional. In both cases, a parametrisation of the set of such L…

We study pathwise invariances of centred random fields that can be controlled through the covariance. A result involving composition operators is obtained in second-order settings, and we show that various path properties including additivity boil down to invariances of the covariance kernel. These results are extended…

The study proves strong cosmic censorship violation for spherically symmetric dust clouds.

In this paper, we study strictly convex affine hypersurfaces centroaffinely congruent to their centre map, in the case when the shape operator has two distinct eigenvalues: one of multiplicity 1, and one nonzero of multiplicity n-1. We show how to construct them from (n-1)-dimensional affine hyperspheres.

We investigate a certain class of solvable metric Lie algebras. For this purpose a theory of twofold extensions associated to an orthogonal representation of an abelian Lie algebra is developed. Among other things, we obtain a classification scheme for indecomposable metric Lie algebras with maximal isotropic centre an…

New proof shows origin-centred balls are unique solutions to curvature problems.

The paper studies convexity of products of squared Euclidean distances.

Under mean radius of curvature flow, a closed convex surface in Euclidean space is known to expand exponentially to infinity. In the 3-dimensional case we prove that the oriented normals to the flowing surface converge to the oriented normals of a round sphere whose centre is determined by the initial surface. To prove…

Paper shows Cox model optimisation leaks patient data in distributed learning.

Conflict sets are loci of intersecting wavefronts emanating from $l$ different surfaces. We show that generically conflict sets are Legendrian: locally they admit the structure of wavefronts. Simple stable singularities for this problem in $\R^n$ occur when $0 \leq n-l \leq 4$. Other related sets, such as kite curves a…

The classifications of holonomy groups in Lorentzian and in Euclidean signature are quite different. A group of interest in Lorentzian signature in n dimensions is the maximal proper subgroup of the Lorentz group, SIM(n-2). Ricci-flat metrics with SIM(2) holonomy were constructed by Kerr and Goldberg, and a single four…

If $K$ is a compact Lie group and $g\geq 2$ an integer, the space $K^{2g}$ is endowed with the structure of a Hamiltonian space with a Lie group valued moment map $Φ$. Let $β$ be in the centre of $K$. The reduction $Φ^{-1}(β)/K$ is homeomorphic to a moduli space of flat connections. When $K$ is simply connected, a dire…

This paper introduces a novel clustering algorithm for heteroscedastic Gaussian data without needing to know the number of clusters.

New formulas limit minimal submanifolds' area in curved spaces.

Lecture notes from the Third International School on Geometry and Physics at the Centre de Recerca Matematica in Barcelona, March 26--30, 2012.

The Kastor-Traschen metric is a time-dependent solution of the Einstein-Maxwell equations with positive cosmological constant $Λ$ which can be used to describe an arbitrary number of charged dynamical black holes. In this paper, we consider the null geodesic structure of this solution, in particular, focusing on the pr…

We discuss a 4[k/2]-dimensional complete hyperkähler submanifold of the (4k-4)-dimensional moduli space of strongly centred SU(2)-monopoles of charge k.

We quantify conditions that ensure that a signed measure on a Riemannian manifold has a well defined centre of mass. We then use this result to quantify the extent of a neighbourhood on which the Riemannian barycentric coordinates of a set of $n+1$ points on an $n$-manifold provide a true coordinate chart, i.e., the ba…

Study proves only origin-centered spheres solve certain curvature problems.

New weight initialisation for ICNNs accelerates learning and improves generalization.

Convexity properties are preserved under radial transformations in hyperbolic and spherical geometries.

The investigation of 3D euclidean symmetry sets (SS) and medial axis is an important area, due in particular to their various important applications. The pre-symmetry set of a surface M in 3-space (resp. smooth closed curve in 2D) is the set of pairs of points which contribute to the symmetry set, that is, the closure …

Study on geodesics in a specific sub-Riemannian structure with two types of behavior.

Study finds Scherk type surfaces as extremals for zero-curvature minimal graphs.

These lecture notes give an introduction to the Kahler-Ricci flow. They are based on lectures given by the authors at the conference "Analytic Aspects of Complex Algebraic Geometry", held at the Centre International de Rencontres Mathematiques in Luminy, in February 2011.

Generalising a seminal result of Epstein and Penner for cusped hyperbolic manifolds, Cooper and Long showed that each decorated strictly convex projective cusped manifold has a canonical cell decomposition. Penner used the former result to describe a natural cell decomposition of decorated Teichmüller space of puncture…

The geometry of causal diamonds or Alexandrov open sets whose initial and final events $p$ and $q$ respectively have a proper-time separation $τ$ small compared with the curvature scale is a universal. The corrections from flat space are given as a power series in $τ$ whose coefficients involve the curvature at the cen…

The Murphy operators in the Hecke algebra H_n of type A are explicit commuting elements whose sum generates the centre. They can be represented by simple tangles in the Homfly skein theory version of H_n. In this paper I present a single tangle which represents their sum, and which is obviously central. As a consequenc…

This survey is based on a series of five lectures, given May 3--7, 2010, at the Centre de Recerca Matematica, Barcelona. The goal of the lectures was to present aspects of the theory of foliation dynamical systems which have particular importance for the classification of foliations of compact manifolds. The lectures e…

We show that the fundamental group of the space of ordered affine-equivalent configurations of at least five points in the real plane is isomorphic to the pure braid group modulo its centre. In the case of four points this fundamental group is free with eleven generators.

New algorithm achieves optimal clustering for sparse centers with high dimensions.

Study shows instability of certain MOTSs with continuous symmetry.

New tests detect high-order interactions without permutations.

In this study, the concept of dual Lorentzian homotetic exponential motions in is discussed and their velocities, accelerations obtained. Also, some geometric results between velocity and acceleration vectors of a point in a spatial motion are obtained. Finally, the theorems related to acceleration and acceleration cen…

Develops non-parametric tests for group symmetry in data.