arXiv research

Study the structure of equidistant decompositions in manifolds.

The paper uses tensor decompositions to improve neural network models for tree data.

Common positive stabilisation found for isotopic contact structures.

In this text we give a decomposition result on polynomial poly-vector fields generalizing a result on the decomposition of homogeneous Poisson structures. We discuss consequences of this decomposition result in particular for low dimensions and low degrees. We provide the tools to calculate simple cubic Poisson structu…

We study the structure of finite quandles in terms of subquandles. Every finite quandle $Q$ decomposes in a natural way as a union of disjoint $Q$-complemented subquandles; this decomposition coincides with the usual orbit decomposition of $Q$. Conversely, the structure of a finite quandle with a given orbit decomposit…

We prove the existence of a local analytic Levi decomposition for analytic Poisson structures and Lie algebroids.

Analyzes canonical reductive decomposition of extrinsic homogeneous submanifolds.

CDFD analyzes circularity and directionality in weighted directed networks.

We discuss structured Schatten norms for tensor decomposition that includes two recently proposed norms ("overlapped" and "latent") for convex-optimization-based tensor decomposition, and connect tensor decomposition with wider literature on structured sparsity. Based on the properties of the structured Schatten norms,…

We present an algebraic investigation of generalized and equiaffine curvature tensors in a given pseudo-Euclidean vector space and study different orthogonal, irreducible decompositions in analogy to the known decomposition of algebraic curvature tensors. We apply the decomposition results to characterize geometric pro…

We prove the existence of a local smooth Levi decomposition for smooth Poisson structures and Lie algebroids near a singular point. In the appendix of this paper, we show an abstract Nash-Moser normal form theorem, which generalizes our Levi decomposition result and which may be helpful in the study of other smooth nor…

In this note we observe that while all overtwisted contact structures on compact 3--manifolds are supported by planar open book decompositions, not all contact structures are. This has relevance to invariants of contact structures and also to the Weinstein conjecture via work of Abbas Cieliebak and Hofer.

Geometrically describes hyperbolic structures on link complements using quantum groups.

A Levi-Malcev type decomposition for $2$-step solvable Lie algebras with a complex structure

The paper develops axioms for uniquely decomposing functions with real arguments.

Lie bialgebra structures are reviewed and investigated in terms of the double Lie algebra, of Manin- and Gauß-decompositions. The standard R-matrix in a Manin decomposition then gives rise to several Poisson structures on the correponding double group, which is investigated in great detail.

We use convex decomposition theory to (1) reprove the existence of a universally tight contact structure on every irreducible 3-manifold with nonempty boundary, and (2) prove that every toroidal 3-manifold carries infinitely many nonisotopic, nonisomorphic tight contact structures.

Adaptive tensor modeling preserves continuity in multidimensional data.

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…

Variables in many massive high-dimensional data sets are structured, arising for example from measurements on a regular grid as in imaging and time series or from spatial-temporal measurements as in climate studies. Classical multivariate techniques ignore these structural relationships often resulting in poor performa…

Optimal Euclidean structure minimizes energy in weighted toroidal graphs.

Handles decompositions reveal new open book structures.

The study provides a structure theorem for a new class of noncompact 3-manifolds.

Study classifies metrics on anti-de Sitter spacetime with specific symmetries.

Estimates CATEs for structured treatments using a new decomposition method.

In this paper an extended CPR decomposition theorem for Finsler symmetric spaces of semi-negative curvature in the context of reductive structures is proven. This decomposition theorem is applied to give a geometric description of the complexification of some infinite dimensional homogeneous spaces.

The paper proposes and discusses semiorthogonal decompositions for moduli spaces of vector bundles.

Examines respectful decompositions of Lie algebras.

A new probabilistic BTD method for tensor data.

A real 3-manifold is a smooth 3-manifold together with an orientation preserving smooth involution, called a real structure. In this article we study open book decompositions on smooth real 3-manifolds that are compatible with the real structure. We call them real open book decompositions. We show that each real open b…

Proves hyperbolic 3-manifolds have angle structures under certain conditions.

Giroux has described a correspondence between open book decompositions on a 3--manifold and contact structures. In this paper we use Heegaard Floer homology to give restrictions on contact structures which correspond to open book decompositions with planar pages, generalizing a recent result of Etnyre.

The paper describes a structural decomposition of a specific type of Schottky groups.

A function group is a finitely generated Kleinian group with an invariant connected component of its region of discontinuity. An extended function group is a finitely generated extended Kleinian group that contains orientation reversing elements and keep invariant a connected components of its region of discontinuity. …

New open books defy positive factorisation in genus one.

Let S be a closed orientable surface of genus at least two, and let C be an arbitrary (complex) projective structure on S. We show that there is a decomposition of S into pairs of pants and cylinders such that the restriction of C to each component has an injective developing map and a discrete and faithful holonomy re…

We construct an open book decomposition compatible with a contact structure given by a rational contact surgery on a Legendrian link in the standard contact $S^3$. As an application we show that some rational contact surgeries on certain Legendrian knots induce overtwisted contact structures.

We show that associating the Euclidean cell decomposition due to Cooper and Long to each point of the moduli space of framed strictly convex real projective structures of finite volume on the once-punctured torus gives this moduli space a natural cell decomposition. The proof makes use of coordinates due to Fock and Go…

The abstract discusses applications of Menke's JSJ decomposition to symplectic fillings of various 3-manifolds.

This work considers a computationally and statistically efficient parameter estimation method for a wide class of latent variable models---including Gaussian mixture models, hidden Markov models, and latent Dirichlet allocation---which exploits a certain tensor structure in their low-order observable moments (typically…

The theory of geometric structures on a surface with nonempty boundary can be developed by using a decomposition of such a surface into hexagons, in the same way as the theory of geometric structures on a surface without boundary is developed using the decomposition of such a surface into pairs of pants. The basic elem…

In this paper we discuss the change in contact structures as their supporting open book decompositions have their binding components cabled. To facilitate this and applications we define the notion of a rational open book decomposition that generalizes the standard notion of open book decomposition and allows one to mo…

AL$\ell_0$CORE tensor decomposition reduces computational cost for sparse count data.

Two methods preserve tensor structure for reduced dimensionality in tensor regression.

Two constructions of contact manifolds are presented: (i) products of S^1 with manifolds admitting a suitable decomposition into two exact symplectic pieces and (ii) fibre connected sums along isotropic circles. Baykur has found a decomposition as required for (i) for all closed, oriented 4-manifolds. As a corollary, w…

Optimal discrete harmonic maps between hyperbolic surfaces are found via minimizing energy.

Decompositions on manifolds appear in various geometric structures. Necessary and sufficient conditions for quotient spaces of decompositions to be manifolds are widely characterized. We characterize necessary and sufficient conditions to be $k$-manifolds $(k = 1, 2)$, which generalize characterizations in the codimens…

We study properties concerning decomposition in cohomology by means of generalized-complex structures. This notion includes the $\mathcal{C}^\infty$-pure-and-fullness introduced by Li and Zhang in the complex case and the Hard Lefschetz Condition in the symplectic case. Explicit examples on the moduli space of the Iwas…