arXiv research

RP-GFRFT unifies fractional order and rotation control for graph signals.

Rotation systems can't always be drawn in surfaces.

We study the problem of learning representations of entities and relations in knowledge graphs for predicting missing links. The success of such a task heavily relies on the ability of modeling and inferring the patterns of (or between) the relations. In this paper, we present a new approach for knowledge graph embeddi…

Odd crossing numbers and even rotation numbers for cycles in plane immersions.

Abstract: Proves generic torus diffeomorphisms act parabolically and non-properly on fine curve graph and have generalized rotation sets.

New Alexander polynomial defined for transverse graphs.

Spherical data is found in many applications. By modeling the discretized sphere as a graph, we can accommodate non-uniformly distributed, partial, and changing samplings. Moreover, graph convolutions are computationally more efficient than spherical convolutions. As equivariance is desired to exploit rotational symmet…

We present a three-dimensional graph convolutional network (3DGCN), which predicts molecular properties and biochemical activities, based on 3D molecular graph. In the 3DGCN, graph convolution is unified with learning operations on the vector to handle the spatial information from molecular topology. The 3DGCN model ex…

The paper classifies CMC free boundary hypersurfaces in rotational domains.

New method synchronizes graphs with probability measures on rotations.

Study homeomorphisms on fine curve graph of surfaces, revealing new types of dynamics.

DeepSphere improves spherical CNNs by balancing efficiency and rotation equivariance.

The study proves conditions for hyperbolic isometries on fine curve graphs of higher genus surfaces.

Study rotational surfaces with prescribed Gauss curvature in 3D space.

Classification of torus homeomorphisms on fine curve graph completed.

In this work, we move beyond the traditional complex-valued representations, introducing more expressive hypercomplex representations to model entities and relations for knowledge graph embeddings. More specifically, quaternion embeddings, hypercomplex-valued embeddings with three imaginary components, are utilized to …

Study curve shortening flows on specific surfaces, proving properties and existence.

Study irrational rotations and construct 2-filling rays on infinite type surfaces.

Predict missing movie ratings or graph embeddings with low rank matrices.

We study constant mean curvature 1/2 surfaces in H2xR that admit a compactification of the mean curvature operator. We show that a particular family of complete entire graphs over H2 admits a structure of infinite dimensional manifold with local control on the behaviors at infinity. These graphs also appear to have a h…

The paper constructs surfaces with prescribed mean curvature in a specific space.

In this paper, we are concerned with hypersurfaces in $H^n\times R$ with constant r-mean curvature, to be called $H_r$-hypersurfaces. We construct examples of complete $H_r$-hypersurfaces which are invariant by parabolic screw motion or by rotation. We prove that there is a unique rotational strictly convex entire $H_r…

Recent machine learning methods make it possible to model potential energy of atomic configurations with chemical-level accuracy (as calculated from ab-initio calculations) and at speeds suitable for molecular dynam- ics simulation. Best performance is achieved when the known physical constraints are encoded in the mac…

We give an explicit calculation of the Wu invariants for immersions of a finite graph into the plane and classify all generic immersions of a graph into the plane up to regular homotopy by the Wu invariant. This result is a generalization of the fact that two plane curves are regularly homotopic if and only if they hav…

Rotors were introduced in Graph Theory by W.Tutte. The concept was adapted to Knot Theory as a generalization of mutation by Anstee, Przytycki and Rolfsen in 1987. In this paper we show that Tristram-Levine signature is preserved by orientation-preserving rotations. Moreover, we show that any link invariant obtained fr…

We investigate Legendrian graphs in $(\R^3, ξ_{std})$. We extend the classical invariants, Thurston-Bennequin number and rotation number to Legendrian graphs. We prove that a graph can be Legendrian realized with all its cycles Legendrian unknots with $tb=-1$ and $rot=0$ if and only if it does not contain $K_4$ as a mi…

New method generates molecular conformations efficiently.

In this article we give necessary and sufficient conditions for two triples of integers to be realized as the Thurston-Bennequin number and the rotation number of a Legendrian theta-graph with all cycles unknotted. We show that these invariants are not enough to determine the Legendrian class of a topologically planar …

We prove two results on the classification of trivial Legendrian embeddings $g: G \rightarrow (S^3,ξ_{std})$ of planar graphs. First, the oriented Legendrian ribbon $R_g$ and rotation invariant $\text{rot}_g$ are a complete set of invariants. Second, if $G$ is 3-connected or contains $K_4$ as a minor, then the unique t…

We prove an existence result for non rotational constant mean curvature ends in $\mathbb{H}^2 \times \mathbb{R}$, where $\mathbb{H}^2$ is the hyperbolic real plane. The value of the curvature is $h \, \in \, (0, 1/2)$. We use Schauder theory and a continuity method for solution of the prescribed mean curvature equation…

Link prediction is an important and frequently studied task that contributes to an understanding of the structure of knowledge graphs (KGs) in statistical relational learning. Inspired by the success of graph convolutional networks (GCN) in modeling graph data, we propose a unified GCN framework, named TransGCN, to add…

Geometric Graph Alignment enhances IoT intrusion detection using NID data.

SE(3)-Transformers maintain equivariance for 3D data under rotations and translations.

New model learns graph neural networks equivariant to various transformations.

The study examines polyhedra with hexagonal and triangular faces, focusing on their 3-regular planar graphs.

We prove that 2-dimensional simplicial complexes whose first homology group is trivial have topological embeddings in 3-space if and only if there are embeddings of their link graphs in the plane that are compatible at the edges and they are simply connected.

New methods prove existence of rotating shapes moving in space.

Geometric GNNs model 3D atomic systems with rotations and translations.

The paper extends graph embedding models to handle multiple relations.

In an ambient space with rotational symmetry around an axis (which include the Hyperbolic and Euclidean spaces), we study the evolution under the volume-preserving mean curvature flow of a revolution hypersurface M generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces …

Duality result connects bounded cohomology to relative Gromov seminorm.

We introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of the peripheral structure of a relatively hyperbolic group, while the later one provides a natural framework for developing a geometric version of small cancel…

A {\em good drawing\/} of $K_n$ is a drawing of the complete graph with $n$ vertices in the sphere such that: no two edges with a common end cross; no two edges cross more than once; and no three edges all cross at the same point. Gioan's Theorem asserts that any two good drawings of $K_n$ that have the same rotations …

We give explicit formulae for fringe lengths of the Calegari-Walker Ziggurats -- i.e. graphs of extremal rotation numbers associated to positive words in free groups. These formulae reveal (partial) integral projective self-similarity in ziggurat fringes, which are low-dimensional projections of characteristic polyhedr…

Study on rotating surfaces in 4D space with matrices.

VDWs enhance graph neural networks for analyzing complex data.

The paper develops a new algorithm for RBMs using dynamical mean-field theory.

A new transform links rotating calorons to solutions of a differential equation.