arXiv research

New method estimates spatial weights matrix for lattice data, improving prediction accuracy.

Researchers calculate complexity of billiard paths in regular polygons.

We show that two uniform lattices of a regular right-angled Fuchsian building are commensurable, provided the chamber is a polygon with at least six edges. We show that in an arbitrary Gromov-hyperbolic regular right-angled building associated to a graph product of finite groups, a uniform lattice is commensurable with…

A new surgery formula for knot lattice homology.

The Phi- relationship also known as Phi-factor appears in a number of lattice structures, mostly considering the lines within several separate circles or polygons. The paper considers a regular hexagonal tessellation as a lattice with the highest specific mechanical stiffness.

Regular integer lattices are characterized by k unit vectors that build up their generator matrices. These have rank k for D-lattices, and are rank-deficient for A-lattices, for E_6 and E_7. We count lattice points inside hypercubes centered at the origin for all three types, as if classified by maximum infinity norm i…

Study Poisson boundaries of building lattices and generalize rigidity results.

Sharp bounds for spanning tree entropy in planar lattices.

In a way similar to the continuous case formally, we define in different but equivalent manners the difference discrete connection and curvature on discrete vector bundle over the regular lattice as base space. We deal with the difference operators as the discrete counterparts of the derivatives based upon the differen…

Data compression speeds up machine learning loss calculations.

We introduce a new discrete system that arises from ellipsoidal billiards and is closely related to the double reflection nets. The system is defined on the lattice of a uniform honeycomb consisting of rectified hypercubes and cross polytopes. In the $2$-dimensional case, the lattice is regular and it incorporates dyna…

We study cocompact lattices with dense projections in a product $G_1 \times G_2$ of locally compact groups and show, under the assumption that each $G_i$ is a closed subgroup of the automorphism group $Aut(T_i)$ of a regular tree satisfying certain local transitivity conditions, that such a lattice is contained in only…

We investigate variations of Brieskorn lattices over non-compact parameter spaces, and discuss the corresponding limit objects on the boundary divisor. We study the associated variation of twistors and the corresponding limit mixed twistor structures. We construct a compact classifying space for regular singular Briesk…

Deep Learning methods, specifically convolutional neural networks (CNNs), have seen a lot of success in the domain of image-based data, where the data offers a clearly structured topology in the regular lattice of pixels. This 4-neighbourhood topological simplicity makes the application of convolutional masks straightf…

Study flat metrics from right prisms, finding non-lattice surfaces with translation coverings.

Investigates methods to regularize quantile regression for accurate predictions.

Regular subgroups of SL3(R) are identified and ruled out.

Paper proves rigidity of Doyle spirals in hexagonal lattice circle packings.

We study tt*-geometry on the classifying space for regular singular TERP-structures, e.g., Fourier-Laplace transformations of Brieskorn lattices of isolated hypersurface singularities. We show that (a part of) this classifying space can be canonically equipped with a hermitian structure. We derive an estimate for the h…

This paper solves PDEs for embedding discrete lattices into smooth manifolds.

Paper recovers lattice signal partitions efficiently.

The paper proves actions of lattices in higher rank groups have cost one.

We introduce a criterion that a given bihamiltonian structure allows a local coordinate system where both brackets have constant coefficients. This criterion is applied to the bihamiltonian open Toda lattice in a generic point, which is shown to be locally isomorphic to a Kronecker odd-dimensional pair of brackets with…

Improved speech recognition for voice assistants by analyzing speech data.

New proof for symmetric spaces with rectangular lattices.

Let $(W,S)$ be a Coxeter system with Davis complex $Σ$. The polyhedral automorphism group $G$ of $Σ$ is a locally compact group under the compact-open topology. If $G$ is a discrete group (as characterised by Haglund--Paulin), then the set $\mathcal V_u(G)$ of uniform lattices in $G$ is discrete. Whether the converse i…

We define an infinite series of translation coverings of Veech's double-n-gon for odd n greater or equal to 5 which share the same Veech group. Additionally we give an infinite series of translation coverings with constant Veech group of a regular n-gon for even n greater or equal to 8. These families give rise to expl…

Study finds the spectrum of a cubic Dirac operator on specific oscillator group manifolds.

L-CNNs preserve gauge symmetry in lattice simulations.

This paper is a contribution to harmonic analysis of compact solvmanifolds. We consider the four-dimensional oscillator group ${\rm Osc}_1$, which is a semi-direct product of the three-dimensional Heisenberg group and the real line. We classify the lattices of ${\rm Osc}_1$ up to inner automorphisms of ${\rm Osc}_1$. F…

Deep convolutional neural networks (CNNs) have shown outstanding performance in the task of semantically segmenting images. However, applying the same methods on 3D data still poses challenges due to the heavy memory requirements and the lack of structured data. Here, we propose LatticeNet, a novel approach for 3D sema…

The study examines knot probabilities in confined lattice polygons.

Novel method for learning Gaussian graphical models from paired data.

New risk measures for incomplete markets without lattice structures.

New framework for detecting complex interactions in multivariate data.

New subgroup found in Lie groups with unusual properties.

Optimal smooth subspaces approximate large data sets efficiently.

We provide examples of towers of covers of cusped hyperbolic 3-manifolds whose exponential homological torsion growth is explicitly computed in terms of volume growth. These examples arise from abelian covers of alternating links in the thickened torus. A corollary is that the spanning tree entropy for each regular pla…

New property identifies arithmetic lattices from nonuniform lattices.

The study explores maps of 2- and 3-uniform tilings on the torus.

Research on graph representation learning has received a lot of attention in recent years since many data in real-world applications come in form of graphs. High-dimensional graph data are often in irregular form, which makes them more difficult to analyze than image/video/audio data defined on regular lattices. Variou…

Research finds bounds for knots in hexagonal lattice and classifies 11-stick knots.

We outline the theory of sets with distributive operations: multishelves and multispindles, with examples provided by semi-lattices, lattices and skew lattices. For every such a structure we define multi-term distributive homology and show some of its properties. The main result is a complete formula for the homology o…

Machine learning classifies topological phases in leaky photonic lattices.

Course on arithmetic lattices at EPFL.

We give a simple example showing that a knot or link diagram that lies in the ${\mathbb{Z}}^2$ lattice is not necessarily the projection of a lattice stick knot or link in the ${\mathbb{Z}}^3$ lattice, and we give a necessary and sufficient condition for when a knot or link diagram that lies in the ${\mathbb{Z}}^2$ lat…

This paper classifies commensurability of Deligne-Mostow lattices.

We explore a simple lattice field model intended to describe statistical properties of high frequency financial markets. The model is relevant in the cross-disciplinary area of econophysics. Its signature feature is the emergence of a self-organized critical state. This implies scale invariance of the model, without tu…