Related papers: Encoding toroidal triangulations
Routing tables in ad hoc and wireless routing protocols can be represented using rooted trees. The constant need for communication and storage of these trees in routing protocols demands an efficient rooted tree coding algorithm. This…
Tensor networks provide compact and scalable representations of high-dimensional data, enabling efficient computation in fields such as quantum physics, numerical partial differential equations (PDEs), and machine learning. This paper…
We introduce a new way to tabulate knots by representing knot diagrams using a pair of planar trees. This pair of trees have their edges labeled by integers, they have no valence 2 vertices, and they have the same number of valence 1…
A triangulation of a surface is irreducible if there is no edge whose contraction produces another triangulation of the surface. In this work we propose an algorithm that constructs the set of irreducible triangulations of any surface with…
A new algorithm for efficient exact maximum likelihood decoding of polar codes (which may be CRC augmented), transmitted over the binary erasure channel, is presented. The algorithm applies a matrix triangulation process on a sparse polar…
Tight triangulations are exotic, but highly regular objects in combinatorial topology. A triangulation is tight if all its piecewise linear embeddings into a Euclidean space are as convex as allowed by the topology of the underlying…
Fourier Ptychography is a recently proposed imaging technique that yields high-resolution images by computationally transcending the diffraction blur of an optical system. At the crux of this method is the phase retrieval algorithm, which…
We introduce a polynomial invariant $V_\tau \in \mathbb{Z}[H_1(M)/\text{torsion}]$ associated to a veering triangulation $\tau$ of a $3$-manifold $M$. In the special case where the triangulation is layered, i.e. comes from a fibration,…
In the paper `On the Dirac-Frenkel Variational Principle on Tensor Banach Spaces', we provided a geometrical description of manifolds of tensors in Tucker format with fixed multilinear (or Tucker) rank in tensor Banach spaces, that allowed…
A common representation of a three dimensional object in computer applications, such as graphics and design, is in the form of a triangular mesh. In many instances, individual or groups of triangles in such representation need to satisfy…
This paper investigates universal polar coding schemes. In particular, a notion of ordering (called convolutional path) is introduced between probability distributions to determine when a polar compression (or communication) scheme designed…
Avraham et al. [AFK+15] presented an alternative approach to parametric search, called \emph{bifurcation}, that performs faster under certain circumstances. Intuitively, when the underlying decider execution can be rolled back cheaply and…
We introduce tensor numerical techniques for solving optimal control problems constrained by elliptic operators in $\mathbb{R}^d$, $d=2,3$, with variable coefficients, which can be represented in a low rank separable form. We construct a…
If a (cusped) surface S admits an ideal triangulation T with no shears, we show an efficient algorithm to give S as a quotient of hypebolic plane by a subgroup of PSL(2, Z). The algorithm runs in time O(n log n), where n is the number of…
Machine learning on trees has been mostly focused on trees as input to algorithms. Much less research has investigated trees as output, which has many applications, such as molecule optimization for drug discovery, or hint generation for…
Let $\cal T$ be a tiling of the plane with equilateral triangles no two of which share a side. We prove that if the side lengths of the triangles are bounded from below by a positive constant, then $\cal T$ is periodic and it consists of…
The bipolar filtration of Cochran, Harvey and Horn presents a framework of the study of deeper structures in the smooth concordance group of topologically slice knots. We show that the graded quotient of the bipolar filtration of…
Let $G$ be a 3-connected planar graph. Define the co-tree of a spanning tree $T$ of $G$ as the graph induced by the dual edges of $E(G)-E(T)$. The well-known cut-cycle duality implies that the co-tree is itself a tree. Let a $k$-tree be a…
This paper contains a classification of countable lower 1-transitive linear orders. The notion of lower 1-transitivity generalises that of 1-transitivity for linear orders, and is essential for the structure theory of 1-transitive trees.…
We describe a smooth structure, called Fr\"olicher space, on CW complexes and spaces of triangulations. This structure enables differential methods for e.g. minimization of functionnals. As an application, we exhibit how an optimized…