arXiv research

Study finds a limiting distribution for free path lengths on flat surfaces with circular obstacles.

Using zippered rectangle coordinates we parametrize a Poincaré section for horocycle flow on the space of genus 2 translation surfaces with one singular cone point of angle $6π$. In addition, we bound the return time under horocycle flow to this Poincaré section by examining a subset of surfaces where a certain sum of …

Very few results are known about the topology of the strata of the moduli space of quadratic differentials. In this paper, we prove that any connected component of such strata has only one topological end. A typical flat surface in a neighborhood of the boundary is naturally split by a collection of parallel short sadd…

In this paper, we define the rectangle condition on the bridge sphere for a $n$-bridge decomposition of a knot whose definition is analogous to the definition of the rectangle condition for Heegaard splittings of $3$-manifolds. We show that the satisfaction of the rectangle condition for a $n$-bridge decomposition can …

Zipper logic is a graph rewrite system, consisting in only local rewrites on a class of zipper graphs. Connections with the chemlambda artificial chemistry and with knot diagrammatics based computation are explored in the article.

Study zippers in hyperbolic 3-manifolds, proving fixed point dichotomy.

Paper finds a counterexample showing rectangle condition doesn't detect strong irreducibility.

The article contains a construction of a self-similar dendryte which cannot be the attractor of any self-similar zipper.

Paper introduces 'zippers' for constructing universal circles.

We propose a construction which transforms a self-similar zipper in $\mathbb R^n$ to a self-affine zipper $\mathbb R^{n+1}$ whose attractor is a smooth curve.

Formula for interleaving distance of rectangle persistence modules.

This thesis classifies pseudo-Anosov homeomorphisms using geometric Markov partitions.

Classifies essential annuli in genus two handlebody-knots, determining hyperbolicity and constructing obstructions.

Optimal weight windows are symmetric rectangles centered at peak.

Curves inscribe rectangles with positive area.

Floer homology applied to inscribing rectangles into curves.

The study describes a cell structure for multisets in a rectangle.

Consider a planar, bounded, $m$-connected region $Ω$, and let $\bordΩ$ be its boundary. Let $\mathcal{T}$ be a cellular decomposition of $Ω\cup\bordΩ$, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair $(S,f)$ where $S$ is a genus $(m-1)$…

The paper improves bounds on how many squares can fit in a rectangle and still have stable homology.

Rectangles can fit on smooth curves, proving a theorem about Klein bottles.

Paper classifies pillow box isometric deformations preserving crease patterns.

We estimate whether there is an embedding from one n-dimensional rectangle into another which expands every k-dimensional area. Our estimate is sharp up to a constant factor in each dimension.

A new ensemble model uses simple hyper-rectangles to improve gradient boosting machine performance.

Equal diagonal energies proven on Liouville surfaces.

The paper proves geometric properties of square tables and saddle surfaces.

In this paper we show that for a given 3-manifold and a given Heegaard splitting there are finitely many preferred decomposing systems of $3g - 3$ disjoint essential disks. These are characterized by a combinatorial criterion which is a slight strengthening of Casson-Gordon's rectangle condition. This is in contrast to…

We give the rectangle condition for strong irreducibility of Heegaard splittings of $3$-manifolds with non-empty boundary. We apply this to a generalized Heegaard splitting of a $2$-fold covering of $S^3$ branched along a link. The condition implies that any thin meridional level surface in the link complement is incom…

The paper proves that any smooth curve can have two similar inscribed rectangles.

In this paper we continue the study started in part I (posted). We consider a planar, bounded, $m$-connected region $Ω$, and let $\bordΩ$ be its boundary. Let $\mathcal{T}$ be a cellular decomposition of $Ω\cup\bordΩ$, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function…

We prove that a bounded open set U in Euclidean n-space has k-width less than C(n) Volume(U)^{k/n}. Using this estimate, we give lower bounds for the k-dilation of degree 1 maps between certain domains in Euclidean space. In particular, we estimate the smallest (n-1)-dilation of any degree 1 map between two n-dimension…

We develop a recursive formula for counting the number of rectangulations of a square, i.e the number of combinatorially distinct tilings of a square by rectangles. Our formula specializes to give a formula counting generic rectangulations, as analyzed by Reading in [5]. Our computations agree with [5] as far as was ca…

We investigate the common underlying discrete structures for various smooth and discrete nets. The main idea is to impose the characteristic properties of the nets not only on elementary quadrilaterals but also on larger parameter rectangles. For discrete planar quadrilateral nets, circular nets, $Q^*$-nets and conical…

Fast BATLLNN speeds up verification of TLL NNs by 400x.

The study generalizes origamis to flat surfaces, exploring their combinatorial and geometric properties.

Given i.i.d samples from some unknown continuous density on hyper-rectangle $[0, 1]^d$, we attempt to learn a piecewise constant function that approximates this underlying density non-parametrically. Our density estimate is defined on a binary split of $[0, 1]^d$ and built up sequentially according to discrepancy crite…

The paper studies geometric structures of polynomial spaces.

We prove that any cyclic quadrilateral can be inscribed in any closed convex $C^1$-curve. The smoothness condition is not required if the quadrilateral is a rectangle.

We discuss differences between genera of smooth and locally-flat non-orientable surfaces in the 4-ball with boundary a given torus knot or 2-bridge knot. In particular, we establish that a result by Batson on the smooth non-orientable 4-genus of torus knots does not hold in the locally-flat category. We further show th…

Casson and Gordon gave the rectangle condition for strong irreducibility of Heegaard splittings [1]. We give a parity condition for irreducibility of Heegaard splittings of irreducible manifolds. As an application, we give examples of non-stabilized Heegaard splittings by doing a single Dehn twist.

We prove that for every smooth Jordan curve $γ$, if $X$ is the set of all $r \in [0,1]$ so that there is an inscribed rectangle in $γ$ of aspect ratio $\tan(r\cdot π/4)$, then the Lebesgue measure of $X$ is at least $1/3$. To do this, we study sets of disjoint homologically nontrivial projective planes smoothly embedde…

ARGEN method improves variable selection and regularization in high-dimensional sparse models.

Let $R$ be a compact, connected, orientable surface of genus $g$ with $n$ boundary components with $g \geq 2$, $n \geq 0$. Let $\mathcal{N}(R)$ be the nonseparating curve graph, $\mathcal{C}(R)$ be the curve graph and $\mathcal{HT}(R)$ be the Hatcher-Thurston graph of $R$. We prove that if $λ: \mathcal{N}(R) \rightarro…

CUR matrix decomposition is a randomized algorithm that can efficiently compute the low rank approximation for a given rectangle matrix. One limitation with the existing CUR algorithms is that they require an access to the full matrix A for computing U. In this work, we aim to alleviate this limitation. In particular, …

Proposes adaptive ridge regression for functional linear models with piecewise shapes.

New method for flexible tubes and structures, enabling rigid-foldability.

We consider the problem of learning a sparse rule model, a prediction model in the form of a sparse linear combination of rules, where a rule is an indicator function defined over a hyper-rectangle in the input space. Since the number of all possible such rules is extremely large, it has been computationally intractabl…

In this work, we study the cellular decomposition of $S$ induced by a filling pair of curves $v$ and $w$, $Dec_{v,w}(S) = S - (v \cup w)$, and its connection to the distance function $d(v,w)$ in the curve graph of a closed orientable surface $S$ of genus $g$. Efficient geodesics were introduced by the first author in j…

Study critical points of Laplace eigenfunctions in polygons.