arXiv research

Characterizes ends and coends of graph pairs using quasi-median graphs.

Study shows mapping class groups are one-ended for surfaces with at least one end.

Every weak Perron number is realized as a stretch factor of a homeomorphism on a surface.

Finite number of eigenvalues found in cylindrical surface.

Paper proves timelike minimal surfaces with any number of ends exist.

The study counts ends on shrinkers using geometric covering methods.

Proves existence of maps with arbitrary ends and conditions for maxfaces.

Suppose $M$ is a complete, embedded minimal surface in $\mathbb{R}^3$ with an infinite number of ends, finite genus and compact boundary. We prove that the simple limit ends of $M$ have properly embedded representatives with compact boundary, genus zero and with constrained geometry. We use this result to show that if …

Finite graphs with specific curvature have limited harmonic functions and ends.

New metrics with constant Q-curvature created by gluing.

We prove a version of the well-known Denjoy-Ahlfors theorem about the number of asymptotic values of an entire function for properly immersed minimal surfaces of arbitrary codimension in R^N. The finiteness of the number of ends is proved for minimal submanifolds with finite projective volume. We show, as a corollary, …

Classifies worst approximable rational numbers using hyperbolic geometry.

Embedding layers are commonly used to map discrete symbols into continuous embedding vectors that reflect their semantic meanings. Despite their effectiveness, the number of parameters in an embedding layer increases linearly with the number of symbols and poses a critical challenge on memory and storage constraints. I…

A manifold's discreteness is tied to its number of ends.

We prove that for every nonnegative integer $g$, there exists a bound on the number of ends of a complete, embedded minimal surface $M$ in $\mathbb{R}^3$ of genus $g$ and finite topology. This bound on the finite number of ends when $M$ has at least two ends implies that $M$ has finite stability index which is bounded …

For any m > 0, we construct properly embedded minimal surfaces in H^2 x R with genus zero, infinitely many vertical planar ends and m limit ends. We also provide examples with an infinite countable number of limit ends. All these examples are vertical bi-graphs.

Uniform bounds on ends for non-branching CD spaces with nonnegative curvature outside a compact set.

Estimates ends of Ricci shrinkers, focusing on smooth and singular cases.

Develops theory linking Schrödinger equations to manifold ends, proving finiteness.

Study of non-metrizable manifolds' ends, generalizing Nyikos's theorem.

Ends and cohomology theory for noncompact spaces.

We present NAVREN-RL, an approach to NAVigate an unmanned aerial vehicle in an indoor Real ENvironment via end-to-end reinforcement learning RL. A suitable reward function is designed keeping in mind the cost and weight constraints for micro drone with minimum number of sensing modalities. Collection of small number of…

New method constructs surfaces with constant mean curvature.

Let f:A-->B be a covering map. We say A has e filtered ends with respect to f (or B) if for some filtration {K_n} of B by compact subsets, A - f^{-1}(K_n) "eventually" has e components. The main theorem states that if Y is a (suitable) free H-space, if K < H has infinite index, and if Y has a positive finite number of …

We combine the DPW method and Opening Nodes to construct embedded surfaces of positive constant mean curvature with Delaunay ends in euclidean space, with no limitation to the genus or number of ends.

In this paper, we investigate the asymptotic behavior of regular ends of flat surfaces in the hyperbolic 3-space H^3. Galvez, Martinez and Milan showed that when the singular set does not accumulate at an end, the end is asymptotic to a rotationally symmetric flat surface. As a refinement of their result, we show that …

We construct Gaussian Harmonic forms of finite Gaussian weighted $L^2$-norm on non-compact surfaces that detect each asymptotically conical end. As an application we prove an extension of the index estimates of self-shrinkers in $[11]$ under the existence of such ends. We show that the Morse index of a self-shrinker is…

Let Y be a noncompact rank one locally symmetric space of finite volume. Then Y has a finite number e(Y) > 0 of topological ends. In this paper, we show that for any natural number n, the Y with e(Y) \leq n that are arithmetic fall into finitely many commensurability classes. In particular, there is a constant c_n such…

Scaling end-to-end reinforcement learning to control real robots from vision presents a series of challenges, in particular in terms of sample efficiency. Against end-to-end learning, state representation learning can help learn a compact, efficient and relevant representation of states that speeds up policy learning, …

The paper extends end concepts to arbitrary groups and spaces.

NSR enables neural networks to reason with continuous numbers and extrapolate.

We consider constant mean curvature surfaces of finite topology, properly embedded in three-space in the sense of Alexandrov. Such surfaces with three ends and genus zero were constructed and completely classified by the authors in arXiv:math.DG/0102183. Here we extend the arguments to the case of an arbitrary number o…

Deploying deep learning (DL) models across multiple compute devices to train large and complex models continues to grow in importance because of the demand for faster and more frequent training. Data parallelism (DP) is the most widely used parallelization strategy, but as the number of devices in data parallel trainin…

A smooth end of a Bryant surface is a conformally immersed punctured disc of mean curvature 1 in hyperbolic space that extends smoothly through the ideal boundary. The Bryant representation of a smooth end is well defined on the punctured disc and has a pole at the puncture. The Willmore energy of compact Bryant surfac…

We propose a novel end-to-end neural network architecture that, once trained, directly outputs a probabilistic clustering of a batch of input examples in one pass. It estimates a distribution over the number of clusters $k$, and for each $1 \leq k \leq k_\mathrm{max}$, a distribution over the individual cluster assignm…

We apply the local removable singularity theorem for minimal laminations and the local picture theorem on the scale of topology to obtain two descriptive results for certain possibly singular minimal laminations of $\mathbb{R}^3$. These two global structure theorems will be applied in forthcoming papers to obtain bound…

We show that any compact half-conformally flat manifold of negative type, with bounded $L^2$ energy, sufficiently small scalar curvature, and a non-collapsing assumption, has all betti numbers bounded. We show that this result is optimal from an analytic perspective by demonstrating singularity models that are 2-ended,…

New theorem bounds group quotient size to subgroups index.

Adversarial methods for imitation learning have been shown to perform well on various control tasks. However, they require a large number of environment interactions for convergence. In this paper, we propose an end-to-end differentiable adversarial imitation learning algorithm in a Dyna-like framework for switching be…

We give a number of examples of pairs of non-compact surfaces which are isoscattering, and which are exceptionally simple in one or more senses. We give examples which are of small genus with a small number of ends, and also examles which are congruence surfaces.

Researchers create minimal surfaces with Scherk ends and find catenoid limits.

The study explores ends in coarse homotopy of proper geodesic spaces.

Representations of sets are challenging to learn because operations on sets should be permutation-invariant. To this end, we propose a Permutation-Optimisation module that learns how to permute a set end-to-end. The permuted set can be further processed to learn a permutation-invariant representation of that set, avoid…

Continuous epimorphisms between certain mapping class groups are induced by homeomorphisms.

The use of deep neural networks in edge computing devices hinges on the balance between accuracy and complexity of computations. Ternary Connect (TC) \cite{lin2015neural} addresses this issue by restricting the parameters to three levels $-1, 0$, and $+1$, thus eliminating multiplications in the forward pass of the net…

End-to-end deep reinforcement learning has enabled agents to learn with little preprocessing by humans. However, it is still difficult to learn stably and efficiently because the learning method usually uses a nonlinear function approximation. Neural Episodic Control (NEC), which has been proposed in order to improve s…

For an immersed minimal surface in $\mathbb{R}^3$, we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we previously obtained bounding the genus and number of ends by the index. Our new estimate resol…

We consider unbranched Willmore surfaces in the Euclidean space that arise as inverted complete minimal surfaces with embedded planar ends. Several statements are proven about upper and lower bounds on the Morse Index - the number of linearly independent variational directions that locally decrease the Willmore energy.…