arXiv research

Lower bound for complexity of finding flex points on cubic curves.

For a real valued periodic smooth function u on R, $n\ge 0$, one defines the osculating polynomial $φ_s$ (of order 2n+1) at a point $s\in R$ to be the unique trigonometric polynomial of degree n, whose value and first 2n derivatives at s coincide with those of u at s. We will say that a point s is a clean maximal flex …

The paper studies geometric loci and their invariants in complex dynamics.

Extends first-order flexes of surfaces to second-order flexes.

Study on choosing points on cubic curves, answering some questions about their flexibility.

FLEX optimizes exploration for nonlinear systems with minimal data.

We show how the rotation and translation fields of a surface, introduced by G. Darboux, may be used to obtain short proofs of a well-known theorem (that reads that the total mean curvature of a surface is stationary under an infinitesimal bending) and a new theorem (that reads that every infinitesimal flex of any simpl…

Using Green's theorem we reduce the variation of the total mean curvature of a smooth surface in the Euclidean 3-space to a line integral of a special vector field and obtain the following well-known theorem as an immediate consequence: the total mean curvature of a closed smooth surface in the Euclidean 3-space is sta…

Osculating spaces of decomposable scrolls (of any genus and not necessarily normal)are studied and their inflectional loci are related to those of their generating curves by using systematically an idea introduced by Piene and Sacchiero in the setting of rational normal scrolls. In this broader setting the extra compon…

Given a polyhedral surface, assume that it is prohibited to change the shape and size of any face but it is permissible to change the dihedral angles between the faces. A polyhedral surface is said to be flexible if it is possible to change its shape under the above restrictions. We prove that flexible polyhedral surfa…

This paper presents a kernel-based discriminative learning framework on probability measures. Rather than relying on large collections of vectorial training examples, our framework learns using a collection of probability distributions that have been constructed to meaningfully represent training data. By representing …

Hashing has been widely adopted for large-scale data retrieval in many domains, due to its low storage cost and high retrieval speed. Existing cross-modal hashing methods optimistically assume that the correspondence between training samples across modalities are readily available. This assumption is unrealistic in pra…

Combines local and global samplers for efficient sampling.

Rigidity is the property of a structure that does not flex. It is well studied in discrete geometry and mechanics, and has applications in material science, engineering and biological sciences. A bar-and-joint framework is a pair $(G,p)$ of graph $G$ together with a map $p$ of the vertices of $G$ into the Euclidean pla…

Estimator calculates surface curvature from point cloud samples.

New tools for constructing fixed point sets in digital topology.

While Multiple Instance (MI) data are point patterns -- sets or multi-sets of unordered points -- appropriate statistical point pattern models have not been used in MI learning. This article proposes a framework for model-based MI learning using point process theory. Likelihood functions for point pattern data derived …

Study on connection points on double regular polygons, providing coordinates and proving non-connection points.

New families of translation surfaces with multiple oblivious points discovered.

This paper reverses a construction by merging boundary critical points into an interior one.

A new model for point processes without intensity function trade-offs.

PINNACLE optimizes point selection for PINNs, improving accuracy.

The minimal number of critical points is studied for smooth functions on closed manifolds.

Convergence to a saddle point for convex-concave functions has been studied for decades, while recent years has seen a surge of interest in non-convex (zero-sum) smooth games, motivated by their recent wide applications. It remains an intriguing research challenge how local optimal points are defined and which algorith…

Investigates point spectra of vector fields and their properties.

We revisit an example of a semi-Riemannian geodesic that was discussed by Musso, Pejsachowicz and Portaluri in 2007 to show that not every conjugate point is a bifurcation point. We point out a mistake in their argument, showing that on this geodesic actually every conjugate point is a bifurcation point. Finally, we pr…

Hard to approximate critical points for simple nonconvex functions.

Algorithm finds periodic points on Veech surfaces.

Estimating boundaries from point clouds with improved accuracy and rigorous error estimates.

The paper classifies circle actions on 6D manifolds with isolated fixed points.

PoPPy is a Point Process toolbox based on PyTorch, which achieves flexible designing and efficient learning of point process models. It can be used for interpretable sequential data modeling and analysis, e.g., Granger causality analysis of multi-variate point processes, point process-based simulation and prediction of…

The paper models user-advertiser interactions using point processes.

A limit point p of a discrete group of Mobius transformations acting on S^n is called a concentration point if for any sufficiently small connected open neighborhood U of p, the set of translates of U contains a local basis for the topology of S^n at p. For the case of Fuchsian groups (n = 1), every concentration point…

The paper develops methods to create private synthetic spatial point patterns.

Self-focal points on ellipsoids of dimension 3 or higher are rare.

We construct non-trapping asymptotically hyperbolic manifolds with boundary conjugate points but no interior conjugate points.

The paper introduces a method to probabilistically select inducing points in sparse Gaussian processes.

Groups with special properties always have fixed points.

The paper proves $\Gamma$-convergence of discrete tangent-point energies to continuous energies and ropelength, with applications to biarc curves.

The study confirms a conjecture about critical points of smooth functions.

New method upsamples sparse, non-uniform point clouds more accurately.

Characterizes Lebesgue points using nearest neighbor methods.

We give a explicit computation of the pointed harmonic volumes of hyperelliptic curves with Weierstrass base points, which are paraphrased into a combinatorial formula.

We propose to explain the predictions of a deep neural network, by pointing to the set of what we call representer points in the training set, for a given test point prediction. Specifically, we show that we can decompose the pre-activation prediction of a neural network into a linear combination of activations of trai…

In recent decades, the use of 3D point clouds has been widespread in computer industry. The development of techniques in analyzing point clouds is increasingly important. In particular, mapping of point clouds has been a challenging problem. In this paper, we develop a discrete analogue of the Teichmüller extremal mapp…

Estimates geodesics on surfaces without conjugate points.

The main result is a direct proof of the implication $(LVKF_{k,3})\Rightarrow( LT_{3k-1,3})$ below. Consider the following statements: ($LVKF_{1,3}$) From any 11 points in $\mathbb{R}^{3}$ one can choose 3 pairwise disjoint triples whose convex hulls have a common point. ($LVKF_{k,3}$) From any $6k + 5$ points in $ \m…

Study shows periodic points of Prym eigenforms in specific genera.