arXiv research

Tripod configurations of plane curves, formed by certain triples of normal lines coinciding at a point, were introduced by Tabachnikov, who showed that $C^2$ closed convex curves possess at least two tripod configurations. Later, Kao and Wang established the existence of tripod configurations for $C^2$ closed locally c…

Counting tripods on a flat torus using lattice point counting.

We establish the Gaussian Double-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose $\mathbb{R}^n$ into three cells of prescribed (positive) Gaussian measure is to use a tripod-cluster, whose interfaces consist of three half-hyperplanes meeting along an $(n-2)$-dimensional plane at $120^{\circ}$ …

Tripod spiders' energy control analyzed for Hooke and Coulomb potentials.

Study polynomial cubic differentials on Riemann surfaces using spectral networks.

Reduces conjecture for Artin groups to simpler cases.

Study three types of uncertainty quantification for binary classification without distributional assumptions.

We establish the Gaussian Multi-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose $\mathbb{R}^n$ into $q$ cells of prescribed (positive) Gaussian measure when $2 \leq q \leq n+1$, is to use a "simplicial cluster", obtained from the Voronoi cells of $q$ equidistant points. Moreover, we prove that…

We use the consistency approach to classify discrete integrable 3D equations of the octahedron type. They are naturally treated on the root lattice $Q(A_3)$ and are consistent on the multidimensional lattice $Q(A_N)$. Our list includes the most prominent representatives of this class, the discrete KP equation and its S…

Let $G$ be a countable group that splits as a free product of groups of the form $G=G_1\ast\dots\ast G_k\ast F_N$, where $F_N$ is a finitely generated free group. We identify the closure of the outer space $P\mathcal{O}(G,\{G_1,\dots,G_k\})$ for the axes topology with the space of projective minimal, \emph{very small} …

Study compares Cox model and RSF for predicting patient survival, finding RSF superior in certain scenarios.

Study of automorphisms and splittings of special groups, showing infinite groups under certain conditions.

New framework for analyzing line fields on surfaces, proving stability under specific conditions.

Paper introduces a meta-critic for accelerating off-policy actor-critic learning.

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

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

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

We present the first provably convergent two-timescale off-policy actor-critic algorithm (COF-PAC) with function approximation. Key to COF-PAC is the introduction of a new critic, the emphasis critic, which is trained via Gradient Emphasis Learning (GEM), a novel combination of the key ideas of Gradient Temporal Differ…

Study critical points of Laplace eigenfunctions in polygons.

Smaller actor-critic models lead to performance degradation and overfitting, highlighting the critic's role in value underestimation.

The paper analyzes an actor-critic algorithm with target networks for deep reinforcement learning.

New PAC-Bayesian approach stabilizes actor-critic learning.

Hard to approximate critical points for simple nonconvex functions.

We prove a version of symmetric criticality for ropelength-critical knots. Our theorem implies that a knot or link with a symmetric representative has a ropelength-critical configuration with the same symmetry. We use this to construct new examples of ropelength critical configurations for knots and links which are dif…

New method improves off-policy critic evaluation in reinforcement learning.

The paper studies critical points in overparameterized neural networks, identifying a star locus and degenerate critical points.

WAVE improves stability in reinforcement learning by adaptively weighting critic's loss.

Actor-critic methods solve reinforcement learning problems by updating a parameterized policy known as an actor in a direction that increases an estimate of the expected return known as a critic. However, existing actor-critic methods only use values or gradients of the critic to update the policy parameter. In this pa…

New findings on Chern flat metrics and their criticality.

We establish regularity results for critical points to energies of immersed surfaces depending on the first and the second fundamental form exclusively. These results hold for a large class of intrinsic elliptic Lagrangians which are sub-critical or critical. They are derived using uniform $ε-$regularity estimates whic…

Critical surfaces are defined by Bachman as topological index 2 surfaces, generalizing incompressible surfaces and strongly irreducible surfaces. In this paper we give a condition to obtain critical Heegaard surfaces by amalgamation. As a special case, we obtain critical Heegaard surfaces by boundary stabilization. It …

Study on CR curves in 3-sphere, focusing on critical curves integration and existence.

Single-timescale actor-critic finds globally optimal policy.

This paper focuses on the problem of topological equivalence of functions with isolated critical points on the boundary of a compact surface $M$ which are also isolated critical points of their restrictions to the boundary. This class of functions we denote by $Ω(M)$. Firstly, we've obtained the topological classificat…

Symmetric critical points lead to symmetry breaking in neural networks.

Upper bounds found for systole function critical points on surface moduli space.

Study classifies ruled surfaces critical to Dirichlet energy.

Study on critical faces convergence in a Poisson point process.

Noninjective monodromy found in polynomial critical point tracking.

New groups found with critical exponents close to but less than max.

Actor-critic methods can achieve incredible performance on difficult reinforcement learning problems, but they are also prone to instability. This is partly due to the interaction between the actor and critic during learning, e.g., an inaccurate step taken by one of them might adversely affect the other and destabilize…

Paper finds critical metrics with pinched curvature are geodesic balls.

Critical hypersurfaces with boundary have unique shapes and properties.

The paper studies the smoothness of critical points of variational integrals on Hessian spaces.

We give an alternative proof of that a critical knot of a Morse-Bott function $f: S^3 \rightarrow \mathbb{R}$ is a graph knot where the critical set of $f$ is a link in $S^3$. Our proof inducts on the number of index-1 critical knots of $f$.

In this paper we investigate complete critical metrics of the $L^{2}$-norm of the scalar curvature. We prove that any complete critical metric with positive scalar curvature has constant scalar curvature and we characterize critical metrics with nonnegative scalar curvature in dimension three and four.

Study shows wealth distribution tails near criticality are not universal.

New rigidity results for critical metrics of a quadratic curvature functional.