Critical nets in k-space have bounded edge lengths and vertices.
problem Understanding the structure and constraints of critical nets in k-dimensional space.
method Analyzing the properties of critical nets under fixed leaf positions and constraints.
result The total length of edges not incident with 1-valent vertices is bounded, and the degree and number of vertices are also bounded.
Critical trajectories in a sphere are found for a specific bending functional.
problem Finding closed trajectories in a sphere for a specific bending functional.
method Existence of infinitely many closed trajectories shown for a given Lagrange multiplier.
result Existence of closed trajectories dependent on a pair of relatively prime natural numbers.
A simple text model shows word lengths follow Zipf's law.
problem Understanding word statistics in large language models.
method A non-linguistic model of text with independent symbol draws.
result Word lengths follow a geometric distribution and Zipf's law.
The paper proves critical lengths for loops on positively curved manifolds.
problem Existence and properties of closed geodesics on positively curved manifolds.
method Analyzing critical lengths of loops and geodesics on Riemannian manifolds with positive sectional curvature.
result Critical lengths attain their maximal value of 2π only for the round metric on the n-sphere.
Critical graphs of quadratic differentials equidistribute in moduli space.
problem Distribution of critical graphs in moduli space.
method Study of Jenkins-Strebel differentials and their critical graphs.
result Critical graphs equidistribute to the Kontsevich measure.
New proofs of Cheeger-like inequalities for coexact 1-forms on hyperbolic manifolds.
problem Cheeger-like inequalities for coexact 1-forms on hyperbolic manifolds.
method Properties of magnetic geodesic flow and behavior at Mañé's critical energy level.
result Improved Cheeger constants and volume dependence in proofs.
The paper studies critical points and flows of a G2-Hilbert functional on manifolds with circle actions.
problem Critical points and flows of the G2-Hilbert functional on manifolds with S1-actions. method Analysis of S1-invariant G2-structures, reduction to a 6-dimensional quotient, and derivation of a negative L2-gradient flow. result The unnormalized flow admits only trivial stationary configurations: flat connection, scalar-flat base metric, and constant fiber length.
Analyticity of critical points for O'Hara's knot energies proved.
problem Analyzing the regularity of critical points for O'Hara's knot energies.
method Cauchy's method of majorants and a Möbius energy-inspired gradient decomposition.
result Smooth critical points of O'Hara's knot energies are analytic.
The paper studies how curves evolve under area constraints and converges to a critical point.
problem Evolution of plane curves with fixed area under elastic energy gradient.
method Local and global existence of the flow, simplicity assumption, Łojasiewicz--Simon inequality.
result The evolving curve's length remains bounded and converges to a critical point.
The paper connects geodesic nets to distance function critical points.
problem Understanding the relationship between geodesic nets and distance function critical points.
method Established a relationship between geodesic nets and critical points of the distance function.
result Bounded the number of balanced points and the length of certain minimizing geodesic nets.
Decouples critic chunk length from policy to improve policy reactivity and performance.
problem Bootstrapping bias and difficulty in extracting optimal policies from chunked critics.
method Optimizes policy against a distilled critic for partial action chunks, allowing shorter chunks for policy.
result Reliably outperforms prior methods on long-horizon offline goal-conditioned tasks.
Unified framework for critical scaling of inverse temperature in self-attention.
problem Conflicting inverse-temperature laws for long-context self-attention.
method Counting gaps and defining an upper-tail accumulation scale.
result Critical inverse-temperature scale determined by gap-counting function.
In 1974, Gehring posed the problem of minimizing the length of two linked curves separated by unit distance. This constraint can be viewed as a measure of thickness for links, and the ratio of length over thickness as the ropelength. In this paper we refine Gehring's problem to deal with links in a fixed link-homotopy …
Study curves evolving on hypersurfaces with free boundaries, preserving length.
problem Evolution of curves on hypersurfaces with free boundaries.
method Nonlocal evolution equation with nonlinear boundary conditions, short-time existence, uniqueness, and parabolic energy estimates.
result Global existence and convergence to critical points proved.
After having given the general variational formula for the functionals indicated in the title, the critical points of the integral of the equi-affine curvature under area constraint and the critical points of the full-affine arc-length are studied in greater detail.
We study metric and analytic properties of generalized lemniscates E_t(f)={z:ln|f(z)|=t}, where f is an analytic function. Our main result states that the length function |E_t(f)| is a bilateral Laplace transform of a certain positive measure. In particular, the function ln|E_t(f)| is convex on any interval free of cri…
Study curves evolving by gradient flow of elastic energy, proving existence, smoothing, and convergence.
problem Evolution of curves with fixed length and clamped boundary conditions.
method Negative L2-gradient flow of elastic energy, existence, parabolic smoothing, constrained Lojasiewicz-Simon gradient inequality. result Convergence to a critical point as time tends to infinity.
We prove upper bounds for the number of critical points in semistable symplectic Lefschetz fibrations. We also obtain a new lower bound for the number of nonseparting vanishing cycles in Lefschetz pencils, and reprove the known lower bounds for the commutator lengths of Dehn twists.
The elastic flow of curves converges smoothly to a critical point.
problem Smooth convergence of elastic flow of curves.
method Application of Lojasiewicz-Simon inequality.
result Smooth convergence to a critical point.
The paper proves stability of critical points for conformally invariant Lagrangians.
problem Stability of critical points for conformally invariant Lagrangians under weak convergence.
method Upper-semi-continuity of Morse index plus nullity established for critical points.
result The sum of Morse indices and nullity is bounded from above by the sum of the Morse indices plus the nullity of the weak limit and bubbles.
Relative cup-length defined for non-Morse functions on manifolds.
problem Defining a lower bound on critical points of non-Morse functions.
method Using local Morse cohomology and cohomology of isolating neighborhoods.
result A lower bound on critical points stronger than absolute cup-length.
The paper classifies and analyzes the stability of elastic curves with fixed endpoints.
problem Classification and stability of pinned elasticae.
method Critical points of the length-penalized elastic bending energy among planar curves with fixed endpoints.
result Explicit parametrization and classification of all critical points with a threshold parameter \(\hatλ \simeq 0.70107\).
Study of closed trajectories in hyperbolic plane with specific curvature constraints.
problem Critical trajectories in hyperbolic plane for a specific energy function.
method Classification of critical trajectories based on momentum causal character, proof of existence of closed trajectories.
result Existence of countably many closed trajectories with time-like momentum.
Study pinches the length of surfaces in a 3D ball.
problem Characterizing minimal surfaces with boundary constraints.
method Pinching condition on second fundamental form.
result Flat equatorial disk and critical catenoid are unique.
The paper calculates minimal ribbonlength for various knots.
problem Tabulation of knots based on energy criterion.
method Direct method for computing minimal ribbonlength.
result Computed minimal ribbonlength for specific knots.
New method simplifies ideal curve flow with length constraint.
problem Analyzing ideal curve flow with length constraint.
method Introduced length constraint to simplify sixth order curvature flow.
result Flow exists for all time and converges to a round circle.
Kernel networks' stability edge linked to Fisher Information singularity.
problem Understanding the stability edge in high-capacity kernel Hopfield networks.
method Statistical manifold analysis and Riemannian geometry.
result The Ridge of Optimization corresponds to the Edge of Stability, revealing a dual equilibrium.
Counting periodic geodesics of bounded length and commutator structure on hyperbolic surfaces.
problem Counting periodic geodesics with specific commutator structure.
method Reduction to counting critical realizations of trivalent graphs.
result Asymptotic count of geodesics with bounded length and commutator structure.
We present new computations of approximately length-minimizing polygons with fixed thickness. These curves model the centerlines of "tight" knotted tubes with minimal length and fixed circular cross-section. Our curves approximately minimize the ropelength (or quotient of length and thickness) for polygons in their kno…
Short note on upper bounds for loop homology classes.
problem Critical values of homology classes in loop spaces of manifolds.
method Analysis of Riemannian and Finsler metrics with positive Ricci curvature.
result Upper bounds for shortest closed geodesics on manifolds.
Breaks down natural image complexity using critical phenomena.
problem Learning the distribution of natural images is challenging.
method Mapped images into a hierarchy of bitplanes, integrating criticality and stochastic processes.
result Generated large, natural-looking images without hidden units.
Study finds minimal length networks connecting three points in Heisenberg group.
problem Finding minimal length networks connecting three points in the Heisenberg group.
method Proved existence of minimal horizontal triods, formulated curve shortening flow, used numerical experiments.
result Characterized and deformed minimal horizontal triods into critical points for length functional.
Let M be a Riemannian 2-sphere. A classical theorem of Lyusternik and Shnirelman asserts the existence of three distinct simple non-trivial periodic geodesics on M. In this paper we prove that there exist three simple periodic geodesics with lengths that do not exceed 20d, where d is the diameter of M. We a…
A Riemannian or Finsler metric on a compact manifold M gives rise to a length function on the free loop space ΛM, whose critical points are the closed geodesics in the given metric. If X is a homology class on ΛM, the minimax critical level cr(X) is a critical value. Let M be a sphere of dimension >2, and fix a metric …
Proves smooth critical points of Möbius energy are analytic.
problem Analyticity of critical points of Möbius energy.
method Cauchy's method of majorants and gradient decomposition.
result Smooth critical points of Möbius energy are analytic.
The paper finds curves minimizing elastic energy pinned at endpoints.
problem Finding curves that minimize elastic energy with fixed endpoints.
method Applying the shooting method to identify and classify critical points.
result Critical points consist of wavelike elasticae, and minimizers have no loops or interior inflection points.
The paper proves that every surface has a spine of minimal area.
problem Whether every closed Riemannian manifold has a spine of minimal area.
method Introduced spine systole, studied minimal spines, classified them on flat tori.
result Proved the existence of a spine of minimal area on every surface.
The ropelength problem asks for the minimum-length configuration of a knotted diameter-one tube embedded in Euclidean three-space. The core curve of such a tube is called a tight knot, and its length is a knot invariant measuring complexity. In terms of the core curve, the thickness constraint has two parts: an upper b…
A new sampler tackles critical phenomena by leveraging scale invariance.
problem Scale invariance at criticality causes sampling difficulties in Monte Carlo simulations.
method RiGCS combines MLMC-HB with generative models to improve sampling efficiency.
result RiGCS achieves significantly higher effective sample size than existing methods.
A comparison of SLDS and LSTM for pedestrian behavior prediction shows SLDS works better with shorter sequences.
problem Time-critical pedestrian behavior prediction in autonomous vehicles.
method Comparison of a switching linear dynamical system (SLDS) and a three-layered bi-directional LSTM neural network.
result SLDS achieves higher accuracy with shorter sequences (10 samples) compared to LSTM's 80% accuracy with 100 samples.
Study critical exponents on hyperbolic surfaces with long boundaries using Weil-Petersson measures.
problem Analyzing critical exponents on hyperbolic surfaces with long boundaries.
method Using spine graph construction and comparing normalized Weil-Petersson and Kontsevich measures.
result Asymptotic convergence-in-mean result of normalized Weil-Petersson measures to normalized Kontsevich measures.
Geodesic nets on flat spheres are studied using Gauss-Bonnet theorem.
problem Existence and non-existence of specific geodesic nets on flat spheres.
method The theorem of Gauss-Bonnet is applied to demonstrate results.
result Existence and non-existence of geodesic nets on regular doubled polygons.
Study electric field and potential of torus knots, focusing on z-axis.
problem Analyze electric field and potential of torus knots.
method Parametrize torus knots, use symmetry, numerical methods, contour integration.
result Electric field is zero only at the origin, extreme points analyzed.
For strong exact magnetic fields the action functional (i.e., the length plus the linear magnetic term) is not bounded from below on the space of closed contractible curves and the lower estimates for critical levels are derived by using the principle of throwing out cycles. It is proved that for almost every energy le…
New model predicts ICU patient stays more accurately.
problem Efficient ICU bed allocation under resource constraints.
method Temporal Pointwise Convolutional Networks (TPC) combining temporal and pointwise convolutions.
result Significant performance improvements over LSTM and Transformer models.
New model predicts ICU patients' stay duration efficiently.
problem Efficient ICU bed allocation under resource constraints.
method Temporal Pointwise Convolution (TPC) model combining temporal and pointwise convolutions.
result Significant performance improvements over LSTM and Transformer models.
Quantum field theory connects deep neural networks to criticality.
problem Understanding the criticality and training dynamics of deep neural networks.
method Constructing quantum field theory for deep neural networks, computing corrections to correlation functions.
result Found precise analogy with O(N) vector model, providing corrections to correlation length. TREP learns pedestrian trajectories efficiently without needing full datasets.
problem Learning fixed-length vector representations of variable-length trajectories.
method Actor-critic sequence-to-sequence autoencoder with spatial-aware objective function.
result TREP efficiently learns trajectory representations without needing full datasets.