Paper proves unique energy-minimizing curves in constrained spaces.
problem Uniqueness of energy-minimizing curves in constrained spaces.
method Investigated energy-minimizing curves with fixed endpoints in a constrained space.
result Proved that the set of points for which the energy-minimizing curve is not unique has no interior points.
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 \(\hatλ \simeq 0.70107\).
We investigate the evolution of open curves with fixed endpoints under the curve shortening flow, which evolves curves in proportion to their curvature. Using a distance comparison of Huisken, we determine the long-term behavior of open curves with fixed endpoints evolving in certain convex domains on surfaces of const…
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.
We prove that the curvature flow of an embedded planar network of three curves connected through a triple junction, with fixed endpoints on the boundary of a given strictly convex domain, exists smooth until the lengths of the three curves stay far from zero. If this is the case for all times, then the evolution exists…
Let M be a possibly non compact smooth manifold. We study genericity in the C^k-topology (3<=k<=+infty) of nondegeneracy properties of semi-Riemannian geodesic flows on M. Namely, we prove a new version of the Bumpy Metric Theorem for a such M and also genericity of metrics that do not possess any degenerate geodesics …
The study of smoothing arcs and curves on surfaces, proving tautness and arc length spectrum properties.
problem Analyzing the geometric and combinatorial effects of smoothing intersections in arcs or curves.
method Geometric and combinatorial analysis, proving tautness and arc length spectrum properties.
result Shortest arcs with self-intersections have exactly or at most one more self-intersection than the self-intersection number.
The computation of the index of the Hessian of the action functional in semi-Riemannian geometry at geodesics with two variable endpoints is reduced to the case of a fixed final endpoint. Using this observation, we give an elementary proof of the Morse Index Theorem for Riemannian geodesics with two variable endpoints,…
Let M be a possibly noncompact manifold. We prove, generically in the C^k-topology (k=2,...,\infty), that semi-Riemannian metrics of a given index on M do not possess any degenerate geodesics satisfying suitable boundary conditions. This extends a result of Biliotti, Javaloyes and Piccione for geodesics with fixed endp…
Curve diffusion flow straightens curves with endpoints on intersecting lines.
problem Straightening open curves with endpoints on intersecting lines.
method Curve diffusion flow with mixed boundary conditions.
result The curve converges to a circular arc of the same length.
New Virasoro-like structures for circle diffeomorphisms with breaks.
problem Constructing Virasoro-like extensions for non-smooth diffeomorphisms.
method Explicit construction of central extensions of Lie groupoids and algebroids.
result Explicit nontrivial central extensions of Lie groupoids and algebroids for broken diffeomorphisms.
It is known that shape injectivity implies homotopical Hausdorff and that the converse does not hold, even if the space is required to be a Peano continuum. This paper gives an alternative definition of homotopical Hausdorff inspired by a new topology on the set of fixed endpoint homotopy classes of paths. This version…
Study a market with uncertain informed traders, finding price impact depends on both asset value and informed trader count distribution.
problem Uncertain participation of informed traders in a market with limit orders.
method Characterized equilibrium by a fixed point integral equation, analyzed large order asymptotics, solved numerically.
result Equilibrium price impact depends on both asset value and distribution of informed traders, not just expected number of informed traders.
Study spectral flow on a warped cylinder with special boundary conditions.
problem Analyzing spectral flow on a warped cylinder with specific boundary conditions.
method Complexifying the twisting bundle, diagonalizing the orthogonal twist, and regrouping conjugate and reflection-paired blocks.
result Explicit formula for RO(O(2))-valued spectral flow, refining ordinary spectral flow. The geodesic equation for the right invariant L2-metric (which is a weak Riemannian metric) on each Virasoro-Bott group is equivalent to the KdV-equation. We prove that the corresponding energy functional, when restricted to paths with fixed endpoints, has no local minima. In particular solutions of KdV don't define…
ACP-UCB1 ranks arms based on upper-tail performance, improving stochastic bandit algorithms.
problem Stochastic bandit algorithms often favor arms with strong upper-tail performance, which is not well-addressed by classical mean-reward criteria.
method ACP-UCB1 combines an adaptive conformal estimate of the upper endpoint with a UCB-type optimism bonus.
result ACP-UCB1 achieves logarithmic upper-quantile regret with per-arm contribution \(O(
icefrac{\log n}{Δ_j^{\mathrm{ACP}}})\).
Prove Gromov's Euclidean endpoint C0 rigidity conjecture for positive mass theorem.
problem Prove Gromov's Euclidean endpoint C0 rigidity conjecture for positive mass theorem. method Prove Gromov's Euclidean endpoint C0 rigidity conjecture for positive mass theorem. result Prove Gromov's Euclidean endpoint C0 rigidity conjecture for positive mass theorem. Optimizer memory affects learning rate sensitivity in shuffle order, impacting fine-tuning noise.
problem Optimizer memory affects the learning rate sensitivity in shuffle order, leading to fine-tuning noise.
method Isolated the mechanism of fixed-clock optimizer memory affecting the learning rate sensitivity in shuffle order, deriving a fit-free way to size the noise.
result Fixed-clock optimizers like AdamW produce a larger first-order noise channel compared to memoryless optimizers, affecting fine-tuning comparisons.
The study bounds slopes for Dehn fillings of two-bridge knots with hyperbolic representations.
problem Bounding slopes for Dehn fillings of two-bridge knots with hyperbolic representations.
method Combining the Riley polynomial with Khoi's surgery-slope formula, and analyzing meridian and longitude translation parameters.
result The set of surgery slopes admitting hyperbolic PSL(2,R) representations is bounded. Characterizes curves for minimal surfaces in de Sitter space.
problem Minimal surfaces in de Sitter space.
method Variational problem to find critical points of center of mass.
result Curves are critical points of center of mass.
The configuration space of the mechanism of a planar robot is studied. We consider a robot which has n arms such that each arm is of length 1+1 and has a rotational joint in the middle, and that the endpoint of the k-th arm is fixed to Ren2(k−1)πi. Generically, the configuration space is diffeomorphic t…
In Carnot-Caratheodory or sub-Riemannian geometry, one of the major open problems is whether the conclusions of Sard's theorem holds for the endpoint map, a canonical map from an infinite-dimensional path space to the underlying finite-dimensional manifold. The set of critical values for the endpoint map is also known …
Let S be an n-punctured sphere, with n≥3. We prove that (3n) is the maximum size of a family of pairwise non-homotopic simple arcs on S joining a fixed pair of distinct punctures of S and pairwise intersecting at most twice. On the way, we show that a square annular diagram A has a corner on …
The paper introduces walks with jumps for modeling neuron activity in hyperbolic space.
problem Encoding neuron activity sequences in hyperbolic space.
method Introducing walks with jumps in hyperbolic geometry to model neuron activity.
result Endpoints of walks with jumps do not fully encode the sequence of jump times.
We consider a family of variational problems on a Hilbert manifold parameterized by an open subset of a Banach manifold, and we discuss the genericity of the nondegeneracy condition for the critical points. Based on an idea of B. White, we prove an abstract genericity result that employs the infinite dimensional Sard--…
FlexServe simplifies deployment of PyTorch models as REST endpoints.
problem Lack of control over model evolution and strict security requirements in operational environments.
method Developed FlexServe, a library to deploy multi-model ensembles with flexible batching.
result Rapid deployment of PyTorch models without intermediate transformations.
We present proofs of basic results, including those developed by Harold Bell, for the plane fixed point problem: does every map of a non-separating plane continuum have a fixed point? Some of these results had been announced much earlier by Bell but without accessible proofs. We define the concept of the variation of a…
It is well known that plane curves with the same endpoints are homotopic. An analogous claim for plane curves with the same endpoints and bounded curvature still remains open. In this work we find necessary and sufficient conditions for two plane curves with bounded curvature to be deformed, one to another, by a contin…
Study on OI surfaces with unique geometric properties.
problem Characterizing and classifying ortho-integral surfaces.
method Analyzing geodesic arcs and cosh-length properties.
result Infinitely many commensurability classes of OI surfaces arise as topologies vary.
Given a rank-two sub-Riemannian structure (M,Δ) and a point x0∈M, a singular curve is a critical point of the endpoint map F:γ↦γ(1) defined on the space of horizontal curves starting at x0. The typical least degenerate singular curves of these structures are called \emph{regular singular curves}; the…
Prognostic scores improve logistic regression analysis in RCTs with binary outcomes.
problem Non-collapsibility in logistic regression analysis of RCTs with binary endpoints.
method Prognostic score adjustment using AI predictions to address non-collapsibility.
result Prognostic score adjustment increases power or reduces sample size for estimating conditional odds ratios.
New models for short rates show longer periods at higher rates.
problem Modeling longer periods of higher interest rates.
method Developed a class of time-homogeneous one-factor Markov diffusion models with specific boundary conditions.
result Explicit expressions for bond prices and transition densities in new probability measure.
New method estimates Schrödinger bridge potentials via empirical risk minimization.
problem Estimating Schrödinger bridge potentials from samples.
method Rewriting Schrödinger system as a fixed-point equation and estimating the potential via empirical risk minimization.
result Uniform concentration of empirical risk around population counterpart under sub-Gaussian assumptions.
Burq-Gérard-Tzvetkov and Hu established Lp estimates (2≤p≤∞) for the restriction of eigenfunctions to submanifolds. The estimates are sharp, except for the log loss at the endpoint L2 estimates for submanifolds of codimension 2. It has long been believed that the log loss at the endpoint can be remov…
For a Riemannian manifold (M,g) with strictly convex boundary ∂M, the lens data consists in the set of lengths of geodesics γ with endpoints on ∂M, together with their endpoints (x−,x+)∈∂M×∂M and tangent exit vectors (v−,v+)∈Tx−M×Tx+M. We show …
Study on Hausdorff dimension of lamination endpoints for fully irreducible automorphisms.
problem Hausdorff dimension of lamination endpoints for fully irreducible automorphisms of free groups.
method Analysis of attracting laminations and ending laminations, using properties of hyperbolic surfaces and free-by-cyclic groups.
result For fully irreducible automorphisms, the set of endpoints of the ending lamination has Hausdorff dimension 0.
The article analyzes the stability of a curve shortening flow for planar networks.
problem Stability analysis of anisotropic curve shortening flow for planar networks.
method Used Lojasiewicz-Simon gradient inequality to derive stability results.
result For initial data close to an energy minimizer, the flow exists globally and converges to a different energy minimum.
We consider two systems of curves (α1,...,αm) and (β1,...,βn) drawn on a compact two-dimensional surface M with boundary. Each αi and each βj is either an arc meeting the boundary of M at its two endpoints, or a closed curve. The αi are pairwise disjoint except for possibly sharing endpoints, and s…
Proves Morse index theorem for geodesics in conic Finsler manifolds.
problem Geodesic index theorem in conic Finsler manifolds with variable endpoints.
method Proves Morse index theorem for geodesics connecting submanifolds in a C7 manifold with a C6 conic pseudo-Finsler metric. result Establishes the Morse index theorem for geodesics in conic Finsler manifolds.
Algorithm samples constrained stochastic differential equations.
problem Sampling stochastic differential equations with complex constraints.
method Pathspace Metropolis-adjusted manifold sampling.
result Demonstrated effectiveness in various constrained conditions.
Researchers found solutions to a complex equation on spheres, overcoming a key difficulty.
problem Finding solutions to a specific equation on spheres with a background metric.
method Constructed a smooth metric invariant under antipodal map, used a noncompact family of solutions, and addressed the loss of ellipticity.
result Provided solutions to the σ2-Yamabe equation for n=27 and beyond, overcoming a main difficulty. Proves a fundamental gap lower bound for horoconvex domains in hyperbolic space.
problem Proving a fundamental gap lower bound for horoconvex domains in hyperbolic space.
method Reduces the problem to a radial-height problem, compares Dirichlet forms with angular operators, and uses Green estimates.
result Establishes a polynomial \(D^{-3}\) scale fundamental gap lower bound.
This work studies the contraction coefficients of Schrödinger bridge problems in linear systems.
problem Optimally controlling the evolution of a system's state density over time.
method Analyzes and improves the convergence rates of dynamic Schrödinger systems via geometric and control-theoretic interpretations.
result New insights into improving computation of worst-case contraction coefficients by preconditioning.
New geometric framework for positive semidefinite matrices of fixed rank.
problem Statistical analysis of positive semidefinite matrices of fixed rank.
method Introducing a manifold S(n,p)∗ with Riemannian geometry and Lie group structure. result Analytical closed forms for geodesics and Fréchet means.
Develops certificates for local population-risk increments using cross-fitted ridge calibration.
problem Certifying local population-risk increments in statistical models.
method Cross-fitted ridge calibration for linear feature classes, separating Taylor fluctuations and remainders.
result Certifies measurable updates from the same sample with penalties dependent on empirical geometry.
We consider quasifuchsian manifolds with "particles", i.e., cone singularities of fixed angle less than π going from one connected component of the boundary at infinity to the other. Each connected component of the boundary at infinity is then endowed with a conformal structure marked by the endpoints of the particle…
TA-CQR predicts regression intervals with exact coverage, splitting miscoverage between endpoints.
problem Predicting regression intervals with exact coverage under reporting constraints.
method TA-CQR uses tail allocation to parameterize the oracle, estimating the allocation by searching quantile cores and applying nonnegative additive split-conformal calibration.
result TA-CQR achieves exact finite-sample marginal coverage under exchangeability, with theoretical guarantees on calibration and length.
QGMS framework detects market endpoints using geometric patterns.
problem Identifying market endpoints in large-scale movements.
method Hybrid of geometric pattern recognition and quantitative modeling.
result Consistently identifies market endpoints before major reversals.