The paper explores gaps in curvature-related metrics and rigidity.
problem Understanding gaps in curvature-related metrics and rigidity.
method Analyzes three types of gaps: spectral, metric-rigidity, and topological-rigidity.
result Proposes open problems in the field.
Study gap-dependent regret bounds for risk-sensitive RL.
problem Risk-sensitive reinforcement learning with entropic risk measure.
method Propose cascaded gaps to adapt to problem structures, derive regret bounds.
result Exponential improvement over existing bounds in appropriate settings.
Proves magnetic gap-labelling conjecture for specific tori.
problem Magnetic spectral gap-labelling conjecture for principal solenoidal tori.
method Proves conjecture in all dimensions for principal solenoidal tori.
result Proves magnetic spectral gap-labelling conjecture.
Paper analyzes origami slope gaps and their distribution, finding a unique pattern.
problem Analyzing slope gaps in origami surfaces.
method Derived slope gap distribution of a specific origami by considering return times under the horocycle flow.
result Found a unique distribution of origami slope gaps, not a sum of scaled Hall distributions.
Random hyperbolic surfaces have nearly optimal spectral gaps.
problem Proving the nearly optimal spectral gap conjecture for random Belyi surfaces.
method Using the Brooks-Makover model, the authors show a spectral gap greater than 1/4 - c/log(n).
result A random hyperbolic surface in the Brooks-Makover model has a spectral gap greater than 1/4 - c/log(n).
Sharp fundamental gap estimate proved for convex domains on spheres.
problem Proving a sharp fundamental gap estimate for convex domains on spheres.
method Proving super log-concavity of the first eigenfunction.
result Sharp fundamental gap bound of 3D2π2 for n≥3. Paper improves volume gap between minimal submanifolds and unit spheres.
problem Volume gap between minimal submanifolds and unit spheres.
method Modified Cheng-Li-Yau coefficients and applied Cheng-Yang eigenvalue estimate for Laplacian.
result Enhanced volume gap between minimal submanifolds and unit spheres.
The article proves a conjecture about the fundamental gap for horoconvex domains in hyperbolic space.
problem Proving a conjecture about the fundamental gap for horoconvex domains in hyperbolic space.
method Establishing conformal log-concavity estimates for the first eigenfunction.
result Proves a conjecture about the fundamental gap for horoconvex domains in hyperbolic space.
Sharp gap theorem found for Yang-Mills connections in 4D.
problem Yang-Mills connections in 4D.
method Exploiting an associated Yamabe-type problem.
result Sharp conformally invariant gap theorem proved.
Kahler-Einstein metrics linked to eigenvalue gaps on Fano manifolds.
problem Existence of Kahler-Einstein metrics on Fano manifolds.
method Characterization via eigenvalue gaps of Cauchy-Riemann and Hamiltonian vector fields.
result Existence of Kahler-Einstein metrics linked to eigenvalue gaps.
Researchers compute gap distributions for saddle connection directions on specific translation surfaces.
problem Computing gap distributions for saddle connection directions on translation surfaces.
method Translation to dynamical question of return times to a transversal under the horocycle flow.
result Gap distributions have support at 0 and quadratic tail decay.
Local gaps in Ricci shrinkers depend only on dimension.
problem Understanding local properties of Ricci shrinkers.
method Proved local versions of Ricci curvature and entropy gap theorems.
result Local gaps depend only on dimension, not global entropy.
Study shows gaps in Bitcoin order book are linked to returns but only in the short term.
problem Understanding the relationship between gaps and returns in Bitcoin order books.
method Examined the dynamics of gaps and returns in a Bitcoin order book without considering long-term causation.
result The causal relationship between gaps and returns is limited to instantaneous causation.
Proves gap rigidity theorem for Hermitian symmetric spaces.
problem Gap rigidity problems in compact Hermitian symmetric spaces.
method Dual analogy to Mok's noncompact case theorem, theorem on higher dimensional submanifolds.
result Proves gap rigidity theorem for diagonal curves in tube type spaces.
Generative Adversarial Privacy (GAP) learns privacy mechanisms from data.
problem Learning optimal privacy mechanisms from data.
method Formulates privacy as a constrained minimax game between privatizer and adversary.
result GAP provides privacy guarantees against strong adversaries.
The paper introduces gapped scale-sensitive dimensions to improve learning rate bounds.
problem Improving lower bounds on rates of convergence in statistical and online learning.
method Introducing and analyzing gapped scale-sensitive dimensions for function classes.
result Gapped dimensions lead to stronger lower bounds on offset Rademacher averages.
The article explores the fundamental gap in Bakry-Emery geometry.
problem The fundamental gap in Bakry-Emery geometry.
method Recalled Bakry-Emery geometry and connected eigenvalues with boundary conditions. Showed a connection between fundamental gap and Bakry-Emery geometry.
result Presented key ideas in Andrews's and Clutterbuck's proof of the fundamental gap conjecture.
The paper calculates gap distributions for translation surfaces, focusing on the double heptagon.
problem Calculating gap distributions for translation surfaces.
method Describes a procedure to find winning holonomy vectors and applies it to the double heptagon.
result Explicitly computed gap distribution for the regular double heptagon translation surface.
Improved gap-dependent bounds for reinforcement learning with linear approximations.
problem Achieving nearly minimax-optimal performance with linear function approximation.
method Developed and analyzed the LSVI-UCB++ algorithm and its concurrent variant.
result First gap-dependent regret bound for nearly minimax-optimal algorithm LSVI-UCB++.
Computing unlinking number is usually very difficult and complex problem, therefore we define BJ-unlinking number and recall Bernhard-Jablan conjecture stating that the classical unknotting/unlinking number is equal to the BJ-unlinking number. We compute BJ-unlinking number for various families of knots and links for w…
New methods reduce bias in estimating optimality gaps for risk-averse stochastic programs.
problem Optimality gap estimation bias in risk-averse stochastic programs.
method Two independent samples, each estimating a different component of the optimality gap.
result Our method reduces bias in estimating optimality gaps for risk-averse problems.
Proves energy gap for Yang-Mills connections on compact manifolds.
problem Proving an energy gap for Yang-Mills connections.
method Used a new approach without Lojasiewicz-Simon gradient inequality.
result Proved an ${L^{ rac{n}{2}}}$-energy gap result.
Deep-Gap predicts crowdsourcing supply-demand gaps using deep learning.
problem Balancing supply and demand in mobile crowdsourcing.
method Residual learning-based deep neural networks trained on time series data and external factors.
result Deep-Gap achieves lowest forecasting errors compared to state-of-the-art methods.
Study fills the Italian pension gap with an optimal investment strategy.
problem Italian pension system reform gap between old and new pensions.
method Stochastic Optimal Control approach to fill the pension gap.
result It is possible to cover the pension gap with additional income from a pension scheme.
New conditions prevent gaps in optimal control problems.
problem Preventing gaps in optimal control problems with state constraints.
method Developed new sufficient conditions not relying on convexity.
result Derived bounds for the size of the relaxation gap.
Proves an energy gap for complex Yang-Mills equations under specific manifold conditions.
problem Proving an energy gap for complex Yang-Mills equations.
method Uses the energy gap result of pure Yang-Mills equations to derive a new result for complex Yang-Mills equations.
result Proves an energy gap for complex Yang-Mills equations under certain manifold conditions.
Federated learning studies separate client data and distribution gaps.
problem Understanding performance differences in federated learning across different datasets.
method Proposed a framework to disentangle out-of-sample and participation gaps.
result Dataset synthesis strategy is crucial for realistic simulations of federated learning generalization.
The paper proves gap theorems for Yang-Mills on manifolds with positive Yamabe.
problem Yang-Mills theory on manifolds with positive Yamabe constant.
method Extending Gursky-Kelleher-Streets results to complete manifolds.
result Equality in gap theorem described in terms of basic instanton.
Study simplicial volume for fixed fundamental groups, finding gaps.
problem Understanding simplicial volume for manifolds with fixed fundamental group.
method Relate gap problem to rationality questions in bounded (co)homology.
result Show existence of gaps in simplicial volume spectrum at zero.
Study shows fundamental gap of horoconvex domains in hyperbolic space has no positive lower bound.
problem Understanding the fundamental gap of horoconvex domains in hyperbolic space.
method Analysis of fundamental gap of geodesic balls as radius goes to infinity.
result Product of fundamental gap and square of diameter has no positive lower bound for horoconvex domains.
Study spectral gaps in hyperbolic rational homology spheres.
problem Finding spectral gaps in hyperbolic rational homology spheres.
method Construction of families of hyperbolic rational homology spheres with coexact 1-form spectral gaps.
result Provided intervals containing limit points of spectral gaps, with the rightmost interval being [0.8196, 0.8277].
New convex domains in hyperbolic space can have lower fundamental gap than constant potentials.
problem Finding convex domains with lower fundamental gap than constant potentials.
method Constructing specific convex domains and potentials with controlled eigenfunctions.
result Fundamental gap of −Δ+V can be strictly smaller than −Δ for convex domains. The paper establishes pressure gaps for manifolds with flat subtori singularities.
problem Understanding phase transitions in nonpositively curved manifolds with flat subtori.
method Derives a pressure gap criterion for closed rank 1 manifolds with specific singular sets and proves Hölder continuity of geometric potentials.
result Geometric potentials have pressure gaps and no phase transitions under certain curvature constraints.
The study finds larger gaps in mean curvature for biharmonic submanifolds in spheres.
problem Understanding gaps in mean curvature for biharmonic submanifolds.
method Analyzing proper biharmonic submanifolds with parallel mean curvature vector field in Euclidean spheres.
result Determining larger gaps in mean curvature for specific biharmonic submanifolds.
Solves clustering contradictions by high-dimensional embedding with wide gaps.
problem Kleinberg's clustering axioms are contradictory.
method Embedding in high-dimensional space with wide gaps between clusters.
result Handles clustering contradictions by design.
Francy enhances GAP for interactive discrete math with web technologies.
problem Lack of abstraction and portability in XGAP for GAP.
method Develops Francy, a graphical semantics package for GAP, integrating web technologies.
result Enhanced usability and accessibility of GAP with rich graphical environment.
Proves effective slope gaps for lattice surfaces.
problem Proving effective slope gaps for lattice surfaces.
method Proves effective slope gap distribution for square torus and general lattice surfaces.
result Effective slope gap distribution result for lattice surfaces.
GSP improves global average pooling for deep metric learning by learning weights and selecting semantic entities.
problem Improving global average pooling for deep metric learning.
method Generalized Sum Pooling (GSP) method that learns weights and selects semantic entities.
result GSP improves metric learning performance on 4 popular benchmarks.
This paper improves Q-learning bounds using reference-advantage decomposition.
problem Improving Q-learning bounds in MDPs with positive suboptimality gaps.
method Develops a novel error decomposition framework to prove gap-dependent regret bounds.
result Establishes logarithmic gap-dependent regret bounds for Q-learning.
Sharp spectral gap estimates on manifolds with integral curvature bounds.
problem Proving spectral gap estimates on manifolds with integral curvature bounds.
method Generalizing previous results to include integral curvature bounds.
result Confirms a conjecture about spectral gap estimates on manifolds with integral curvature bounds.
Abstract: Unknown status of Jacobian Conjecture, proof has a gap.
problem Status of Jacobian Conjecture
method Analysis of proof of theorem 2.1
result Proof of theorem 2.1 contains a gap
Fundamental gap vanishes for convex domains in hyperbolic space.
problem Behavior of fundamental gap in convex domains in hyperbolic space.
method Proof for Laplace operator with Dirichlet boundary conditions.
result Fundamental gap can be arbitrarily small for domains of any diameter.
Uniform spectral gap found for stable commutator length in hyperbolic 2-orbifolds.
problem Understanding stable commutator length in 3-manifolds.
method Explicit quasimorphisms for generic case, hyperbolic geometry for exceptional case.
result Explicit uniform gap of 1/36 for all orbifolds except a sphere with three cone points.
Study calculates eigenvalues and eigenfunctions for spherical triangles and finds fundamental gap behavior.
problem Understanding eigenvalues and gaps in spherical triangles.
method Explicit computation of Dirichlet eigenvalues and eigenfunctions for spherical lunes and triangles.
result Fundamental gap of spherical triangles increases as the angle of the lune decreases.
The paper proves lower bounds for Gaussian-weighted curvature integrals of self-shrinkers.
problem Proving lower bounds for Gaussian-weighted \(L^2\)-curvature integrals of self-shrinkers.
method Combining normal coordinate functions with weighted Poincaré inequalities and first-eigenvalue estimates.
result Explicit lower bounds in terms of entropy for closed self-shrinkers, leading to curvature gaps.
The paper proves gap theorems for Yang-Mills theory on specific manifolds.
problem Proving gap theorems for Yang-Mills theory on four-dimensional manifolds.
method Applying weighted Poincaré inequalities to complete manifolds.
result Obtained gap theorems on Euclidean space and characterized the BPST instanton.
We prove the Fundamental Gap Conjecture, which states that the difference between the first two Dirichlet eigenvalues (the spectral gap) of a Schrödinger operator with convex potential and Dirichlet boundary data on a convex domain is bounded below by the spectral gap on an interval of the same diameter with zero poten…
Paper introduces a method to control early classification accuracy gaps.
problem Maintaining accuracy in early classification without full input processing.
method Statistical framework for a calibrated stopping rule.
result Reduces up to 94% of timesteps while controlling accuracy gaps.