Higher curvature corrections to Bianchi identities conflict with exceptional generalised geometry.
problem Incompatibility of higher curvature corrections with exceptional generalised geometry.
method Analysis of higher derivative corrections to Bianchi identities in supergravity and M-theory.
result Higher curvature corrections imply an L∞ algebra for the gauge field, terminating at level ten. Geometric formalism views optimization algorithms as discrete connections, revealing their algebraic curvature and flatness properties.
problem Understanding and optimizing the behavior of iterative optimization algorithms.
method Introducing a geometric and operator-theoretic formalism where optimization algorithms are encoded by coupled channels (drift and diffusion) whose algebraic curvature measures the deviation from ideal reversibility.
result Flat connections correspond to methods whose updates commute up to higher order, achieving minimal numerical dissipation and preserving stability.
The paper refines classical covariance asymptotics using geometric information geometry.
problem Deviation of finite-sample behavior from classical predictions in curved models.
method Develops a curvature-aware refinement by viewing parametric families as Riemannian manifolds with Fisher-Rao metric.
result Derives an \(n^{-2}\) correction to the leading \(n^{-1}I(θ)^{-1}\) covariance term for score-root estimators.
Develops a curvature-corrected tangent space method for manifold-valued data.
problem Generalizing real-valued data approximation to manifold-valued data.
method Systematic approach to developing global-geometry aware, computationally feasible approximation schemes.
result Proposes CC-tHOSVD for low-rank approximation of manifold-valued data.
Researchers define geodesic curvature for 3D sub-Riemannian curves, improving distance calculations.
problem Improving sub-Riemannian distance calculations for 3D curves.
method Introducing geodesic curvature kζ for smooth horizontal curves in 3D contact sub-Riemannian manifolds. result Geodesic curvature appears as the first corrective term in the Taylor expansion of sub-Riemannian distance.
Defines rho numbers for metrics with positive scalar curvature.
problem Computing index classes for manifolds with boundary and cuspidal parabolics.
method Uses orbital integrals, delocalized eta invariants, and index theorems.
result Defines higher rho numbers for metrics with positive scalar curvature.
Geometrically refines Cramér-Rao bound using extrinsic manifold curvature.
problem Improving estimator efficiency in non-asymptotic settings.
method Incorporates curvature-aware corrections based on extrinsic geometry of statistical model manifold.
result Meaningful tightening of estimator variance bounds.
Expectation Propagation (EP) provides a framework for approximate inference. When the model under consideration is over a latent Gaussian field, with the approximation being Gaussian, we show how these approximations can systematically be corrected. A perturbative expansion is made of the exact but intractable correcti…
Develops a novel fast bootstrap for dependent data with higher-order accuracy.
problem Estimation of parametric and semi-parametric models for dependent data.
method i.i.d. resampling of smoothed moment indicators, asymptotic refinements under mild assumptions.
result Higher-order correct asymptotic confidence distributions and confidence intervals.
We propose a formulation of a Lorentzian quantum geometry based on the framework of causal fermion systems. After giving the general definition of causal fermion systems, we deduce space-time as a topological space with an underlying causal structure. Restricting attention to systems of spin dimension two, we derive th…
We report analytical results for the development of the viscous fingering instability in a cylindrical Hele-Shaw cell of radius a and thickness b. We derive a generalized version of Darcy's law in such cylindrical background, and find it recovers the usual Darcy's law for flow in flat, rectangular cells, with correctio…
We study heterotic supergravity at O(α′), first described in detail in 1989 by Bergshoeff and de Roo. In particular, we discuss an ambiguity of a connection choice on the tangent bundle. It is well known that at O(α′) the Hull connection gives a consistent supergravity theory with supersymmetry …
The paper proves inequalities for scalar curvature on various manifolds.
problem Proving inequalities for scalar curvature on different types of manifolds.
method Analyzing Riemannian manifolds with nonnegative Ricci curvature and applying Cohn-Vossen-type inequalities.
result Sharp asymptotic scalar-curvature flux upper bound of 8π in dimension three.
Quantum codes on hyperbolic lattices outperform Euclidean ones with higher rates and lower overhead.
problem Improving quantum error correction performance with hyperbolic lattices.
method Unified framework using Hyperbolic Cycle Basis algorithm for CSS codes construction and benchmarking.
result Achieved higher encoding rates and lower qubit overhead in hyperbolic quantum error correction codes.
Stability of capillary hypersurfaces with higher order mean curvature.
problem Stability of capillary hypersurfaces with constant higher order mean curvature.
method Generalization of classical stability theory for capillary hypersurfaces.
result Results on stability for capillary hypersurfaces with higher order mean curvature.
New CRB derived for curved models using extrinsic geometry.
problem Estimate curved statistical families accurately.
method Vector generalization of CRB with curvature correction using SDP and SOS relaxations.
result Directional curvature correction provides more accurate estimation.
The purpose of this erratum is to correct a mistake in the proof of Theorem 4.1 of our paper \cite{CF}.
The paper provides estimates for higher-order Ricci curvature along Kähler-Ricci flows.
problem Estimating higher-order curvature along Kähler-Ricci flows on compact Kähler manifolds.
method Proving uniform bounds for Ricci curvature and scalar curvature in various orders and norms.
result A geometric obstruction causes a specific third-order derivative of Ricci curvature to blow up at rate et/2. Corrected a mistake in a paper about minimal surfaces.
problem A mistake in a paper about complete minimal surfaces with finite total curvature.
method No new method introduced, just correcting an error.
result Corrected a mistake in a previously published paper.
The paper generalizes Riemann curvature for manifolds with discontinuous metrics.
problem Generalizing Riemann curvature for manifolds with discontinuous metrics.
method Proposes a generalized Riemann curvature tensor combining angle defects and jumps in second fundamental forms.
result The generalized curvature tensor approximates classical curvature for smooth approximations of metrics.
New bounds for neural networks on curved manifolds improve generalization.
problem Existing generalization theories fail to account for non-Euclidean manifold structures.
method Derive covering number bounds incorporating manifold-specific properties like curvature.
result Sharp Rademacher complexity bounds for neural networks on compact manifolds.
In this paper, we use localization algebras to study higher rho invariants of closed spin manifolds with positive scalar curvature metrics. The higher rho invariant is a secondary invariant and is closely related to positive scalar curvature problems. The main result of the paper connects the higher index of the Dirac …
Unified geometric interpretation of statistical estimation inequalities.
problem Curvature corrections in parametric statistical estimation.
method Cartan-geometric jet bundle formulation and jet prolongations.
result Unified geometric interpretation of higher-order information inequalities.
New proof shows symmetry for certain curved surfaces in higher dimensions.
problem Understanding symmetries in curved surfaces evolving over time.
method Generalized previous result to higher dimensions, proving symmetry condition.
result Uniformly 3-convex translating solitons must have SO(n−1) symmetry. Improved GEC models use scored data from large pretraining to outperform.
problem Addressing data sparsity in Grammatical Error Correction.
method Derive example-level scores from a smaller, higher-quality dataset and incorporate delta-log-perplexity into training schedules.
result Models trained on scored data achieve state-of-the-art results.
We construct new examples of algebraic curvature tensors so that the Jordan normal form of the higher order Jacobi operator is constant on the Grassmannian of subspaces of type (r,s) in a vector space of signature (p,q). We then use these examples to establish some results concerning higher order Osserman and highe…
We prove a general relative higher index theorem for complete manifolds with positive scalar curvature towards infinity. We apply this theorem to study Riemannian metrics of positive scalar curvature on manifolds. For every two metrics of positive scalar curvature on a closed manifold and a Galois cover of the manifold…
This note corrects conditions in Proposition 3.4 and Theorem 5.2(ii) and comments on imprecisions in Propositions 4.2 and 4.4 in Fissler and Ziegel (2016).
This note corrects a mistake in the original book in the evolution equations of total curvature for the curve-shrinking flow in an ambient Ricci Flow. The resulting upper bound for the evolution of total curvature is an exponential bound in time. The change involves the multiplicative constant. Here we show that it dep…
The paper proves an obstruction to immersed curves with specific curvature.
problem Existence of immersed curves with prescribed curvature.
method Proofs and corrections of submitted manuscript.
result An obstruction to the existence of immersed curves of prescribed curvature.
Proves higher regularity for anisotropic inverse mean curvature flow.
problem Higher regularity of solutions to anisotropic inverse mean curvature flow.
method Proves Harnack estimate and constructs smooth solutions from C1 initial sets. result Smooth solutions become smooth outside a compact set.
Study examines corrections to heterotic geometry on SU(3) manifolds.
problem Analyzing α′2 corrections to heterotic supersymmetry algebra. method Derives integrability condition and pure gauge correction from graviton equation of motion.
result Curvature of tangent bundle connection acquires a (0,2) component, disrupting semi-classical intuition.
Study loop corrections in random feature models affecting training and test errors.
problem Analyzing loop corrections in random feature models to understand training and test errors.
method Statistical physics and effective field theory approach to study loop corrections.
result Derived loop corrections to training error, test error, and generalization gap.
Introduce a variance-weighted batch distribution for diverse sampling in diffusion models.
problem Independent sampling in diffusion models.
method Introduce a variance-weighted batch distribution.
result Sampler with a transparent probabilistic target.
Study of Brown--York mass for four-dimensional asymptotically flat manifolds.
problem Calculating mass for hypersurfaces in four-dimensional asymptotically flat manifolds.
method Intrinsic definition of mean curvature, expansion analysis for large uniformly convex hypersurfaces.
result Shape-dependent correction to ADM mass for nearly round surfaces vanishes under certain conditions.
An updated version with a few corrections.
Study proves rigidity theorems for ancient solutions to mean curvature flow with convex image.
problem Rigidity of ancient solutions to mean curvature flow with convex Gauss image.
method Refined curvature estimates.
result Better rigidity theorems for ancient solutions in higher codimension.
The study proves a neighborhood theorem for mean curvature flow in higher dimensions.
problem Proving a canonical neighborhood theorem for mean curvature flow in higher dimensions.
method Proved a canonical neighborhood theorem for mean curvature flow of compact submanifolds in RN with a pinching condition. result Proved a canonical neighborhood theorem for mean curvature flow in dimensions n≥5. It has been pointed out to the author by David Glickenstein that the proof of the (closely related) Lemmas 1.2 and 3.2 in the title paper is incorrect. The statements of both Lemmas are correct, and the purpose of this note is to give a correct argument. The argument is of some interest in its own right.
This paper extends a 3D result to higher dimensions for manifolds with positive curvature.
problem Proving higher-dimensional manifolds with positive curvature operator and strictly convex boundary are diffeomorphic to the Euclidean disk.
method Using the positive curvature operator and strictly convex boundary conditions to deduce the manifold's diffeomorphism to the Euclidean disk.
result Compact n-manifolds with positive curvature operator and strictly convex boundary are diffeomorphic to the standard n-dimensional Euclidean disk.
Study on uniqueness of hypersurfaces in hyperbolic space with constant mean curvature.
problem Uniqueness of hypersurfaces with constant higher order mean curvature in hyperbolic space.
method Generalization of Bernstein theorem and proof of Bernstein type results for immersed hypersurfaces.
result Rigidity of horospheres and equidistant spheres in terms of their higher order mean curvatures.
Study finds infinitely many non-radial solutions for negative scalar curvature in higher dimensions.
problem Prescribing scalar and boundary mean curvature in a ball with negative scalar curvature.
method Analyzes the existence of infinitely many non-radial positive solutions in dimensions 5 and above.
result First existence result for negative scalar curvature in higher dimensions.
Study proves mean curvature flows on spheres in higher dimensions.
problem Existence of mean curvature flows on spheres.
method Generalized previous results to higher dimensions, proving existence of flows.
result Existence of infinitely many eternal weak mean curvature flows in Sn+1 connecting specific hypersurfaces. Examines algebraic conditions for positive sectional curvature in 4D and higher.
problem Determining when the sectional curvature of a Riemannian manifold is positive.
method Analyzes algebraic conditions for sectional positivity in 4D and higher dimensions.
result Characterizes a dense open subset of operators in 4D for sectional positivity.
Recently deep neural networks have been successfully used for various classification tasks, especially for problems with massive perfectly labeled training data. However, it is often costly to have large-scale credible labels in real-world applications. One solution is to make supervised learning robust with imperfectl…
We derive pointwise curvature estimates for graphical mean curvature flows in higher codimensions. To the best of our knowledge, this is the first such estimates without assuming smallness of first derivatives of the defining map. An immediate application is a convergence theorem of the mean curvature flow of the graph…
Defines new Roe algebras for cylindrical spaces, solving metric curvature problems.
problem Existence and classification of metrics with positive scalar curvature on spaces with cylindrical ends.
method Variant of Roe algebras for cylindrical spaces, relating to relative higher index theory.
result Defines higher rho-invariants and provides a concise proof of a related result.
Generalized Huber's theorem for specific manifold curvature types.
problem Finite point conformal compactification on manifolds with certain curvature integrability.
method Generalization of Huber's theorem to higher dimensions with $L^ rac{n}{2}$ integrable Ricci curvatures.
result Validated finite point conformal compactification theorem for new class of manifolds.