Survey of weak form's role in equation learning, parameter estimation, and coarse graining.
problem Noise robustness, accuracy, and computational efficiency in weak form applications.
method Survey and recent developments in weak form versions of equation learning, parameter estimation, and coarse graining.
result Surprising noise robustness, accuracy, and computational efficiency in weak form applications.
Survey on smooth function and form density in Riemannian Sobolev spaces.
problem Density of smooth functions and forms in Sobolev spaces on Riemannian manifolds.
method Careful examination of weak covariant derivatives and partial derivatives.
result Equivalence of weak covariant derivatives to weak partial derivatives.
This paper is devoted to give a complete unified study of several weak forms of $\ddb-$Lemma on compact complex manifolds.
The paper studies symplectic forms on projective limits of Banach bundles and their Darboux Theorem.
problem Conditions for weak symplectic forms on projective limits of Banach bundles.
method Analyzing projective sequences of Banach bundles and applying Darboux Theorem.
result Necessary and sufficient conditions for the Darboux Theorem on projective limits of Banach manifolds.
New Lie 2-algebra structure for multiplicative forms on quasi-Poisson groupoids.
problem Understanding Lie 2-algebra structures on geometric stacks.
method Construction of graded weak Lie 2-algebras from multiplicative forms and differential forms.
result Established a morphism between Lie 2-algebras and weak Lie 2-algebras of multiplicative forms.
Study confirms Indian stock market is weak form inefficient.
problem Impact of stock market efficiency on investment returns.
method Runs test, Autocorrelation test, Autoregression test on daily stock indices.
result Indian stock market is weak form inefficient and can be outperformed.
Study on Kähler manifolds proves weak decompositions and relates harmonic forms.
problem Analyzing harmonic forms on Kähler manifolds.
method Proves weak W1,2 Bott-Chern and Dolbeault decompositions. result Strict relation between W1,2 Bott-Chern harmonic forms and the W1,2 Bott-Chern decomposition. In this paper, we study general (α,β)-metrics which α is a Riemannian metric and β is an one-form. We have proven that every weak Landsberg general (α,β)-metric is a Berwald metric, where β is a closed and conformal one-form. This show that there exist no generalized unicorn metric in this class of general $(…
We detail the construction of a weak Poisson bracket over a submanifold of a smooth manifold M with respect to a local foliation of this submanifold. Such a bracket satisfies a weak type Jacobi identity but may be viewed as a usual Poisson bracket on the space of leaves of the foliation. We then lift this weak Poisson …
The paper finds the Finsler structure of Apollonian weak metric on unit disc.
problem Understanding the Finsler structure of Apollonian weak metric on the unit disc.
method Analyzing the deformation of hyperbolic Poincaré metric by a closed 1-form.
result The Apollonian weak-Finsler structure has bounded below S-curvature and flag curvature K satisfying −∞<K<−1. Weak dual pairs defined in Dirac-Jacobi geometry, proving equivalence and leaf correspondence theorems.
problem Defining and studying weak dual pairs in Dirac-Jacobi structures.
method Adopting omni-Lie algebroid approach, proving equivalence and leaf correspondence theorems.
result Existence of self-dual pairs and alternative proof of normal form theorem.
Geometric analysis proves weak KAM solutions constant under specific conditions.
problem Conditions for weak KAM solutions to be constant.
method Geometric and differential analysis of Hamilton-Jacobi equations.
result Weak KAM solutions are constant if and only if the 1-form is harmonic.
Develops analysis of weak immersions with bounded second fundamental forms in critical Sobolev space.
problem Analyzing weak immersions with bounded second fundamental forms in a critical Sobolev space.
method Develops analysis of Lipschitz immersions with bounded second fundamental forms in W2n−1,2 space. result Proves existence of C1 differential structure from weak immersions with bounded second fundamental forms. Characterizes when differential forms have weak exterior derivatives based on limiting behavior of integration over simplices.
problem Characterizing differential forms with weak exterior derivatives.
method Uses integration over simplices to characterize the limiting behavior of differential forms.
result Proves a direct analogue of the Bourgain-Brezis-Mironescu characterization for differential forms.
Equivalent bicategories constructed from action Lie groupoids.
problem Equivalence of bicategories constructed from action Lie groupoids.
method Localizing at equivariant weak equivalences, surjective submersive equivariant weak equivalences, and all weak equivalences.
result Weak equivalences between action Lie groupoids are isomorphic to compositions of nice forms of equivariant weak equivalences.
Self-test loss functions improve data-driven modeling of weak-form operators and gradient flows.
problem Challenges in selecting test functions for data-driven modeling involving weak-form operators and gradient flows.
method Introducing self-test loss functions that depend on unknown parameters and are quadratic.
result Self-test loss functions conserve energy for gradient flows and coincide with log-likelihood ratios for stochastic differential equations.
The paper classifies special hypersurfaces in space forms.
problem Investigating gradient Yamabe solitons in space forms.
method Using the weak Omori-Yau principle for the drifted Laplacian.
result Gradient Yamabe solitons are fully classified under certain conditions.
The paper explores criteria for positivity of forms and proves their strong positivity in specific cases.
problem Understanding positivity of exterior forms on complex vector spaces.
method Dimensionality reduction and criteria based on Hermitian matrices.
result Strong positivity of certain forms proven by duality.
Faster weak supervision framework using triplet methods.
problem Computational inefficiency in weak supervision models.
method Closed-form solution for latent variable models, avoiding iterative methods.
result Orders of magnitude faster than previous approaches.
Online algorithm identifies PDEs from noisy data snapshots.
problem Identifying PDEs from sequential solution snapshots.
method Combines weak-form discretization with online proximal gradient descent.
result Efficiently identifies and tracks systems with time-varying coefficients.
The Efficient Market Hypothesis has been a staple of economics research for decades. In particular, weak-form market efficiency -- the notion that past prices cannot predict future performance -- is strongly supported by econometric evidence. In contrast, machine learning algorithms implemented to predict stock price h…
The study explores new metric structures on manifolds, linking them to Einstein metrics.
problem Characterizing and understanding weak K-contact manifolds and their properties.
method Analyzing weak K-contact manifolds and their properties, including the parallel Ricci tensor and generalized Ricci soliton structures.
result Sufficient conditions for weak K-contact manifolds with specific properties to be Einstein manifolds.
Study weak super Ricci flow through neckpinch in metric measure spaces.
problem Understanding Ricci flow behavior at singularities.
method Introduce weak super Ricci flow and show conditions for continuation.
result Weak super Ricci flow properties at singularities.
The aim of this paper is to introduce new forms of the weak and Omori-Yau maximum principles for linear operators, notably for trace type operators, and show their usefulness, for instance, in the context of PDE's and in the theory of hypersurfaces. In the final part of the paper we consider a large class of non-linear…
Study on Nijenhuis tensor forms and vanishing properties.
problem Understanding the properties of Nijenhuis tensor.
method Provided strong and weak forms of the square of Nijenhuis tensor, and identified vanishing results.
result Identified new vanishing results for the square of Nijenhuis tensor.
Boosting combines weak classifiers to form highly accurate predictors. Although the case of binary classification is well understood, in the multiclass setting, the "correct" requirements on the weak classifier, or the notion of the most efficient boosting algorithms are missing. In this paper, we create a broad and ge…
For a finite dimensional symplectic manifold (M,ω) with a symplectic form ω, corresponding loop space (LM=C∞(S1,M)) admits a weak symplectic form Ωω. We prove that the loop space over $\mbr^n$ admits Darboux chart for the weak symplectic structure Ωω. Further, we show that inclusion map from the symp…
The scarcity of data annotated at the desired level of granularity is a recurring issue in many applications. Significant amounts of effort have been devoted to developing weakly supervised methods tailored to each individual setting, which are often carefully designed to take advantage of the particular properties of …
Paper develops equivariant basic cohomology for Lie groupoids.
problem Equivariant cohomology for Lie groupoids with weak actions.
method Using Kan fibrations and fiber structures, constructing models and comparing with existing theories.
result Equivariant basic cohomology theory for orbifolds and Lie groupoids.
Atiyah classes of DG manifolds of positive amplitude are invariant under weak equivalences.
problem Defining and studying Hochschild cohomology of DG manifolds of positive amplitude.
method Using poly-differential operators and derived intersection, proving invariance under weak equivalences.
result Hochschild cohomology of DG manifolds of positive amplitude is invariant under weak equivalences.
Proves harmonic coordinates for weak immersions in even dimensions.
problem Existence of harmonic coordinates for weak immersions in Sobolev spaces.
method Analyzes weak immersions in critical Sobolev spaces and uses smallness conditions on the second fundamental form.
result Global harmonic coordinates exist for weak immersions in even dimensions under certain conditions.
Regression problems assume every instance is annotated (labeled) with a real value, a form of annotation we call \emph{strong guidance}. In order for these annotations to be accurate, they must be the result of a precise experiment or measurement. However, in some cases additional \emph{weak guidance} might be given by…
Construct algorithms for Frobenius manifolds and residue pairings on Calabi-Yau varieties.
problem Construct algorithms for Frobenius manifolds and residue pairings on Calabi-Yau varieties.
method Analyze a dGBV algebra and introduce weak primitive forms.
result Explicit algorithms for Frobenius manifolds and residue pairings.
We show that any isometric immersion of a flat plane domain into R3 is developable provided it enjoys the little Hölder regulairty c1,2/3. In particular, isometric immersions of local C1,α regularity with α>2/3 belong to this class. The proof is based on the existence of a weak notion of second …
ASTRA uses unlabeled data and weak rules to train deep models effectively.
problem Learning with weak supervision rules is challenging due to their heuristic and noisy nature.
method ASTRA framework that considers contextualized representations and pseudo-labels for unlabeled data, and a rule attention network to aggregate labels.
result Significant improvements over state-of-the-art baselines on text classification benchmarks.
The study proves a theorem on Riemannian manifolds for wedge products of weakly convergent differential forms.
problem Analyzing the limiting behavior of wedge products of weakly convergent differential forms on Riemannian manifolds.
method Formulating and proving compensated compactness theorems for wedge products of differential forms on closed Riemannian manifolds.
result The theorem generalizes the div-curl lemma for vectorfields and applies to critical regularity exponents.
Conformal Killing forms are a natural generalization of conformal vector fields on Riemannian manifolds. They are defined as sections in the kernel of a conformally invariant first order differential operator. We show the existence of conformal Killing forms on nearly Kaehler and weak G_2-manifolds. Moreover, we give a…
We develop a general framework on Dirichlet spaces to prove a weak form of the Bakry-Émery estimate and study its consequences. This estimate may be satisfied in situations, like metric graphs, where generalized notions of Ricci curvature lower bounds are not available.
Proves a weak version of Perdomo Conjecture on minimal hypersurfaces.
problem Establishing a lower bound for the squared length of the second fundamental form on minimal hypersurfaces.
method Analyzes closed embedded, non-totally geodesic minimal hypersurfaces in Sn+1, proving a positive constant δ(n) depending only on n. result Introduces a positive constant δ(n) such that ∫MS≥δ(n)mVol(Mn) for any minimal hypersurface Mn in Sn+1. We are concerned with the global weak continuity of the Cartan structural system -- or equivalently, the Gauss--Codazzi--Ricci system -- on semi-Riemannian manifolds with lower regularity. For this purpose, we first formulate and prove a geometric compensated compactness theorem on vector bundles over semi-Riemannian m…
Estimates ends of Ricci shrinkers, focusing on smooth and singular cases.
problem Understanding the structure of ends in Ricci shrinkers, especially singular ones.
method Analyzes general and asymptotically conical ends, applies to weak convergence.
result No new conical end can form in the limit of sequences of Ricci shrinkers.
We prove the existence of a unique global weak solution to the full bosonic string heat flow from closed Riemannian surfaces to an arbitrary target under smallness conditions on the two-form and the scalar potential. The solution is smooth with the exception of finitely many singular points. Finally, we discuss the con…
Study improves cryptocurrency price prediction using unlabeled text data.
problem Predicting cryptocurrency returns from unlabelled text data.
method Introduced weak learning approach to finetune BERT on unlabeled text data.
result Finetuning pretrained NLP models with weak labels enhances forecast accuracy.
WSINDy for PDEs robustly identifies models from noisy data.
problem Identifying nonlinear dynamics from noisy partial differential equations data.
method Weak formulation of PDEs, Fourier-based model identification, sequential-thresholding least-squares.
result WSINDy enables robust identification of PDEs in noisy conditions.
Unified approach for learning with weak labels across various tasks.
problem Learning with noisy or incomplete labels in diverse machine learning settings.
method Implicit posterior models for joint label inference.
result Unified training objective for various machine learning tasks.
New varifold example shows decomposition failure.
problem Curvature varifolds cannot always be decomposed.
method Constructed a specific curvature varifold.
result Found a varifold with a non-preserved weak second fundamental form under decomposition.
Pairwise "same-cluster" queries are one of the most widely used forms of supervision in semi-supervised clustering. However, it is impractical to ask human oracles to answer every query correctly. In this paper, we study the influence of allowing "not-sure" answers from a weak oracle and propose an effective algorithm …
WENDy now estimates nonlinear ODEs with noisy data.
problem Estimating parameters of nonlinear ODEs with noisy data.
method WENDy-MLE algorithm for maximum likelihood estimation of nonlinear-in-parameters ODEs.
result WENDy-MLE outperforms other methods in accuracy, speed, and domain of convergence.