This paper is devoted to the study of geometric structures modeled on homogeneous spaces G/P, where G is a real or complex semisimple Lie group and is a parabolic subgroup. We use methods from differential geometry and very elementary finite-dimensional representation theory to construct sequences of invar…
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48 results for “Bernstein-Gelfand-Gelfand resolutions”
Constructs BGG resolutions for symplectic case.
problem Exactness of BGG resolutions in singular infinitesimal characters.
method Penrose transform over Lagrangian Grassmannian.
result Exactness of constructed complex over big affine cell.
Researchers create exact sequences for isotropic 2-Grassmannian.
problem Constructing exact sequences for isotropic 2-Grassmannian.
method Using Penrose transform over a double fibration.
result The constructed sequences are analogues of the Bernstein-Gelfand-Gelfand resolutions.
For smooth manifolds equipped with various geometric structures, we construct complexes that replace the de Rham complex in providing an alternative fine resolution of the sheaf of locally constant functions. In case that the geometric structure is that of a parabolic geometry, our complexes coincide with the Bernstein…
Study Bernstein-Gelfand-Gelfand complexes on Lipschitz domains, computing cohomology and applying to elasticity models.
problem Cohomology of BGG complexes on bounded Lipschitz domains.
method Computes cohomology of conformal deformation and Hessian complexes in Sobolev spaces, allowing multiple input complexes.
result Establishes conformal Korn inequality and proposes generalizations of continuum models with microstructures.
New differential complexes on symplectic manifolds.
problem Developing calculus on symplectic manifolds.
method Coupling a symplectic manifold to a vector bundle with a constrained curvature.
result Construction of new differential complexes.
We give a complete construction of the Bernstein-Gelfand-Gelfand complex on real or complex projective space using minimal ingredients.
Study projective and almost conformally symplectic structures on manifolds.
problem Relations between projective and almost conformally symplectic structures.
method Single almost conformally symplectic connection with totally trace-free torsion.
result Generalizes Fedosov structures and encodes variability of connections in projective class.
We give a simple construction of the Bernstein-Gelfand-Gelfand sequences of natural differential operators on a manifold equipped with a parabolic geometry. This method permits us to define the additional structure of a bilinear differential cup product on this sequence, satisfying a Leibniz rule up to curvature terms.…
Solves index problem for curved BGG sequences in parabolic geometry.
problem Index theory of curved Bernstein-Gelfand-Gelfand sequences.
method Utilizes K-homology and noncommutative geometry.
result Solves the index problem for BGG-sequences on flat parabolic geometry.
New finite element method for complex forms in any dimension.
problem Discretization of complex forms in arbitrary dimensions.
method Finite element discretization of -form-valued -forms on triangulations.
result Generalizes existing finite element methods for various tensor fields.
For a real or complex semisimple Lie group and two nested parabolic subgroups , we study parabolic geometries of type . Associated to the group , we introduce a class of relative natural bundles and relative tractor bundles and construct some basic invariant differential operators on …
Lecture notes on BGG complexes using Lie groups and algebras.
problem Constructing BGG complexes on open domains.
method Representation theory of semisimple Lie groups and Lie algebras.
result Introduction of BGG complexes with Lie group and algebra insights.
The study extends hypoellipticity to filtered manifolds and applies it to BGG sequences.
problem Analyzing hypoellipticity on general filtered manifolds.
method Extending Rockland criterion to pseudodifferential calculus, constructing parametrix, generalizing BGG machinery.
result Generalized BGG sequences are Rockland in a graded sense.
This is an expanded version of a series of two lectures given at the IMA summer program "Symmetries and Overdetermined Systems of Partial Differential Equations". The main part of the article describes the Riemannian version of the prolongation procedure for certain overdetermined system obtained recently in joint work…
New path integrals for elasticity derived from differential complex theory.
problem Deriving path integrals for elasticity equations.
method Using Bernstein-Gelfand-Gelfand (BGG) construction and properties of the de Rham complex, derived path integral operators for elasticity.
result Path integral operators for elasticity satisfying and .
The paper constructs discrete Hessian and divdiv complexes on triangulations and proves their cohomology isomorphic to continuous versions.
problem Discrete construction of Hessian and divdiv complexes on triangulations.
method Construction of discrete Hessian and divdiv complexes using finite elements and Dirac measures on triangulations.
result The cohomology of the constructed complexes is isomorphic to the continuous de Rham cohomology.
Mapping Low-Resolution Images To Multiple High-Resolution Images Using Non-Adversarial Mappingeess.IV
SR-NAM maps low-res images to multiple high-res images realistically.
problem Mapping low-resolution images to multiple high-resolution images realistically.
method SR-NAM using Non-Adversarial Mapping (NAM) technique and a degradation model.
result Realistic degradation and down-sampling of high-resolution images.
Deep learning speeds up whole heart MRI to 30 seconds.
problem Long acquisition times in whole heart MRI.
method Deep learning, specifically a 3D residual U-Net, to reconstruct high-resolution images from low-resolution data.
result Super-resolution images show better edge sharpness and fewer artefacts than low-resolution images.
CR Killing operator derived from tractor calculus for CR structures.
problem Analyzing CR structures and their deformations.
method Tractor calculus and BGG operators applied to compatible almost CR structures.
result CR Killing operator is a first BGG operator for the modified adjoint tractor connection.
Researchers create compatibility complexes for Einstein metrics.
problem Constructing solutions to the conformal-to-Einstein operator.
method Using a method that leverages symmetries and geometric properties.
result At most one independent solution exists under genericity assumptions.
The paper proposes a method to improve fine-resolution predictions using coarse-resolution data.
problem Limited supervision for fine-resolution predictions with scarce data.
method Attention-based regularization on multi-view coarse-resolution data.
result The method improves fine-resolution predictions using coarse-resolution data.
Super-resolution improves MRI resolution and accuracy for biomarker assessment.
problem Inadequate SNR for accurate quantification in high-resolution MRI.
method Utilized deep learning super-resolution to maintain SNR for T2 relaxation time biomarkers while generating high-resolution images.
result Super-resolution successfully maintains high-resolution and accurate biomarkers for MRI.
This study prioritizes temporal resolution over spatial in energy systems models due to higher influence.
problem The impact of spatial and temporal resolution on energy system models.
method Global sensitivity analysis to compare structural aspects, spatial, and temporal resolution.
result Temporal resolution has a higher influence on all results parameters compared to spatial resolution.
Deep learning improves 3D microscopy resolution without matched target images.
problem Anisotropic resolution in volumetric fluorescence microscopy.
method Cycle-consistent generative adversarial network trained on unpaired 2D images.
result Enhanced axial resolution and restored details between imaging planes.
There are 2^n possible resolutions of a smooth pseudodiagram with n precrossings. If we consider piecewise-linear (PL) pseudodiagrams and resolutions that themselves are PL, certain resolutions of the pseudodiagram may not exist in three-space. We investigate this situation and its impact on the weighted resolution set…
Recently M. Kreck introduced a class of stratified spaces called p-stratifolds [M. Kreck, Stratifolds, Preprint]. He defined and investigated resolutions of p-stratifolds analogously to resolutions of algebraic varieties. In this note we study a very special case of resolutions, so called optimal resolutions, for p-str…
High-resolution signal recovery via generalized sampling and functional principal component analysismath.ST
The paper presents a method to recover high-resolution signals from low-resolution measurements.
problem Recovering high-resolution signals from low-resolution indirect measurements.
method Combining generalized sampling and functional principal component analysis.
result High-resolution recovery is possible under certain conditions and with a sufficiently large training set.
A resolution of the St. Petersburg paradox is presented. In contrast to the standard resolution, utility is not required. Instead, the time-average performance of the lottery is computed. The final result can be phrased mathematically identically to Daniel Bernoulli's resolution, which uses logarithmic utility, but is …
The `Folk Theorem' that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type is proved in the context of manifolds with corners. This procedure is shown to capture the simultaneous resolution of all isotropy types in a `resolution tower' which projects …
DeepDownscale uses deep learning to upscale weather forecasts.
problem Computational expense of high-resolution weather models limits their use.
method Supervised deep learning to learn high-resolution from low-resolution forecasts.
result Significant improvement in weather forecast quality.
Develops a multi-resolution multi-task framework for integrating noisy, varying data.
problem Integrating evidence from multiple observation processes with varying resolutions and noise levels.
method Multi-resolution Multi-task Gaussian Processes (MRGP) framework, shallow and deep Gaussian Process mixtures.
result Generalizes and outperforms state-of-the-art GP compositions, offering efficient corrections and approximations.
Study contact resolutions for Jacobi structures, providing examples and impossibility results.
problem Understanding contact resolutions of Jacobi structures.
method Examining various classes of Jacobi structures and their contact properties.
result Identified conditions under which contact resolutions exist and those where they do not.
WrapNet optimizes inference for low-resolution neural networks by using 8-bit additions.
problem Reducing multiplication complexity in low-resolution neural networks.
method Adapting neural networks to use low-resolution (8-bit) additions in accumulators, with a cyclic activation layer and overflow penalty regularizer.
result Achieves comparable classification accuracy to 32-bit counterparts using low-resolution additions.
Study proposes CNN for reconstructing high-res urban DEMs.
problem Lack of high-res urban DEM datasets for flood modeling.
method Multi-scale CNN model trained on urban DEMs of varying resolutions.
result CNN-based method produces superior high-res urban DEMs.
This paper simplifies diffusion models for high resolution images.
problem Applying diffusion models to high resolution images is challenging.
method Adjust noise schedule, scale specific parts, add dropout, and use downsampling.
result Achieved state-of-the-art image generation performance.
A refined form of the `Folk Theorem' that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type was established by the authors in the context of manifolds with corners; the canonical construction induces fibrations on the boundary faces of the resolution…
Parabolic conformally symplectic structures III; Invariant differential operators and complexesmath.DG
This is the last part of a series of articles on a family of geometric structures (PACS-structures) which all have an underlying almost conformally symplectic structure. While the first part of the series was devoted to the general study of these structures, the second part focused on the case that the underlying struc…
New deep learning method improves 4D Flow MRI super-resolution under domain shift.
problem Domain shift in low-resolution 4D Flow MRI data.
method Distributional deep learning framework for domain generalization.
result Framework significantly outperforms traditional methods in real data applications.
Bayesian framework for solar magnetogram super-resolution with uncertainty quantification.
problem Uncertainty in super-resolving solar magnetic field images.
method Bayesian decomposition of uncertainties into epistemic and aleatoric.
result Generation of maps measuring the range of possible high-resolution explanations.
Unified proofs for various resolution theorems using simpler techniques.
problem Proving the cell-like, -, and -resolution theorems.
method Unified proofs employing simpler extensions and topological methods.
result Simpler proofs for the resolution theorems.
Generates high-resolution images from low-resolution inputs.
problem Generating realistic images from low-resolution inputs.
method Latent Adversarial Generator (LAG) using perceptual loss.
result Samples of high-resolution images from low-resolution inputs.
3D CNNs improve brain tumor segmentation using multi-resolution features.
problem Brain tumor segmentation in MR images.
method Three 3D CNN architectures combining fine and coarse features.
result Multi-resolution architectures outperform single-resolution networks.
A refined form of the `Folk Theorem' that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type is proved in the context of manifolds with corners. This procedure is shown to capture the simultaneous resolution of all isotropy types in a `resolution stru…
Unique shrinking solitons found on cone resolutions.
problem Uniqueness of shrinking Kähler-Ricci solitons on cone resolutions.
method Analyzing asymptotic conical properties and using Esparza's result.
result At most one complete shrinking Kähler-Ricci soliton exists on resolutions of Kähler cones.
MeshfreeFlowNet generates high-resolution spatio-temporal solutions from low-resolution inputs.
problem Generating high-resolution spatio-temporal solutions from low-resolution inputs.
method Physics-constrained deep learning framework using fully convolutional encoders.
result Significantly outperforms existing baselines in super-resolution of turbulent flows.
In the paper titled "Bockstein basis and resolution theorems in extension theory" (arXiv:0907.0491v2), we stated a theorem that we claimed to be a generalization of the Edwards-Walsh resolution theorem. The goal of this note is to show that the main theorem from (arXiv:0907.0491v2) is in fact equivalent to the Edwards-…
Developed AI models for multi-gas detection in near IR spectrums.
problem Detecting multiple gases in near IR spectrums.
method Used Monte Carlo KNN and multi-resolution CNN, synthesized near IR spectrums, optimized kernel sizes and channels.
result Multi-resolution CNN outperforms other models.