Geometric heat-flow theory detects Lagrangian coherent structures.
problem Detecting Lagrangian coherent structures in advective-diffusive systems.
method Transform Eulerian advection-diffusion to Lagrangian coordinates, then solve the resulting geometric heat equation.
result LCSs are boundaries of metastable sets under Lagrangian diffusion.
A new method uses coherent structure coloring to estimate model parameters more accurately than traditional methods.
problem Challenges in estimating model parameters from Lagrangian data in turbulent flows.
method Use coherent structure coloring (CSC) field to assess model skill and estimate model parameters.
result Error in the CSC field can accurately determine model parameters, while conventional methods fail.
Method identifies coherent structures in sparse flow data.
problem Identifying coherent structures in sparse particle trajectory data.
method Spectral graph theory and hierarchical clustering.
result Algorithm successfully identifies coherent structures in various flows.
Method detects coherent structures from sparse flow data using graph theory.
problem Detecting coherent structures from sparse flow data.
method Graph coloring and spectral graph drawing algorithms applied to kinematic dissimilarity of trajectories.
result Robustly detects coherent structures using significantly less data than existing methods.
A membrane technique, in which the symplectic and Ricci forms are integrated over surfaces in a complexification of the phase space, as well a ``creation" connection with zero curvature over lagrangian submanifolds, is used to obtain a unified quantization including a noncommutative algebra of functions, its representa…
A new method learns stable, temporally coherent features in point clouds.
problem Flickering and halo structures in inferred solutions for time-varying point clouds.
method Proposes a novel temporal loss function and super-resolution method.
result Demonstrates flexibility and effectiveness on large, deforming point sets.
The study of special Lagrangian classes and semistable Mukai vectors on K3 surfaces.
problem Counting special Lagrangian classes and semistable Mukai vectors for K3 surfaces.
method Analyzing flat surfaces and K3 surfaces, using asymptotics and stability conditions.
result Exact leading term in the asymptotics of the number of semistable Mukai vectors.
Study shows how heat leaks from material sets in low diffusivity scenarios.
problem Understanding heat leakage from material sets in low diffusivity limits.
method Generalized leading-order asymptotics for time-dependent diffusion processes.
result Diffusive transport out of a material set is proportional to the surface area of the set boundary.
Formulates mechanics for probability distributions on statistical manifold.
problem Formulating mechanics for probability distributions on statistical manifold.
method Information-geometric formulation of Classical Mechanics on statistical manifold, using dually-flat connection and Hilbert bundle structure.
result Provides coherent formalism for Lagrangian and Hamiltonian mechanics on statistical bundle.
New potentials found for sheaves on Calabi-Yau 4-folds.
problem Understanding sheaves on Calabi-Yau 4-folds.
method Derived Quot-stacks and Lagrangian distributions.
result Globally defined −1-shifted potentials on sheaves. Tensor analysis improves structural health monitoring of complex aerospace systems.
problem Highly redundant and correlated sensor data in structural health monitoring.
method Tensor-based learning for multi-way structural data analysis.
result Tensor-based approach successfully detects damage in aeroservoelastic models.
New language models improve document coherence.
problem Existing language models fail to account for discourse structure.
method Introduced Document-Context Language Models (DCLM) using multi-level recurrent neural networks.
result DCLM models yield better document coherence than word-level models.
As we have proved in [L], the geodesic flows associated with the flat metrics on T^2 minimize the polynomial entropy. In this paper, we show that, among the geodesic flows that are Bott integrable and dynamically coherent, the geodesic flows associated to flat metrics are local strict minima for the polynomial entropy.…
The study examines Kähler structures on coadjoint orbits of Lie groups using coherent and squeezed states.
problem Does the coadjoint orbits of Lie groups support a Kähler structure?
method Examined three Lie groups: Weyl-Heisenberg, SU(2), and SU(1,1). Used coherent and squeezed states to explore Kähler structures.
result Coherent states provide Kähler embeddings, while squeezed states only symplectic embeddings.
Model criticism tool evaluates text coherence and structure in generated long-form text.
problem Evaluate the high-level structure of generated text for coherence, coreference, and topicality.
method Apply model criticism in latent space to compare real and generated data distributions.
result Transformer-based models struggle with maintaining structural coherence and coreference.
Researchers identify bi-Lagrangian structures on specific nilpotent Lie algebras.
problem Identifying bi-Lagrangian structures on nilpotent Lie algebras.
method Study of bi-Lagrangian structures on nilmanifolds of dimension ≤ 6, focusing on symplectic forms and complementary Lagrangian foliations.
result Determine which 6-dimensional nilpotent Lie algebras admit a bi-Lagrangian structure.
Paper solves flat bi-Lagrangian structure problems in ray space.
problem Existence of flat bi-Lagrangian structures in ray space.
method Established geometric conditions for flat canonical connections.
result Complete solutions to two problems regarding flat bi-Lagrangian structures.
Study of Lagrangian Engel structures on symplectic 4-manifolds.
problem Understanding the geometry of Engel structures on symplectic 4-manifolds.
method Solving equivalence problems and classifying homogeneous examples.
result Determine all compact, homogeneous examples of Lagrangian Engel structures.
The paper studies coherent sheaves on subvarieties of Hopf manifolds.
problem Understanding coherent sheaves on subvarieties of Hopf manifolds.
method Proves a version of GAGA theorem, shows natural algebraic structure, and uses quotient and embedding properties.
result Any reflexive coherent sheaf on M is filtrable. Solves division problem for L. Hörmander's systems.
problem Division problem for L. Hörmander's overdetermined systems.
method Formulates and proves divisibility criterion, coherence theorem.
result Establishes effective divisibility criterion and extends coherence theorem.
A realization of coherent state Lie algebras by first-order differential operators with holomorphic polynomial coefficients on Kähler coherent state orbits is presented. Explicit formulas involving the Bernoulli numbers and the structure constants for the semisimple Lie groups are proved.
Kernel methods detect coherent structures in dynamical data.
problem Detecting coherent structures in complex dynamical systems.
method Kernel-based dimensionality reduction techniques and eigendecompositions of RKHS operators.
result Coherent sets of particle trajectories can be computed by kernel CCA.
Paper introduces quantile coherency to measure dependence in economic time series.
problem Measuring general dependence structures in economic time series.
method Defined quantile coherency estimators and discussed their asymptotic properties.
result Demonstrated the usefulness of quantile coherency in assessing time series models.
Extract coherent structures from Jupiter's clouds and snow events without flow models.
problem Finding coherent structures in fluidic systems from image data.
method Image processing with anisotropic, directed diffusion operator from a directed affinity matrix.
result Demonstrated coherence in Jupiter's clouds and snow events without flow modeling.
Let M be a K3 surface or an even-dimensional compact torus. We show that the category of coherent sheaves on M is independent from the choice of the complex structure, if this complex structure is generic.
It is shown that the heat operator in the Hall coherent state transform for a compact Lie group K is related with a Hermitian connection associated to a natural one-parameter family of complex structures on T∗K. The unitary parallel transport of this connection establishes the equivalence of (geometric) quantizati…
Authors characterize Lagrangian cone structures from (2,3,5)-distributions.
problem Characterizing Lagrangian cone structures from (2,3,5)-distributions. method Characterization via pseudo-product structures of type G2. result Completion of duality between (2,3,5)-distributions and Lagrangian cone structures. Study special Lagrangian moduli spaces with boundary.
problem Understanding geometric structures on moduli spaces of special Lagrangians.
method Investigates geometric structures and constructs special affine structures and a Hessian metric.
result Constructs a pair of special affine structures and a Hessian metric on the moduli space.
This article explains A∞-algebras and Hochschild homology.
problem Understanding A∞-algebras and Hochschild homology. method Elementary construction and detailed proofs of results.
result Unified discussion of algebraic results from various sources.
Study characterizes SU(3)-structures with special Lagrangian 3-folds as minimal.
problem Characterizing SU(3)-structures with special Lagrangian 3-folds.
method Derivation of mean curvature formulas and analysis of SU(3)-structures.
result Obtained formulas for mean curvature and characterized SU(3)-structures.
Introduces internal Lagrangians for differential equations and connects them to presymplectic structures.
problem Understanding the geometry of differential equations and their solutions.
method Develops a spectral sequence related to internal Lagrangians and investigates connections to presymplectic structures.
result Interprets a term in Vinogradov's spectral sequence for gauge theories.
Flexible Lagrangians in Weinstein domains lead to new constructions and exotic structures.
problem Understanding regularity and flexibility of Lagrangian manifolds.
method Introducing and discussing regularity and flexibility for Lagrangian manifolds with Legendrian boundary in Weinstein domains.
result Many closed n-manifolds can be exact Lagrangian submanifolds of T∗Sn with exotic Weinstein structures. Study canonical curves and Kropina metrics in Lagrangian contact geometry.
problem Characterize canonical curves and their relationship to Lagrangian contact structures.
method Construct Fefferman-type spaces, analyze chains and null-chains, use Kropina metrics, apply Fermat principle.
result Chains and null-chains in integrable Lagrangian contact structures are geodesics of Kropina metrics.
The paper establishes a connection between superminimal surfaces and Lagrangian submanifolds in twistor spaces.
problem Understanding the geometric relationship between superminimal surfaces and Lagrangian submanifolds in twistor spaces.
method Proves a bijective correspondence between superminimal surfaces and Lagrangian submanifolds of twistor spaces, using specific constructions and projections.
result Produces many Lagrangian submanifolds of twistor spaces that are also minimal.
We introduce length dilatation structures on metric spaces, tempered dilatation structures and coherent projections and explore the relations between these objects and the Radon-Nikodym property and Gamma-convergence of length functionals. Then we show that the main properties of sub-riemannian spaces can be obtained f…
Construct special Lagrangian pair of pants in n dimensions.
problem Constructing special Lagrangian pair of pants in general dimensions.
method Inside the cotangent bundle of Tn with the Euclidean structure. result Special Lagrangian pair of pants constructed in general dimensions.
Coherence Pursuit is a fast, simple, robust PCA algorithm.
problem Principal Component Analysis with robustness to outliers.
method Coherence Pursuit algorithm: computes Gram matrix, identifies strong coherence with rest of data.
result Outliers are distinguished from inliers by strong mutual coherence with data points.
The paper explores bi-Lagrangian structures in Teichmüller theory.
problem Exploring geometric structures on manifolds and their applications in Teichmüller theory.
method Review and introduction of bi-Lagrangian structures, focusing on symplectic and Lagrangian foliations.
result Complexification of real-analytic Kähler manifolds has a natural complex bi-Lagrangian structure.
This note is mostly an exposition of an unpublished result of Deligne, which introduces an analogue of perverse t-structure on the derived category of coherent sheaves on a Noetherian scheme with a dualizing complex. Construction extends to the category of coherent sheaves equivariant under an action of an algebraic …
New dataset and models generate piano music with coherent structure across multiple timescales.
problem Generating coherent musical structure with neural networks is challenging.
method Used notes as an intermediate representation to model and synthesize music across multiple timescales.
result Trained models capable of transcribing, composing, and synthesizing audio waveforms with coherent musical structure.
Defines algebraic structures in Lagrangian Floer cohomology using differential forms.
problem Modeling algebraic structures in Lagrangian Floer cohomology.
method Defines two algebra structures using differential forms and a closed-open map, showing they coincide.
result Two algebra structures for the 2-dimensional Clifford torus coincide.
Coherent diversification of tech fields correlates with higher labor productivity.
problem Understanding how firms' technological diversification impacts productivity.
method Analyzed patent data of 70k firms over 2004-2013, defined coherent diversification as network of related tech fields.
result Firms with coherent diversification structure outperform those with scattered diversification in labor productivity.
Study minimal Lagrangian submanifolds of complex hyperquadric.
problem Classify minimal Lagrangian submanifolds of complex hyperquadric.
method Use non-integrable almost product structures and local angle functions.
result Classify minimal Lagrangian submanifolds with constant sectional curvatures and those with coinciding local angle functions.
The paper studies Lagrangian surfaces in a specific Riemannian product space.
problem Characterizing and classifying Lagrangian surfaces in a particular geometric space.
method Analyzes various types of Lagrangian surfaces and their properties.
result Classification of different types of Lagrangian surfaces.
Holomorphic symplectic structure on Lagrangian moduli space.
problem Understanding the structure of Lagrangian submanifolds in hyperKähler manifolds.
method Proving the existence of a natural holomorphic symplectic structure on the relative Albanese over the moduli space.
result The relative Albanese over the moduli space of complex Lagrangian submanifolds has a natural holomorphic symplectic structure.
This paper generalizes L2 cohomology theory for complex manifolds.
problem Developing a L2 cohomology theory for Hodge modules on infinite covering spaces.
method Formulating a conjectural generalization of L2-Mixed Hodge structures using Saito's Mixed Hodge Modules.
result Partial results in the conjectural generalization of L2-Mixed Hodge structures.
New geometric structures that relate the lagrangian and hamiltonian formalisms defined upon a singular lagrangian are presented. Several vector fields are constructed in velocity space that give new and precise answers to several topics like the projectability of a vector field to a hamiltonian vector field, the comput…
The paper studies financial market coherence and incoherence using phase oscillators.
problem Understanding the dynamics of financial market coherence and incoherence.
method A coupled dynamical system of phase oscillators to model financial price fluctuations.
result Financial markets exhibit a coexistence of coherent and incoherent collective behavior.