Models of 2-nondegenerate CR hypersurfaces in C^N are characterized and their defining equations simplified.
problem Characterizing and simplifying the defining equations of 2-nondegenerate CR hypersurfaces.
method Characterization of 2-nondegenerate models, derivation of normal forms, computation of CR invariants, derivation of infinitesimal symmetries.
result The moduli space of 2-nondegenerate CR hypersurfaces in C^N is infinite dimensional for N>3.
We study CR hypersurfaces in C^4 with constant rank Levi form and find their defining equations.
problem CR hypersurfaces in C^4 with constant rank Levi form and their defining equations.
method Obtained a complete normal form for models of real analytic uniformly 2-nondegenerate CR hypersurfaces in C^4.
result Found explicit formulas for infinitesimal symmetries of homogeneous 2-nondegenerate models.
The study identifies two sources of invariants in 2--nondegenerate CR geometries.
problem Characterizing fundamental invariants of 2--nondegenerate CR geometries.
method Analyzes the harmonic curvature and the difference in complex structures.
result Nontrivial examples of CR geometries can be obtained as deformations of models.
In this article, we solve the equivalence problem for 2--nondegenerate CR geometries that have (at every point) a homogeneous space G/H as a maximally symmetric model for G simple real Lie group of CR automorphisms. This completes the classification of real submanifolds in complex space that are maximally symmetric…
Classifies homogeneous CR hypersurfaces in low dimensions with maximal symmetry.
problem Classifying CR hypersurfaces with maximal symmetry in low dimensions.
method Introduced modified CR symbols to organize local invariants, classified hypersurfaces through modified symbols, and used Lie group structures.
result Found nine model structures among locally homogeneous 2-nondegenerate hypersurfaces in C4. Study finds maximal symmetry groups for CR structures with specific properties.
problem Determining the maximal dimension of symmetry groups for CR structures.
method Proved the sharp upper bound for the dimension of symmetry groups for homogeneous, 2-nondegenerate CR manifolds.
result The maximal dimension is n2+7 for n≥3. The paper explores CR structures and their leaf spaces in semi-Riemannian manifolds.
problem Classifying CR structures and their leaf spaces.
method Using unit tangent bundles and dynamical Legendrian contact structures.
result New examples of 2-nondegenerate CR structures are provided.
New CR hypersurfaces in complex space with specific properties.
problem Constructing CR hypersurfaces with arbitrary nilpotent symbols.
method Introduced a class of CR hypersurfaces with methods applicable to all cases with N>5. result Solved equivalence problem for structures with a single Jordan block symbol.
In our earlier articles we studied tube hypersurfaces in C3 that are 2-nondegenerate and uniformly Levi degenerate of rank 1. In particular, we showed that the vanishing of the CR-curvature of such a hypersurface is equivalent to the Monge equation with respect to one of the variables. In the present paper…
We apply E. Cartan's method of equivalence to classify 7-dimensional, 2-nondegenerate CR manifolds M up to local CR equivalence in the case that the cubic form of M satisfies a certain symmetry property with respect to the Levi form of M. The solution to the equivalence problem is given by a parallelism on a prin…
Develops new approach to recover CR structures from their Levi foliations.
problem Recovering CR structures from their Levi foliations for nonregular symbols.
method Reduction to dynamical Legendrian contact structure on leaf space.
result New geometric interpretation of CR prolongation conditions.
Complete normal forms for specific real hypersurfaces in complex space are constructed.
problem Constructing complete normal forms for real hypersurfaces in C3. method Utilizing equivariant moving frames for systematic symbolic manipulation.
result Complete normal forms for 5-dimensional real hypersurfaces in C3 are found. We explicitly determine the structure equations of 5-dimensional Levi 2-nondegenerate CR hypersurfaces, using our recently constructed canonical Cartan connection for this class of CR manifolds. We also give an outline of the basic properties of absolute parallelisms and Cartan connections, together with a brief discus…
The class IV2 of 2-nondegenerate constant Levi rank 1 hypersurfaces M5⊂C3 is governed by Pocchiola's two primary invariants W0 and J0. Their vanishing characterizes equivalence of such a hypersurface M5 to the tube MLC5 over the real light cone in R3. Whe…
We continue our study, initiated in an earlier article, of a class of rigid hypersurfaces in C3 that are 2-nondegenerate and uniformly Levi degenerate of rank 1, having zero CR-curvature. We drop the restrictive assumptions of the earlier paper and give a complete description of the class. Surprisingly, th…
We extend the notion of a fundamental negatively Z-graded Lie algebra mx=⨁p≤−1mxp associated to any point of a Levi nondegenerate CR manifold to the class of k-nondegenerate CR manifolds (M,D,J) for all k≥2 and call this invariant the core …
Let M be a CR manifold of hypersurface type, which is Levi degenerate but also satisfying a k-nondegeneracy condition at all points. This might be only if dim M is greater than or equal to 5 and if dim M = 5, then k= 2 at all points. We prove that for any 5-dimensional, uniformly 2-nondegenerate CR manifold M there exi…
Defines pre-Kähler structures and their properties.
problem Geometry of Levi degenerate CR hypersurfaces.
method Introduces pre-Kähler structures, holomorphic degeneration, and finite-nondegeneracy.
result Symmetry algebra is finite-dimensional if and only if structure is finitely nondegenerate.
3D projective structures can be metrized with conformal structures.
problem Weyl metrizability of 3D projective structures.
method Interpreting Weyl metrizability as CR submanifolds in 7D.
result Beltrami's theorem extends to conformal structures in 3D.
An absolute parallelism for 2-nondegenerate CR manifolds M of hypersurface type was recently constructed independently by Isaev-Zaitsev, Medori-Spiro, and Pocchiola in the minimal possible dimension (dimM=5), and for dimM=7 in certain cases by the first author. We develop a bigraded analog of Tanaka's prolo…
Study para-CR structures relaxing complex conjugation constraint, finding homogeneous models.
problem Para-CR structures with relaxed complex conjugation constraint.
method Cartan's method of equivalence, PDEs analysis, homogeneous models determination.
result Determined all concerned homogeneous models and their symmetries.
Real analytic (Cω) surfaces S2 in R3∋(x,y,u) graphed as {u=F(x,y)} with Fxx=0 whose Gaussian curvature vanishes identically: \[ 0 \,\equiv\, F_{xx}\,F_{yy} - F_{xy}^2, \] possess, under the action of the affine transformation group ${\sf Aff}_3(\mathbb{R}) = {\s…
We study the local equivalence problem for real-analytic (Cω) hypersurfaces M5⊂C3 which, in coordinates (z1,z2,w)∈C3 with w=u+iv, are rigid: \[ u \,=\, F\big(z_1,z_2,\overline{z}_1,\overline{z}_2\big), \] with F independent of v. Specifically, we study th…
Study para-CR structures in 5D with degenerate Levi form, revealing geometric conditions for conic graphs and Lorentzian ODEs.
problem Investigate invariant properties of para-CR structures in 5D with degenerate Levi form.
method Analyze basic invariants and their vanishing conditions to establish geometric interpretations and necessary conditions.
result Vanishing of the third basic invariant N(G,H)≡0 implies contact projective geometries on quotient spaces. Study normal forms for a specific type of complex hypersurface.
problem Tackle the normal forms of a special class of rigid hypersurfaces in complex space.
method Apply Chen-Merker method to find relative invariants and construct a bridge between Poincaré and Cartan normal forms.
result Establish the Poincaré-Moser complete normal form for the hypersurface.
The paper introduces BCART models for aggregate claim amount, improving frequency-severity and joint modeling.
problem Modeling aggregate claim amount with frequency-severity and joint dependencies.
method Developed three types of BCART models: frequency-severity, sequential, and joint models. Used various distributions for claim severity data.
result Weibull distribution outperforms gamma and lognormal for right-skewed, heavy-tailed claim severity data.
The paper uses model-based trees to create interpretable surrogate models for complex machine learning models.
problem Interpreting complex machine learning models.
method Using model-based trees to partition feature space and create interpretable models.
result Model-based trees generate optimal surrogate models that balance interpretability and performance.
Gauge Flow Models use a learnable Gauge Field in Generative Flow Models.
problem Improving generative model performance.
method Integrates a learnable Gauge Field into Flow ODEs.
result Gauge Flow Models outperform traditional Flow Models in Flow Matching experiments.
The study examines how model predictions hold up under model extensions.
problem Model predictions may not be robust under model extensions, limiting their applicability.
method The study uses causal ordering to assess robustness of qualitative model predictions and characterizes model extensions that preserve predictions.
result Conditions and techniques are provided to assess robustness of model predictions under model extensions.
Revises Bayesian model averaging for foundation models.
problem Ensemble pre-trained and lightly-finetuned foundation models for improved classification performance.
method Introduces trainable linear classifiers and computationally cheaper model averaging scheme (OMA).
result Ensembled models can better predict on various datasets.
Paper introduces symmetric divergence link models for probability distributions.
problem Symmetric divergence measures for probability distributions.
method Two general classes of link models: one for survival functions and another for cumulative probability distribution functions.
result Advantages of symmetric divergence measures over asymmetric measures for model averaging and feature assessment.
New method to handle credit portfolio model uncertainties.
problem Model risk in credit portfolio models.
method Demonstrates comprehensive yet easy-to-implement approach to uncertainty in model parameters.
result Comprehensive method to deal with model uncertainties.
The paper tests stock return models and uses LSTM to predict stock returns.
problem Validating stock return models and predicting stock returns.
method Used Fama-French three-factor, four-factor, and five-factor models; also used LSTM model.
result Fama-French five-factor model shows better validity for stock returns.
Researchers review challenges in interpreting additive models, especially neural additive models.
problem Challenges in interpreting additive models, particularly neural additive models.
method Review of generalized additive models and discussion of nonidentifiability.
result Challenges in claiming interpretability or suitability for safety-critical applications of additive models.
Novel hybrid modeling combines ML and physics for real-time diagnosis.
problem Real-time diagnosis of complex systems.
method Combines machine learning and physics-based models to create reduced-order models.
result Generated models are two orders of magnitude simpler, improving efficiency.
CRS model improves ranking data modeling with theoretical guarantees.
problem Lack of rich, multimodal models for ranking data.
method Contextual Repeated Selection (CRS) model for multimodal ranking data.
result CRS model significantly outperforms existing methods in various ranking contexts.
Sigma models linked to Gross-Neveu models via quiver varieties.
problem Understanding the relationship between sigma models and Gross-Neveu models.
method Exploring the mathematical correspondence between sigma models and Gross-Neveu models, including their geometric and trigonometric/elliptic deformations.
result Sigma models are mathematically equivalent to Gross-Neveu models under certain conditions.
Interpretable machine learning has become a strong competitor for traditional black-box models. However, the possible loss of the predictive performance for gaining interpretability is often inevitable, putting practitioners in a dilemma of choosing between high accuracy (black-box models) and interpretability (interpr…
Simple models are preferred over complex models, but over-simplistic models could lead to erroneous interpretations. The classical approach is to start with a simple model, whose shortcomings are assessed in residual-based model diagnostics. Eventually, one increases the complexity of this initial overly simple model a…
Matryoshka hides secret models in a carrier model, achieving high capacity and robustness.
problem Stealing functionality of private ML data by hiding models in a carrier model.
method Parameter sharing approach exploiting the learning capacity of the carrier model.
result Hides a 26x larger secret model or 8 secret models in the carrier model.
Eigen-stratified models reduce model size and improve performance.
problem Large model size in Laplacian-regularized stratified models.
method Formulate eigen-stratified models with linear combinations of bottom eigenvectors of the graph Laplacian.
result Significant reduction in model size with eigen-stratified models.
Seq2Seq models speed up epidemic model predictions.
problem Complex epidemic models are computationally expensive.
method Used deep seq2seq models as surrogates for complex models.
result Surrogates predict scenarios up to several thousand times faster.
This work develops scalable model selection methods with fast update and selection.
problem Efficient model selection for large pools of candidate models.
method Isolated model embedding, which supports asymptotically fast update and selection.
result Standardized Embedder achieves competitive model selection performances.
Paper proposes BMPO to optimize policies using bidirectional models.
problem Model-based reinforcement learning's reliance on forward model accuracy.
method Develops BMPO using both forward and backward models for policy optimization.
result BMPO outperforms state-of-the-art methods in sample efficiency and asymptotic performance.
Copulas outperform marginal models in multivariate risk forecasting, reducing model risk by narrowing down the set of models.
problem Model risk in multivariate risk forecasting, especially during crises.
method Comprehensive empirical study comparing Copula-GARCH models with fixed marginals, copulas, or neither.
result Model risk is almost entirely due to copula choice, not marginal models.
BayesBlend blends multiple models' predictions for better insurance loss predictions.
problem Improving insurance loss predictions by combining multiple models.
method Pseudo-Bayesian model averaging, stacking, and hierarchical stacking.
result BayesBlend provides a user-friendly way to blend model predictions and estimate weights.
The paper identifies when larger models improve predictions and proposes a switcher model.
problem Understanding when larger models benefit from added complexity.
method Numerical studies on T5 architecture to analyze predictive uncertainty and model performance.
result Large models improve on examples where small models are uncertain, but not on certain examples.
Improved diffusion model generation speed with speculative sampling.
problem Generating samples from computationally expensive diffusion models.
method Extending speculative sampling to diffusion models, using fast draft models for candidate token generation.
result Significant speedup in generation, halving the number of function evaluations.