We give a new proof for the local existence of a smooth isometric embedding of a smooth -dimensional Riemannian manifold with nonzero Riemannian curvature tensor into -dimensional Euclidean space. Our proof avoids the sophisticated arguments via microlocal analysis used in earlier proofs. In Part 1, we introduce …
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169,341 papers · 148 categories
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48 results for “differentially positive systems”
Study of a 3D system on Wallach spaces, finding interrelations with invariant metrics.
problem Understanding the dynamics of a 3D system on Wallach spaces.
method Analyzing the normalized Ricci flow on generalized Wallach spaces.
result Characterized interrelations between the normalized Ricci flow and invariant metrics on Wallach spaces.
Paper proves autodiff systems are correct for non-differentiable functions.
problem Correctness of autodiff systems for non-differentiable functions in deep learning.
method Investigation of PAP functions and introduction of intensional derivatives.
result Intensional derivatives always exist and coincide with standard derivatives for almost all inputs.
Paper derives explicit expression of Alekseev-Meinrenken diffeomorphism.
problem Understanding the Alekseev-Meinrenken diffeomorphism.
method Via the Stokes phenomenon of meromorphic linear systems of ODEs with Poncaré rank 1.
result Explicit expression of the Alekseev-Meinrenken diffeomorphism.
It is shown that existence of a global solution to a particular nonlinear system of second order partial differential equations on a complete connected Riemannian manifold has topological and geometric implications and that in the domain of positivity of such solution its reciprocal is the radial function of only one o…
New geometric system from Hessian operators offers solutions to geometric problems.
problem Solving geometric problems using Hessian operators.
method Introducing a new differential-geometric system based on -Hessian operators.
result Deduced an a priori -estimate for solutions to the Dirichlet problem for -Hessian equations.
We search for Riemannian metrics whose Levi-Civita connection belongs to a given projective class. Following Sinjukov and Mikes, we show that such metrics correspond precisely to suitably positive solutions of a certain projectively invariant finite-type linear system of partial differential equations. Prolonging this …
In the paper, some concepts of modern differential geometry are used as a basis to develop an invariant theory of mechanical systems, including systems with gyroscopic forces. An interpretation of systems with gyroscopic forces in the form of flows of a given geodesic curvature is proposed. For illustration, the proble…
New system solves curvature for ample vector bundles, proving Griffiths conjecture.
problem Proving Griffiths conjecture on vector bundle positivity.
method Proposes Hermitian-Yang-Mills elliptic system for curvature.
result Solutions provide metrics with positive curvature in Griffiths sense.
Graph neural controlled differential equations learn graph dynamics from vertex observations.
problem Predicting future states of dynamical systems on graphs with limited vertex data.
method Incorporates graph topology information into NCDE to predict graph dynamics.
result Informed NCDE requires fewer parameters and lower MAE compared to previous methods.
Complex contact manifolds, varieties of minimal rational tangents, and exterior differential systemsmath.AG
The abstract discusses classification theorems for complex contact manifolds and their geometric properties.
problem Classifying complex contact manifolds and understanding their geometric properties.
method Analyzing contact lines and varieties of minimal rational tangents.
result Partial classification theorems for projective complex contact manifolds.
The paper proves existence of solutions for Einstein-type elliptic systems on AE manifolds.
problem Analyzing semi-linear systems of partial differential equations motivated by the conformal formulation of Einstein constraint equations.
method Proving existence theorems under suitable conditions, including smallness assumptions on free parameters.
result Existence of far from CMC (near CMC) Yamabe positive (Yamabe non-positive) solutions for charged dust coupled to the Einstein equations.
A Deep Autoencoder System for Differentiation of Cancer Types Based on DNA Methylation Stateq-bio.QM
Deep autoencoder predicts cancer types from DNA methylation patterns.
problem Differentiating cancer types based on DNA methylation states.
method Deep learning system with CpG island state classification and statistical methods.
result Overall Sensitivity of 88.24%, Specificity of 83.33%, Accuracy of 84.75%.
New proof of Minkowski identities for hypersurfaces in curved spaces.
problem Proving Minkowski identities for hypersurfaces in constant curvature manifolds.
method Using a differential system and position vector field.
result New proof of Minkowski identities for hypersurfaces.
The coupled system, where one is a degenerate parabolic equation and the other has not a diffusion term arises in the modeling of European options with liquidity shocks. Two implicit-explicit (IMEX) schemes that preserve the positivity of the differential problem solution are constructed and analyzed. Numerical experim…
This paper introduces belief propagation for Gaussian reciprocal processes and links it to stability theory.
problem Efficient inference algorithms for reciprocal processes, especially those with a single loop network.
method Introduces belief propagation for Gaussian reciprocal processes and links it to stability theory.
result Established a link between convergence analysis of belief propagation for Gaussian reciprocal processes and stability theory for differentially positive systems.
Paper proves positivity for differential classes, similar to earlier work on stable curves.
problem Positivity of divisor classes for families of stable differentials.
method Establishes analogous positivity result to Cornalba-Harris for stable differentials.
result Analogous positivity result for divisor classes of stable differentials.
Introduces exterior differential systems for grad students.
problem No specific problem stated; introduces a topic.
method No specific method mentioned; brief introduction.
result Introduces exterior differential systems.
On characteristic forms of positive vector bundles, mixed discriminants and pushforward identitiesmath.AG
The paper proves positivity of characteristic forms for certain vector bundles.
problem Characterizing positivity conditions for vector bundles.
method Operator theory, pushforward identities, and differential forms.
result Schur polynomials in Chern forms of Nakano and Griffiths positive vector bundles are positive as differential forms.
Noncommutative geometry connects higher order connections to quantization.
problem Quantization in noncommutative geometry.
method Introducing natural linear differential operators and Spencer operators.
result Higher order connections are equivalent to quantization.
Deep density methods improve filtering in high-dimensional systems.
problem Nonlinear filtering in high-dimensional systems.
method Two deep density methods based on Feynman-Kac formulas and neural networks.
result Logarithmic deep backward stochastic differential equation filter outperforms classical methods in high dimensions.
Guyon-Lekeufack model accurately predicts market volatility.
problem Modeling and predicting market volatility accurately.
method Path-dependent volatility model with weighted past price returns and squared volatility.
result Wellposedness of the coupled system of stochastic differential equations for all parameter values.
New method recovers graph latent positions under edge differential privacy.
problem Recovering latent graph information from privatized graphs.
method Applying geometric insights to adjust statistical inference for privatized graphs.
result Achieves consistent recovery of latent positions under local edge differential privacy constraints.
Study differential properties of symmetric real matrices with a specific metric.
problem Differential geometric properties of symmetric real matrices.
method Trace metric on non-singular symmetric real matrices, isometries of positive definite matrices.
result Description of the full group of isometries for positive definite matrices.
New framework establishes positivity of DNTK for PINNs.
problem Establishing positivity of NTK for PINNs with multiple differential operators.
method Proposed Differential Neural Tangent Kernel (DNTK) for PINNs.
result Positivity of infinite width DNTK for various activation functions and differential operators.
Energy-momentum's local conservation laws and generalized isometric embeddings of vector bundlesmath.DG
This text is the extended version of a talk given at the conference Geometry, Topology, QFT and Cosmology hold from May 28 to May 30, 2008 at the Observatoire de Paris. Using exterior differential systems, I provide a positive answer to the generalized isometric embedding problem of vector bundles, and show how conserv…
Market microstructure model with speculators who deduce asset value from prices.
problem Modeling market microstructure with agents who deduce asset value from prices.
method Control-stopping games and coupled control-stopping problems (RBSDEs).
result Existence of a solution to the system of coupled control-stopping problems.
The paper provides a different proof of the result of Brendle-Schoen on the differential sphere theorem. It is shown directly that the invariant cone of curvature operators with positive (or non-negative) complex sectional curvature is preserved by the Ricci flow. This implies, by a result of Böhm-Wilking, that the nor…
New curves generalize flat metrics from quadratic to q-differentials.
problem Determining flat metrics from curve lengths.
method Introduced q-simple curves to generalize results from quadratic to q-differentials.
result Lengths of q-simple curves uniquely determine non-positively curved Euclidean cone metrics induced by q-differentials.
E-NFs generate molecules and their positions while preserving Euclidean symmetries.
problem Generating molecules with their positions while preserving Euclidean symmetries.
method Integrating E(n) graph neural networks into a differential equation to create an invertible equivariant function.
result E-NFs significantly outperform baselines and existing methods in log-likelihood for particle systems and molecules.
Quasi-Lie schemes for PDEsmath.AP
Extends quasi-Lie systems to PDEs for integrability analysis.
problem Analyzing integrability conditions for PDEs.
method Develops a procedure to construct quasi-Lie systems for PDEs through quasi-Lie schemes.
result Obtains -dependent superposition rules and integrability conditions.
Alternative proof for non-existence of complete curves in differential strata.
problem Non-existence of complete algebraic curves in strata of holomorphic differentials.
method Using positivity of divisor classes on moduli spaces of curves.
result Alternative proof confirming Gendron's result on non-existence.
This paper deals with an optimal position management problem for a market maker who has to face uncertain customer order flows in an illiquid market, where the market maker's continuous trading incurs a stochastic linear price impact. Although the execution timing is uncertain, the market maker can also ask its OTC cou…
In the present article the geometry of semi-Riemannian manifolds with nonholonomic constraints is studied. These manifolds can be considered as analogues to the sub-Riemannian manifolds, where the positively definite metric is substituted by a nondegenerate metric. To study properties of the exponential map the Christo…
In this paper, we prove a differential Harnack inequality for positive solutions of time-dependent heat equations with potentials. We also prove a gradient estimate for the positive solution of the time-dependent heat equation.
System for automatic differentiation in a functional array-processing language.
problem Efficient automatic differentiation in functional array languages.
method Automatic differentiation in a higher-order functional array-processing language with source-to-source support and global optimizations.
result The system outperforms state-of-the-art tools on machine learning and computer vision benchmarks.
The study constructs differential systems on Riemann surfaces and explores their monodromy properties.
problem Constructing holomorphic differential systems with specific monodromy properties.
method Exploring the monodromy of holomorphic differential systems on Riemann surfaces.
result Holomorphic maps from Riemann surfaces to quotient spaces exist without factoring through elliptic curves.
This paper is being replaced by another of the author's that contains a brief summary of the problem of positivity of Green's functions, heat kernels, and principal eigenvalues of higher-order elliptic differential operators.
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.
Differential equations in a tangent category I: Complete vector fields, flows, and exponentialsmath.CT
Abstract: Defines differential equations in tangent categories, providing conditions for completeness and new perspectives.
problem Defining and working with differential equations in abstract tangent categories.
method Introduces curve objects and dynamical systems, providing conditions for completeness and exploring exponential maps.
result Provides abstract conditions for dynamical systems to be complete and introduces differential exponential rig.
The paper generalizes Monge-Ampère equations and their solutions in differential geometry.
problem Understanding the structure of Monge-Ampère equations and their solutions.
method Generalizing Monge-Ampère equations to higher-order systems and proving their solutions correspond to integral manifolds of exterior differential systems.
result The Korteweg-de Vries (KdV) equation and Cauchy-Riemann equations are examples of generalized Monge-Ampère equations.
This paper establishes three relations between the Toda field theory associated to a simple Lie algebra and the integral curves of the standard differential system on the corresponding complete flag variety. The motivation comes from the viewpoint on the Toda field theories as Darboux integrable differential systems as…
The purpose of this paper is to connect two subjects: the theory of quantum integrable systems (complete commutative rings of differential operators), and differential Galois theory. We define quantum completely integrable systems (QCIS), algebraically integrable QCIS, the differential Galois group of a QCIS. We show t…
We study the interplay between the differential Galois group and the Lie algebra of infinitesimal symmetries of systems of linear differential equations. We show that some symmetries can be seen as solutions of a hierarchy of linear differential systems. We show that the existence of rational symmetries constrains the …
Two privacy-preserving rating collection methods for recommender systems.
problem Collecting user ratings while maintaining privacy.
method Modified Laplace mechanism and randomized response.
result Both mechanisms are differentially private and preserve data utility.
ADVI simplifies model fitting for probabilistic models.
problem Fitting complex models to large datasets is computationally challenging.
method Automatic differentiation variational inference (ADVI).
result ADVI automatically derives efficient variational inference algorithms.
Affine Processesmath.PR
We put forward a complete theory on moment explosion for fairly general state-spaces. This includes a characterization of the validity of the affine transform formula in terms of minimal solutions of a system of generalized Riccati differential equations. Also, we characterize the class of positive semidefinite process…
We discover a fundamental exterior differential system of Riemannian geometry; indeed, an intrinsic and invariant global system of differential forms of degree associated to any given oriented Riemannian manifold of dimension . The framework is that of the tangent sphere bundle of . We generalise to a R…