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-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
169,341 papers · 148 categories
Trend · papers per month
48 results for “Differential systems”
Paper connects Toda field theories to differential systems on flag varieties.
problem Connecting Toda field theories to differential systems on flag varieties.
method Establishing isomorphisms and quotient relations between jet spaces and flag varieties.
result Toda field theories are related to differential systems on flag varieties through quotient actions.
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.
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.
Study the connection between differential Galois groups and symmetries of linear differential equations.
problem Understanding the relationship between differential Galois groups and symmetries in systems of linear differential equations.
method Investigate the hierarchy of linear differential systems and the constraints on differential Galois groups due to rational symmetries.
result Rational symmetries impose constraints on the differential Galois group, dependent on the Maclaurin series of the symmetry.
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.
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.
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…
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.
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…
Extends exterior diff. sys. to Lie algebroids with examples.
problem Invariant inverse problem of the calculus of variations
method Extends exterior differential systems to Lie algebroids, defines integral manifolds.
result Defines integral manifolds for exterior diff. systems on Lie algebroids.
New tensors and invariants found in 3D Riemannian geometry.
problem Understanding intrinsic properties of 3D Riemannian manifolds.
method Applying a fundamental exterior differential system to 3D Riemannian geometry.
result Discovery of new global tensors and intrinsic invariants.
In this paper we study the differential systems on Leibniz algebroids. We introduce a class of almost metriplectic manifolds as a special case of Leibniz manifolds. Also, the notion of almost metriplectic algebroid is introduced. These types of algebroids are used in the presentation of associated differential systems.…
In this paper we analyze the tangential symmetries of Darboux integrable decomposable exterior differential systems. The decomposable systems generalize the notion of a hyperbolic exterior differential system and include the classic notion of Darboux integrability for first order systems and second order scalar equatio…
Flat systems of up to 2 dimensions have flat subsystems.
problem Characterizing flat subsystems in flat systems of differential dimension 2.
method Analyzing subsystems of a flat system of differential dimension at most 2.
result Flat subsystems of a flat system of differential dimension at most 2 exist and can have independent time-uniform outputs.
A complete solution to the multiplier version of the inverse problem of the calculus of variations is given for a class of hyperbolic systems of second-order partial differential equations in two independent variables. The necessary and sufficient algebraic and differential conditions for the existence of a variational…
On Driftless Systems with m controls and 2m or 2m-1 states that are Flat by Pure Prolongationmath.OC
New findings on flatness for specific driftless systems.
problem Determining flatness for driftless systems with m inputs and 2m or 2m-1 states.
method Using pure prolongation, the paper presents new sufficient conditions for flatness.
result The conditions proposed broaden the class of recognized flat systems.
Paper solves complex control problems using novel SDEs.
problem Solving stochastic differential games for nonlinear systems.
method Uses Deep Forward-Backward SDEs with neural networks.
result Numerical solution validated on two example systems.
Neural differential equations combine deep learning and differential equations for modeling complex systems.
problem Modeling complex systems with high capacity and efficiency.
method Combining neural networks and differential equations, focusing on neural ordinary, controlled, and stochastic differential equations.
result NDEs offer high-capacity function approximation, strong priors, and handle irregular data efficiently.
New Lie systems derived from Goursat distributions with applications to differential equations.
problem Analyzing Lie systems associated with Goursat distributions and their applications.
method Analyzing bracket-generating distributions and their relation to Lie systems, focusing on reductions and reconstructions.
result Lie systems associated with Goursat distributions can be reduced and solutions reconstructed from reduced systems.
This is an elementary introduction to exterior differential systems motivated by two examples: minimal submanifolds and the isometric embedding problem. The two main goals of the lectures are: 1. To explain how to find an appropriate geometric setting for studying a given system of pde. 2. To explain the Cartan algorit…
A theorem of Maurer-Cartan type for Lie algebroids is presented. Suppose that any vector subbundle of a Lie algebroid is called interior differential system (IDS) for that Lie algebroid. A theorem of Cartan type is obtained. Extending the classical notion of exterior differential system (EDS) to Lie algebroids, a theor…
Survey uses Milnor fibrations to classify first integrals of differential systems.
problem Classifying first integrals of differential systems using geometric-topological methods.
method Utilizing Milnor fibrations and connections with harmonic morphisms to provide topological and geometric descriptions.
result Geometric-topological classifications of first integrals for both isolated and non-isolated singularities.
Characterizes differential forms and vector fields with constant coefficients on manifolds.
problem Understanding constant coefficient differential forms and vector fields on manifolds.
method Analyzes differential forms and vector fields of specific degrees, proving obstructions and characterizing solutions to partial differential systems.
result Characterizes differential forms and vector fields with constant coefficients of various degrees on smooth manifolds.
Proposes a privacy-preserving recommendation system using matrix factorization and differential privacy.
problem Privacy leakage in recommendation systems when anonymizing user data is not sufficient.
method Uses matrix factorization and differential privacy via the Gaussian mechanism.
result Demonstrates excellent utility for privacy-preserving recommendation systems.
Stabilized neural differential equations enforce constraints on dynamical systems.
problem Ensuring dynamical systems preserve known constraints like conservation laws.
method SNDEs with a stabilization term to enforce manifold constraints.
result SNDEs outperform existing methods and broaden constraint types.
We propose definitions of homogeneity and projective equivalence for systems of ordinary differential equations of order greater than two, which allow us to generalize the concept of a spray (for systems of order two). We show that the Euler-Lagrange fields of parametric Lagrangians of order greater than one which are …
The systems of complex analytic second order ordinary differential equations whose solutions close up to become rational curves (after analytic continuation) are characterized by the vanishing of an explicit differential invariant, and turn out to provide an infinite dimensional family of integrable systems.
In this paper we show that there are applications that transform the movement of a pendulum into movements in . This can be done using Euler top system of differential equations. On the constant level surfaces, Euler top system reduces to the equation of a pendulum. Those properties are also considered in…
Global regularity and solvability of left-invariant differential systems on compact Lie groupsmath.AP
Abstract: Studies differential systems on compact Lie groups, extending Greenfield and Wallach's methods.
problem Global properties of left-invariant differential systems on compact Lie groups.
method Abstract: Extends Greenfield and Wallach's methods to systems, obtaining characterizations for regularity, range closeness, and cohomology spaces.
result Abstract: Derives generalizations of results and global versions of Caetano and Cordaro's result.
The problem of feedback equivalence for control systems is considered. An algebra of differential invariants and criteria for the feedback equivalence for regular control systems are found.
We establish a link between the study of completely integrable systems of partial differential equations and the study of generic submanifolds in C^n. Using the recent developments of Cauchy-Riemann geometry we provide the set of symmetries of such a system with a Lie group structure. Finally we determine the precise u…
Exterior differential systems are given, and their Cartan characters calculated, for Maxwell and SU(2)-Yang-Mills equations in dimensions from three to six.
In this paper we present a far-reaching generalization of E. Vessiot's analysis of the Darboux integrable partial differential equations in one dependent and two independent variables. Our approach provides new insights into this classical method, uncovers the fundamental geometric invariants of Darboux integrable syst…
For For a given PDE system, or an exterior differential system possessing a Lie group of internal symmetries the orbit reduction procedure is introduced. It is proved that the solutions of the reduced exterior differential system are in one-to-one correspondence with the moduli space of regular solutions of the prolong…
We give a new mechanism for constructing Backlund transformations by using symmetry reduction of differential systems. We then characterize a family of Backlund transformations between Darboux integrable systems where the Backlund transformation can be constructed by the proposed symmetry reduction method.
Monodromy map from differential systems to character variety is generically immersive for complex -representations.
problem Characterizing when the monodromy map is immersive for differential systems.
method Analyzing the space of -differential systems on a compact Riemann surface and the character variety of -representations.
result The monodromy map is an immersion at the generic point when the complex dimension of is at least three.
The properties of the Riemann extensions of nonriemannian spaces defined by the first order systems of differential equations are considered.
New integrable matrix PDEs derived from Frölicher-Nijenhuis brackets.
problem Developing integrable systems from tensor field properties.
method Using Frölicher-Nijenhuis brackets to generate bi-differential graded algebras and PDE systems.
result New integrable nonlinear matrix PDEs and systems are derived.
Analyzes Lie and Cartan's work on differential equations and their solutions.
problem Integration of partial differential equations and existence of solutions.
method Lie groupoids, Lie pseudo-groups, Grassmannian contact structures, local equivalence problem.
result Necessary criteria for the existence of solutions to partial differential equations.
Systems of partial differential equations lie at the heart of physics. Despite this, the general theory of these systems has remained rather obscure in comparison to numerical approaches such as finite element models and various other discretisation schemes. There are, however, several theoretical approaches to systems…
New formula for instantaneous frequency in unbalanced systems.
problem Estimating frequency in unbalanced electrical systems.
method Utilizes affine differential geometry to link frequency and voltage derivatives.
result Proposes a new formula for instantaneous frequency estimation.
Develops a mathematical framework for causal fermion systems in infinite dimensions.
problem Analysis of causal fermion systems in infinite-dimensional settings.
method Introduces Banach manifold structure and expedient differential calculus.
result Establishes Hölder continuity of causal Lagrangian and integrated causal Lagrangian.
We generalize the classical Lie results on a basis of differential invariants for a one-parameter group of local transformations to the case of arbitrary number of independent and dependent variables. It is proved that if universal invariant of a one-parameter group is known then a complete set of functionally independ…
On the port-Hamiltonian representation of systems described by partial differential equationsmath.OC
We consider infinite dimensional port-Hamiltonian systems. Based on a power balance relation we introduce the port-Hamiltonian system representation where we pay attention to two different scenarios, namely the non-differential operator case and the differential operator case regarding the structural mapping, the dissi…
In this paper, the symmetry group of a differential system of n quadratic homogeneous first order ODEs of n variables is studied. For this purpose, we consider the action of both point and contact transformations to signify the corresponding Lie algebras. We also find the independent differential invariants of these ac…
We approach the construction of Backlund transformations for Darboux integrable hyperbolic partial differential equations in the plane through the reduction of exterior differential systems. For example it is shown that all the Backlund transformations in arXiv:0707.4408v2 can be constructed using symmetry reduction.