We introduce two numerical conjugacy invariants for dynamical systems -- the complexity and weak complexity indices -- which are well-suited for the study of "completely integrable" Hamiltonian systems. These invariants can be seen as "slow entropies", they describe the polynomial growth rate of the number of balls (fo…
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48 results for “complex integrable systems”
Study connects landslide flow to integrable systems for harmonic maps.
problem Understanding the holonomy of complex landslide flow.
method Integrable systems approach to harmonic maps into symmetric spaces.
result Holonomy of complex landslide flow derived from harmonic map holonomy.
Elliptic systems are characterized by Darboux integrability.
problem Characterizing elliptic differential systems with holomorphic solutions.
method Using a complex manifold and associated holomorphic Pfaffian system.
result Elliptic systems are Darboux integrable under generic conditions.
Based on the classical Plücker correspondence, we present algebraic and geometric properties of discrete integrable line complexes in . Algebraically, these are encoded in a discrete integrable system which appears in various guises in the theory of continuous and discrete integrable systems. Geometrically, the e…
In this paper we deal with the classical question of existence of polynomial in momenta integrals for geodesic flows on the 2-torus. For the quasi-linear system on coefficients of the polynomial integral we consider the region (so called elliptic regions) where there are complex-conjugate eigenvalues. We show that for …
Overview of integrable systems with symmetries, focusing on toric and semitoric systems.
problem Classifying and understanding integrable systems with symmetries.
method Using decorated polygons and controlled bifurcations in one-parameter families of systems.
result Construction of explicit semitoric systems with prescribed invariants.
Abstract integrable systems found on a specific manifold.
problem Holomorphic integrable systems on hyperkähler manifolds.
method Study of manifold, canonical abstract integrable system, traditional integrable systems.
result Canonical abstract integrable system on .
Study pseudo-holomorphic disks in real analytic hypersurfaces using exterior differential systems.
problem Existence of pseudo-holomorphic disks in non-integrable real analytic hypersurfaces.
method Theory of exterior differential systems.
result Non-existence of certain equivalent structures in the non-integrable case.
Paper models uncertainty in electricity and gas markets to assess its impact.
problem Addressing uncertainties in coupled electricity and gas markets.
method Integrated and stochastic optimisation approaches for large-scale energy systems.
result Quantifies the value of encoding uncertainty in models.
The paper extends geometric study to systems of PDEs, introducing variational bi-complex for conservation laws.
problem Geometric study of systems of semi-linear hyperbolic PDEs in three variables.
method Introduces variational bi-complex to define form-valued conservation laws and provides a method for generating conservation laws.
result Generates infinitely many conservation laws for systems of three equations.
Developed a symplectic integrator for complex manifolds.
problem Simulating Hamiltonian systems on specific manifolds.
method Partitioned Runge--Kutta methods for Hamiltonian systems on products of Hamiltonian manifolds, with derived symplecticity conditions.
result Derived algebraic conditions for symplecticity of methods.
SRNNs learn dynamics of physical systems from data.
problem Learning dynamics of complex, noisy Hamiltonian systems.
method SRNNs model Hamiltonian functions with neural networks, using symplectic integration and optimization.
result SRNNs reliably learn dynamics of complex and noisy Hamiltonian systems.
In the spirit of Klein's Erlangen Program, we investigate the geometric and algebraic structure of fundamental line complexes and the underlying privileged discrete integrable system for the minors of a matrix which constitute associated Plücker coordinates. Particular emphasis is put on the restriction to Lie circle g…
Proves local bi-integrability of bi-Hamiltonian systems via bi-Poisson reduction.
problem Local bi-integrability of bi-Hamiltonian systems.
method Bi-Poisson reduction to prove local bi-integrability.
result Constructs a complete set of functions in bi-involution for bi-Hamiltonian systems.
Paper constructs solutions for a class of overdetermined systems.
problem Constructing solutions for a class of overdetermined systems.
method Resolution of the solution sheaf, sufficient condition for global exactness, gluing techniques, local solvability of the Treves complex.
result Obtained a sufficient condition for global exactness, leading to gluing techniques for local solutions.
New method integrates sparse parametric and nonparametric techniques for complex system modeling.
problem Lack of accurate modeling for complex biological systems due to nonlinearities.
method Sparse nonparametric estimation framework combining parametric and nonparametric techniques.
result Accurately captures nonlinearities in complex systems without prior information.
This research shows that Mishchenko-Fomenko subalgebras are completely integrable on all regular orbits.
problem The complete integrability of Mishchenko-Fomenko subalgebras on regular adjoint orbits.
method The approach incorporates the theory of regular -triples and associated Slodowy slices, as developed by Kostant.
result Each Mishchenko-Fomenko subalgebra yields a completely integrable system on all regular orbits.
For integrable Hamiltonian systems with two degrees of freedom whose Hamiltonian vector fields have incomplete flows, an analogue of the Liouville theorem is established. A canonical Liouville fibration is defined by means of an "exact" 2-parameter family of flat polygons equipped with certain pairing of sides. For the…
We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semi-simple Lie algebra and finite order automorphisms. For example, the non-linear Schrödinger equation, the n-wave equation, and the sigma-model are soliton flows; and the equation…
Develops a measure-theoretic framework for complex co-occurrence data.
problem Modeling and interpreting complex co-occurrences in high-dimensional data.
method Introduces measure-theoretic probability and conditional probability, investigates E-integrals.
result Establishes a rigorous measure-theoretic foundation for co-occurrence modeling.
Integral points are potentially dense in character varieties of quasi-projective varieties.
problem Density of integral points in character varieties of quasi-projective varieties.
method Reduction to Riemann surfaces and use of Corlette-Simpson work.
result Integral points have Zariski-dense orbit under the mapping class group.
Study of complex diffeomorphisms and their groups to find first integrals for holomorphic foliations.
problem Finding first integrals for holomorphic foliations under irreducibility conditions.
method Investigate groups of complex diffeomorphisms with irreducibility property and apply to foliations.
result Existence of first integrals for foliations under certain irreducibility conditions.
Symplectic groupoids create Poisson integrators for complex systems.
problem Creating efficient integrators for non-linear Poisson structures.
method Recursive solutions of Hamilton-Jacobi equation, interpreted as Lagrangian bisections.
result Constructs Poisson integrators using symplectic groupoids.
We consider an embedding of a -dimensional CW complex into the -sphere, and construct it's dual graph. Then we obtain a homogeneous system of linear equations from the -dimensional CW complex in the first homology group of the complement of the dual graph. By checking that the homogeneous system of linear equa…
Framework integrates Markov and causal models for accurate counterfactual inference.
problem Lack of counterfactual inference in Markov models and identification in causal models.
method Defines structural causal models in terms of Markov process parameters and equilibrium dynamics, enabling consistent counterfactual inference.
result Proposed framework alleviates identifiability issues and improves accuracy of counterfactual inference.
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.
The paper extends Newlander-Nirenberg theorem to domains with boundary.
problem Extending Newlander-Nirenberg theorem to domains with boundary.
method Analyzing formally integrable complex structures on domains with boundary.
result Existence of global holomorphic coordinate systems on the closure of a bounded strictly pseudoconvex domain.
Proposes neural networks that preserve physical system dynamics.
problem Learning accurate representations of dynamical systems.
method Variational integrator networks designed to preserve geometric structure.
result Accurately learns dynamical systems from noisy observations.
Let L be a link in an integral homology three-sphere. We give a description of the Heegaard Floer homology of integral surgeries on L in terms of some data associated to L, which we call a complete system of hyperboxes for L. Roughly, a complete systems of hyperboxes consists of chain complexes for (some versions of) t…
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.
We started from computer experiments with simple one-dimensional ergodic dynamical systems called interval exchange transformations. Correlators in these systems decay as a power of time. In the simplest non-trivial case the exponent is equal to 1/3. We found a formula connecting characteristic exponents with explicit …
We study integrable geodesic flows on Stiefel varieties given by the Euclidean, normal (standard), Manakov-type, and Einstein metrics. We also consider natural generalizations of the Neumann systems on with the above metrics and proves their integrability in the non-commutative sense b…
We develop a global Poincaré residue formula to study period integrals of families of complex manifolds. For any compact complex manifold equipped with a linear system of generically smooth CY hypersurfaces, the formula expresses period integrals in terms of a canonical global meromorphic top form on . Two…
The SL(2)-character variety X of a closed surface M enjoys a natural complex-symplectic structure invariant under the mapping class group G of M. Using the ergodicity of G on the SU(2)-character variety, we deduce that every G-invariant meromorphic function on X is constant. The trace functions of closed curves on M de…
The study examines stability of Hamiltonian Poisson integrators on both integrable and non-integrable systems.
problem Investigating stability properties of Hamiltonian Poisson integrators.
method Examples of Lotka-Volterra dynamics and numerical investigations of a non-integrable system are used.
result The existence of a modified Hamiltonian is crucial for the stability of Hamiltonian Poisson integrators.
We consider 3-webs, hyper-para-complex structures and integrable Segre structures on manifolds of even dimension and generalise the second heavenly Plebański equation in the context of higher-dimensional hyper-para-complex structures. We also characterise the Segre structures admitting a compatible hyper-para-complex s…
In this paper, multi-component generalizations to the Camassa-Holm equation, the modified Camassa-Holm equation with cubic nonlinearity are introduced. Geometric formulations to the dual version of the Schrödinger equation, the complex Camassa-Holm equation and the multi-component modified Camassa-Holm equation are pro…
Computational Intelligence (CI) is a sub-branch of Artificial Intelligence paradigm focusing on the study of adaptive mechanisms to enable or facilitate intelligent behavior in complex and changing environments. There are several paradigms of CI [like artificial neural networks, evolutionary computations, swarm intelli…
This paper uses a path integral approach to model complex economic systems with many agents.
problem Modeling economic systems with a large number of interacting agents.
method Develops a path integral formalism to describe the behavior of a large number of agents in an economic system.
result The method provides an analytical treatment of business cycle models with many agents, revealing various phases and interactions.
Combines higher complex structures with flat connections to link to -algebras.
problem Linking higher complex structures to -algebras via flat connections.
method Introduces -parabolic connections and studies their curvature.
result Establishes a direct link between flat connections and higher complex structures.
The paper proves conditions for Darboux integrability in diagonal hydrodynamic systems.
problem Conditions for Darboux integrability in diagonal hydrodynamic systems.
method Proof of conditions using Laplace transformation sequences and geometric interpretations.
result Diagonal systems of hydrodynamic type are Darboux integrable if and only if the corresponding systems for commuting flows are Darboux integrable.
Develops methods to solve complex and real Hessian equations.
problem Solving complex and real Hessian equations on various domains.
method Introduces an ansatz to reduce PDEs to systems of ODEs, integrating via abelian integrals.
result Constructs entire solutions of arbitrary subcritical phase for dHYM/LYZ and special Lagrangian equations.
Deep Learning-Based Residual Useful Lifetime Prediction for Assets with Uncertain Failure Modescs.LG
This research improves asset life prediction by integrating deep learning with mixture distributions.
problem Predicting residual useful life for assets with multiple failure modes.
method Integrates mixture (log)-location-scale distribution with deep learning.
result Proposed models outperform existing methods in predicting residual useful life.
Survey of open problems in finite-dimensional integrable systems.
problem Open problems in finite-dimensional integrable systems.
method None specified; survey of existing open problems.
result Many open problems were identified from a conference.
The Mishchenko-Fomenko conjecture says that for each real or complex finite-dimensional Lie algebra $\goth g$ there exists a complete set of commuting polynomials on its dual space $\goth g^*$. In terms of the theory of integrable Hamiltonian systems this means that the dual space $\goth g^*$ endowed with the standard …
New Y-systems for Miquel dynamics are Möbius invariant.
problem Miquel dynamics circle centers are not Möbius invariant.
method Introduced new Y-systems involving only intersection points.
result New Y-systems are Möbius invariant and satisfy the transformation group principle.
We analyze here Hamiltonian stationary surfaces in the complex projective plane as (local) solutions to an integrable system, formulated as a zero curvature on a loop group. As an application, we show in details why such tori are finite type solutions, and eventually describe the simplest of them: the homogeneous ones.
Investigates integrable systems with linear periodic integral for e(3) Lie algebra.
problem Analyzes singularities and topological properties of integrable systems.
method Examines singularities of Liouville foliation, bifurcation diagram, transformations of Liouville tori, and isoenergy surfaces.
result Discovers topological properties of integrable systems with linear periodic integral.