The paper defines equations for constructing projective structures on 3-manifolds.
problem Constructing projective structures on 3-manifolds.
method Defining a system of real valued equations and inequalities.
result These equations can be used to detect properly convex structures on 3-manifolds.
Proves local solvability for G2-structures with Poisson equations.
problem Local solvability of Poisson equations for G2-structures. method Proves local solvability for G2-structures with Poisson equations. result Local solvability of Poisson equations for closed G2-structures. Solutions to a differential equation link to contact structures.
problem Linking solutions of a specific differential equation to contact structures.
method Established a correspondence between solutions of Noth's equation and diffeomorphisms of contact structures.
result Established a correspondence between solutions of Noth's equation and diffeomorphisms of contact structures of type G2. Paper discovers structural dynamics equations from only acceleration data.
problem Discovering equations from only acceleration measurements in structural dynamics.
method Library-based approach with Approximate Bayesian Computation (ABC) prioritizing parsimonious models.
result Efficacy demonstrated in four structural dynamics examples, including linear and nonlinear systems.
Formula connects G2-structure geometry to Poisson equation.
problem Solvability conditions for G2-structures in a Poisson equation. method Developed a Gauss-Codazzi-like formula for G2-structures. result Necessary and sufficient conditions for solvability in cohomogeneity one.
Superintegrable systems on surfaces are classified geometrically.
problem Classifying superintegrable systems on conformal surfaces.
method Geometric structures on conformal surfaces, conformal covariant structural equations.
result Explicit set of algebraic equations defining superintegrable systems on all constant curvature surfaces.
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…
New equations for Cosserat media motions derived from bundle automorphisms.
problem Modeling deformations in Cosserat media.
method Euler-type equations derived from SO(3)-bundle automorphisms.
result Presented new equations for Cosserat media motions.
Global invariant for path structures and differential equations defined on torus.
problem Global invariant for path structures and differential equations.
method Computed as a secondary invariant from a Cartan connection on a canonical bundle.
result Formula for global invariant of second order differential equations on torus.
In this review article we discuss four recent methods for computing Maurer-Cartan structure equations of symmetry groups of differential equations. Examples include solution of the contact equivalence problem for linear hyperbolic equations and finding a contact transformation between the generalized Hunter-Saxton equa…
Extends Zeitlin's model to 3-D axisymmetric Euler equations.
problem Preserving geometric structure in 3-D Euler equations.
method Axisymmetric discretization of 3-D Euler equations on the 3-sphere.
result First discretization of 3-D Euler equations preserving geometric structure.
Solves constant pre-factor problem for tt*-Toda equations using asymptotic data and symplectic structures.
problem Constant pre-factor problem for the tt*-Toda equations.
method Explicit evaluation using asymptotic data and introduction of symplectic structures.
result Preservation of symplectic structures by Riemann-Hilbert correspondence for wider class of solutions.
Study on G2 structures with torsion and existence of solutions.
problem Existence and properties of strong G2-structures with torsion.
method Investigation of the twisted G2 equation and analysis of invariant structures.
result Non-existence of non-trivial solutions on compact solvmanifolds.
Maps dBKP solutions to MS system solutions, defining Einstein-Weyl structures.
problem Constructing solutions and structures for dBKP and MS systems.
method Map construction and spectral characterisation of reductions.
result Defines Einstein-Weyl structures for dBKP and BMS systems.
Monograph explores algebraic structures related to Yang-Baxter equation.
problem Yang-Baxter equation and its solutions in algebra.
method Investigation of skew braces, quandles, racks, and Rota-Baxter groups.
result Interrelations and applications of these structures to knot theory.
We discuss relations between the para-CR structures and differential equations (both ODEs and PDEs of finite type).
In this paper we show that the dimensionally reduced Seiberg-Witten equations lead to a Higgs field and study the resulting moduli spaces. The moduli space arising out of a subset of the equations, shown to be non-empty for a compact Riemann surface of genus g >= 1, gives rise to a family of moduli spaces carrying a hy…
We consider structural equation models in which variables can be written as a function of their parents and noise terms, which are assumed to be jointly independent. Corresponding to each structural equation model, there is a directed acyclic graph describing the relationships between the variables. In Gaussian structu…
Abstract: Study Hamiltonian systems on almost cosymplectic manifolds, extending contact Hamiltonian systems.
problem Extend Hamiltonian systems to almost cosymplectic manifolds.
method Determine Hamiltonian vector field on odd-dimensional almost cosymplectic manifolds.
result Extend equations of motion to generalized transitive almost cosymplectic structures.
In differential-geometric language, vortex-lines equations on extended phase space of a system may be written as iγ˙dσ=0, where σ is a differential 1-form. This is the structure, to give a paradigmatic example, of the Hamilton equations. Here, we study equations of the same structure, where σ is a differen…
Geometrodynamics derived from Riemannian manifolds using geospin matrix.
problem Formulating dynamics on Riemannian manifolds using Cartan structural equations.
method Introducing four real dynamical variables and applying them to Cartan structural equations.
result Rewritten Cartan structural equations in a real geometrodynamical form.
The paper proves quasi-Einstein structures on manifolds admit Killing vector fields and provides new examples.
problem Classifying quasi-Einstein structures and understanding their properties.
method Analyzing quasi-Einstein equations and exploring their connections to Hitchin's equations.
result A class of quasi-Einstein structures on closed manifolds must admit a Killing vector field.
Study solves complex Hessian equation on Hermitian manifolds.
problem Solving Hessian equations on Hermitian manifolds with mixed structure.
method Derive a priori estimates and solve Dirichlet problem under conditions.
result Solvability of the Dirichlet problem for mixed Hessian equations.
We apply the technique of integrable extensions to the symmetry pseudo-group of the dKP-hyper CR interpolating equation. This allows us to find a covering for this equation and to construct multi-valued Einstein-Weyl structures.
The paper studies hyperkähler structures and adapted complex structures using the Monge-Ampère equation.
problem Finding hyperkähler structures and adapted complex structures in tangent bundles.
method Analyzing the asymptotic expansion of the Monge-Ampère equation and using gauge transformations.
result Explicit computation of 4th order terms in the asymptotic expansion and equivalence to gauge transformations.
Using nonlinear pde techniques, we construct a new family of globally smooth tt* structures. This includes tt* structures associated to the (orbifold) quantum cohomology of a finite number of complex projective spaces and weighted projective spaces. The existence of such "magical solutions" of the tt* equations, namely…
Study establishes monodromy equivalence for Lamé-type equations and constructs cone spherical metrics.
problem Investigating monodromy equivalence and finite-gap structures of Lamé-type equations.
method Analyzing finite-gap structures and constructing cone spherical metrics.
result Established monodromy equivalence between classical and generalized Lamé-type equations, derived finite-gap structures, and constructed cone spherical metrics.
New framework tackles geometric structure existence and classification.
problem Existence and classification of geometric structures.
method Developed a new framework of relative algebroids.
result New framework addresses geometric structure problems.
This article is a local analysis of integrable GL(2)-structures of degree 4. A GL(2)-structure of degree n corresponds to a distribution of rational normal cones over a manifold M of dimension (n+1). Integrability corresponds to the existence of many submanifolds that are spanned by lines in the cones. These GL(2)-stru…
We describe a method for solving the Maurer-Cartan structure equation associated with a Lie algebra that isolates the role of the Jacobi identity as an obstruction to integration. We show that the method naturally adapts to two other interesting situations: local symplectic realizations of Poisson structures, in which …
Scroll structures on solutions of 4D integrable equations are involutive and governed by a dispersionless hierarchy.
problem Characterizing the geometry of solutions to 4D integrable equations.
method Defining rational normal scrolls and showing their involutivity.
result Involutive scroll structures are governed by a dispersionless integrable hierarchy.
We show existence and uniqueness of solutions to the Monge-Ampere equation on compact almost complex manifolds with non-integrable almost complex structure.
The paper studies Einstein-type manifolds with structural conditions.
problem Investigating geometric structures on Riemannian manifolds.
method Unified approach to various geometric structures and curvature conditions.
result Rigidity results for Einstein-type manifolds under specific curvature conditions.
Affine manifolds linked to integrable equations and geometric structures.
problem Understanding the geometric and algebraic properties of affine manifolds.
method Analyzing the Kahlerian tangent bundle and multi-dimensional consistency of the TED equation.
result Affine manifolds are related to self-dual Einstein spaces and Hessian structures.
The paper constructs invariant Calabi-Yau structures on complexified symmetric spaces.
problem Constructing invariant Calabi-Yau structures on complexified symmetric spaces.
method Solutions of a Monge-Ampère type equation.
result Existence of solutions to the Monge-Ampère type equation.
Solves geometric problems using fully nonlinear equations and Morse theory.
problem Geometric problems, specifically Loewner-Nirenberg and Yamabe problems.
method Investigates structure of fully nonlinear equations and applies Morse theory techniques.
result Constructs admissible metrics under weak conditions and demonstrates topological obstructions.
Study proves structure results for homogeneous spaces supporting specific equations.
problem Proving structure results for homogeneous spaces supporting specific equations.
method Analyzing homogeneous spaces with non-constant solutions to two general classes of equations involving the Hessian and an invariant 2-tensor.
result Generalizes rigidity results for gradient Ricci solitons and warped product Einstein metrics.
New conformal geometry method solves Einstein-Weyl equations.
problem Solving Einstein-Weyl equations in 4D spacetimes.
method Combining conformal and complex geometry techniques.
result Reduced Einstein-Weyl equations to a single conformally invariant scalar equation.
We show that supersymmetric flux vacua with intermediate SU(2) structure is closely related to some special classes of half-flat structures. More concretely, solutions of the SUSY equations IIA possess a symplectic half-flat structure, whereas solutions of the SUSY equations IIB admit a half-flat structure which is in …
The hyper-CR Einstein-Weyl structures on R3 can be described in terms of the solutions to the dispersionless Hirota equation. In the present paper we show that simple geometric constructions on the associated twistor space lead to deformations of the Hirota equation that have been introduced recently by B. Krugliko…
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.
We explore variational Poisson-Nijenhuis structures on nonlinear PDEs and establish relations between Schouten and Nijenhuis brackets on the initial equation with the Lie bracket of symmetries on its natural extensions (coverings). This approach allows to construct a framework for the theory of nonlocal structures.
An AH (affine hypersurface) structure is a pair comprising a projective equivalence class of torsion-free connections and a conformal structure satisfying a compatibility condition which is automatic in two dimensions. They generalize Weyl structures, and a pair of AH structures is induced on a co-oriented non-degenera…
We consider Riemannian 4-manifolds (X,gX) with a Spin^c-structure and a suitable circle bundle Y over X such that the Spin^c-structure on X lifts to a spin structure on Y. With respect to these structures a spinor φ on X lifts to an untwisted spinor ψ on Y and a U(1)-gauge field A for the Spin^c-st…
Paper solves Hessian equations on Kähler manifolds.
problem Solving Hessian equations on Kähler manifolds.
method Combines elementary symmetric functions; provides sufficient and necessary condition.
result Generalizes results for Hessian and Hessian quotient equations.
In \cite{LZ2} it is proved that for certain class of perturbations of the hyperbolic equation ut=f(u)ux, there exist changes of coordinate, called quasi-Miura transformations, that reduce the perturbed equations to the unperturbed one. We prove in the present paper that if in addition the perturbed equations posse…
Graphical notation simplifies complex polynomial constraints in linear models.
problem Complex polynomial constraints in linear structural equation models are impractical.
method Developed a graphical notation to represent these constraints.
result The graphical notation simplifies the representation of many polynomial constraints.
The paper proves isometric embedding equations in low Sobolev regularity.
problem Proving isometric embedding equations in low Sobolev regularity.
method Proving Cartan's and Gauss's equations for C0∩H21 frames and deducing the Gauss equation for C1∩W1+32,3 isometric embeddings. result Gauss equation holds for C1∩W1+32,3 isometric embeddings.