arXiv research

We find the complete set of fundamental invariants for systems of ordinary differential equations of order $\ge 4$ under the group of point transformations generalizing similar results for contact invariants of a single ODE and point invariants of systems of the second and the third order. It turns out that starting fr…

Invariants defined for braid systems under Hurwitz equivalence.

New invariants for 3-manifolds using special G-systems.

Proves partial-dual genus polynomial is a knot invariant weight system.

New invariant for virtual links using multi-switches and algebraic systems.

Study Vassiliev invariants for virtual knots, expanding quantum theory.

Study non-asymptotic bounds on correlation in high-dimensional linear systems, revealing invariant subspaces and bottlenecks.

The paper explores weight systems and their applications to graph and embedded graph invariants.

The paper studies invariant complex manifolds in holomorphic slow-fast systems.

We give a construction of Kirby weight systems associated to sl(2) and valued into the finite field Z/pZ. We show that it is possible to apply this sequence of weight systems on the universal invariant of framed link. We also show that the corresponding sequence admits a Fermat limit, which defines an asymptotic ration…

Paper defines invariants for elliptic Weyl groups and connects them to Frobenius structures.

The paper describes solutions for linear control systems on Lie groups.

Study finds Stäckel equivalence for superintegrable systems via invariant quadrics.

In integrable hydrodynamic systems, coordinates exist where generators and symmetries are simple.

Proposes local coordinate frames for improving model performance in complex dynamical systems.

Invariant reduction preserves Poisson structures in PDEs.

Constructs a 4-invariant for graphs at c = 3/8.

Let $\mathcal{S}$ be an integrable Pfaffian system. If it is invariant under a transversally free infinitesimal action of a finite dimensional real Lie algebra $g$ and consequently invariant under the local action of a Lie group $G$, we show that the vertical variational cohomology of $\mathcal{S}$ is equal to the Lie …

All DeFi markets are essentially CFMMs with increasing invariants.

Invariant measures found for contact Hamiltonian systems split into Reeb and Liouville dynamics.

We apply Arnold's theory of generic smooth plane curves to Stark-Zeeman systems. This is a class of Hamiltonian dynamical systems that describes the dynamics of an electron in an external electric and magnetic field, and includes many systems from celestial mechanics. Based on Arnold's $J^+$-invariant, we introduce inv…

New Y-systems for Miquel dynamics are Möbius invariant.

Enhances Vassiliev knot invariants using chord diagrams.

New weight systems derived from a specific Lie algebra for knot invariants.

Study generalizes non-interaction theorems for relativistic systems.

The paper calculates a specific weight system for chord diagrams with a particular graph structure.

Study magnetic curvature on Lie groups, extending Milnor's work.

We use Polyak's skein relation to give a new proof that Milnor's string link homotopy invariants are finite type invariants, and to develop a recursive relation for their associated weight systems. We show that the obstruction to the triviality of these weight systems is the presence of a certain kind of spanning tree …

This work generalizes Hamiltonian mechanics using closed differential forms.

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.

New cohomology theory shows compact Lie group actions are Morita invariant.

The description of invariants of surfaces with respect to the motion groups is reduced to the description of invariants of parameterized surfaces with respect to the motion groups. Existence of a commuting system of invariant partial differential operators (derivatives) and a finite system of invariants, such that any …

New solution to Einstein-Maxwell equations invariant under dilations.

In this paper, we first study the Poisson reductions of controlled Hamiltonian (CH) system and symmetric CH system by controllability distributions. These reductions are the extension of Poisson reductions by distribution for Poisson manifolds to that for phase spaces of CH systems with external force and control. We g…

In this article, we define an independence system for a classical knot diagram and prove that the independence system is a knot invariant for alternating knots. We also discuss the exchange property for minimal unknotting sets. Finally, we show that there are knot diagrams where the independence system is a matroid and…

We discuss the problem of the existence of a regular invariant Lagrangian for a given system of invariant second-order differential equations on a Lie group $G$, using approaches based on the Helmholtz conditions. Although we deal with the problem directly on $TG$, our main result relies on a reduction of the system on…

The paper develops algorithms and topological invariants for distinguishing dynamic systems.

A framework for reducing PDEs by symmetry, preserving key structures.

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…

We discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom. Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. We a…

We introduce systems of objects and operators in linear monoidal categories called $\hat Ψ$-systems. A $\hat Ψ$-system satisfying several additional assumptions gives rise to a topological invariant of triples (a closed oriented 3-manifold $M$, a principal bundle over $M$, a link in $M$). This construction generalizes …

In this note, we consider generalizations of the asymptotic Hopf invariant, or helicity, for Hamiltonian systems with one-and-a-half degrees of freedom and symplectic diffeomorphisms of a two-disk to itself.

We compute the characteristic Cartan connection associated with a system of third order ODEs. Our connection is different from Tanaka normal one, but still is uniquely associated with the system of third order ODEs. This allows us to find all fundamental invariants of a system of third order ODEs and, in particular, de…

The paper introduces a new differential-geometric system which originates from the theory of $m$-Hessian operators. The core of this system is a new notion of invariant differentiation on multidimensional surfaces. This novelty gives rise to the following absolute geometric invariants: invariant derivatives of the surf…

Study curvature and torsion in Gaussian distribution's dual coordinate system.

New variational principles found for conformal geodesics.

Explicitly describes pluriclosed metrics on compact Lie groups.

Category theory generalizes finite type invariants using diagrams systems.