arXiv research

In this article, we give proofs on the Arnold Lagrangian intersection conjecture on the cotangent bundles, Arnold-Givental Lagrangian intersection conjecture and the Arnold fixed point conjecture.

Explain Arnold's proof of the Morse index theorem using Maslov index.

In [R2] and [RO] the Arnold conjecture for closed symplectic manifolds with trivial second homotopy group was proved. This proof used surgery and cobordism theory. Here we give a purely cohomological proof of this result.

Proves Weinstein's and Arnold's conjectures using contact instantons.

The paper confirms Arnold's conjecture about hyperbolic polynomials.

The Hessian Topology is a subject with interesting relations with some classical problems of analysis and geometry. In this article we prove a conjecture on this subject stated by V.I. Arnold concerning the number of connected components of hyperbolic homogeneous polynomials of degree $n$. The proof is constructive and…

A new KAN variant uses sinusoidal activations to approximate functions.

Nearby pinwheels are isotopic, solving Arnold's conjecture.

The Arnold conjecture is proven for integers using Floer theory.

We prove that the Lusternik-Schnirelmann category $cat(M)$ of a closed symplectic manifold $(M, ω)$ equals the dimension $dim(M)$ provided that the symplectic cohomology class vanishes on the image of the Hurewicz homomorphism. This holds, in particular, when $π_2(M)=0$. The Arnold conjecture asserts that the number of…

New integral expression quantizes Arnold strangeness.

In order to generalize finite element methods to differential forms, Arnold, Falk, and Winther constructed two families of spaces of polynomial differential forms on a simplex $T$, the $\mathcal P_rΛ^k(T)$ spaces and the $\mathcal P_r^-Λ^k(T)$ spaces, where $k$ is the degree of the form and $r$ is the degree of its coe…

Kolmogorov-Arnold Networks promise scalable performance in high dimensions.

New integer-valued functions for Legendrian knots.

This note is purely expository. In the course of the Kolmogorov-Arnold solution of Hilbert's 13th problem on superpositions there appeared the notion of basic embedding. A subset K of R^2 is basic if for each continuous function f:K->R there exist continuous functions g,h:R->R such that f(x,y)=g(x)+h(y) for each point …

Revisits the connection between neural networks and the Kolmogorov-Arnold theorem.

We define the generalized connected sum for generic closed plane curves, generalizing the strange sum defined by Arnold, and completely describe how the Arnold invariants $J^{\pm}$ and $\mathit{St}$ behave under the generalized connected sums.

Arnold-Liouville systems cannot be bi-Hamiltonian generically.

The paper proves that most metrics satisfy a strong version of Arnold's conjecture for Laplace eigenvalues.

Solves an Arnold trivium problem using calculus and topology.

Proves Arnold-Thom conjecture for surfaces' arrival times.

Explains Arnold's J+ invariant for curves, using basic math.

A new Kolmogorov-Arnold network improves function approximation and optimization.

SVGP KAN integrates uncertainty quantification into Kolmogorov-Arnold networks.

Recently V. Arnold introduced Strangeness and $J^{\pm}$ invariants of generic immersions of an oriented circle to $\R^2$. Here these invariants are generalized to the case of generic immersions of an oriented circle to an arbitrary surface $F$. We explicitly describe all the invariants satisfying axioms, which naturall…

Combines geometric hydrodynamics with magnetic systems to derive new equations and prove well-posedness.

Proves Arnold conjecture for singular symplectic manifolds using novel techniques.

Study shows almost all Arnold stable solutions have no conjugate points.

In this paper, we define finite type invariants for cyclic equivalence classes of nanophrases and construct the universal ones. Also, we identify the universal finite type invariant of degree 1 essentially with the linking matrix. It is known that extended Arnold's basic invariants to signed words are finite type invar…

Unified framework for Arnold-type invariants via dual complexes and finite-difference structures.

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…

We prove the Arnold conjecture for closed symplectic manifolds with $π_2(M)=0$ and $\cat M=\dim M$. Furthermore, we prove an analog of the Lusternik-Schnirelmann theorem for functions with ``generalized hyperbolicity'' property.

Kolmogorov-Arnold Networks achieve optimal convergence rates in nonparametric regression.

Kolmogorov-Arnold Networks improve deep learning adaptivity and can approximate Besov functions optimally.

Quotients $Y=X/conj$ of complex surfaces by anti-holomorphic involutions $conj\: X\to X$ tend to be completely decomposable when they are simply connected, i.e., split into connected sums, $n CP^2\#m\barCP2$, if $w_2(Y)\ne0$, or into $n(S^2\times S^2)$ if $w_2(Y)=0$. If $X$ is a double branched covering over $CP^2$, th…

Extends Arnold's linking theory to higher dimensions and submanifolds.

Proposes a new neural network architecture combining MLP and basis functions.

We present some results supporting the Iwase-Sakai conjecture about coincidence of the topological complexity $TC(X)$ and monoidal topological complexity $TC^M(X)$. Using these results we provide lower and upper bounds for the topological complexity of the wedge $X\vee Y$. We use these bounds to give a counterexample t…

The aim of this paper is to compare stratifications of moduli spaces given by group actions in the case of similarity of matrices introduced by Arnold and the author's stratification by projective orbifolds, and its relation to deformations o elements in the moduli space.

S2KAN integrates symbolic primitives into neural network activations for improved interpretability.

A 3D metric conformally related to Arnold cat fast dynamo metric: ${ds_{A}}^{2}=e^{-λz}dp^{2}+e^{λz}dq^{2}+dz^{2}$ is shown to present a behaviour of non-dynamos where the magnetic field exponentially decay in time. The Riemann-Christoffel connection and Riemann curvature tensor for the Arnold and its conformal counter…

KANEL combines models for early hit enrichment in virtual screening.

Kolmogorov-Arnold Networks offer interpretable models for energy applications.

KANs replace fixed MLP weights with learnable edge functions, improving accuracy and interpretability.

Kolmogorov-Arnold Networks enable ultrafast online learning with fixed-point quantization.

We prove conjectures of Rene Thom and Vladimir Arnold for C^2 solutions to the degenerate elliptic equation that is the level set equation for motion by mean curvature. We believe these results are the first instances of a general principle: Solutions of many degenerate equations behave as if they are analytic, even wh…

Presented are two neural network architectures for convex functions, demonstrating competitive performance.

Recently Arnold's $\St$ and $J^{\pm}$ invariants of generic planar curves have been generalized to the case of generic planar wave fronts. We generalize these invariants to the case of wave fronts on an arbitrary surface $F$. All invariants satisfying the axioms which naturally generalize the axioms used by Arnold are …