arXiv research

Ancient solutions to biharmonic heat equation bounded by polynomial dimensions.

Researchers compute A-polynomials of manifolds using symplectic properties and cluster algebras.

Simpler equations derived for knot polynomials coefficients, forming a ring.

We prove that the N-colored Jones polynomial for the torus knot T_{s,t} satisfies the second order difference equation, which reduces to the first order difference equation for a case of T_{2,2m+1}. We show that the A-polynomial of the torus knot can be derived from this difference equation. Also constructed is a q-hyp…

Graphical notation simplifies complex polynomial constraints in linear models.

Study on Monge-Ampère equations with polynomial growth rates.

In this paper we study both analytic and numerical solutions of option pricing equations using systems of orthogonal polynomials. Using a Galerkin-based method, we solve the parabolic partial diferential equation for the Black-Scholes model using Hermite polynomials and for the Heston model using Hermite and Laguerre p…

Classifies polynomial growth solutions to drift-harmonic equations on asymptotically paraboloidal manifolds.

We discuss the relation between knot polynomials and the KP hierarchy. Mainly, we study the scaling 1-hook property of the coloured Alexander polynomial: $\mathcal{A}^\mathcal{K}_R(q)=\mathcal{A}^\mathcal{K}_{[1]}(q^{\vert R\vert})$ for all 1-hook Young diagrams $R$. Via the Kontsevich construction, it is reformulated …

Study polynomial growth harmonic functions on infinite penny graphs.

Solving polynomial equations finds circle packings on surfaces.

We study the dependence of solutions of equations of the form $a_0 + a_1 z^{\ell_1} + ... + a_m z^{\ell_m} = 0$, on the exponents $\ell_1, ..., \ell_m$. We apply our results to equations that appear in graph theory, the theory of 3-manifolds fibering over the circle, and the theory of free-by-cyclic groups. In particul…

Ancient solutions on a strip are constant if polynomial, and have finite-dimensional space for slower growth.

The paper constructs quantum invariants for knotoid diagrams.

In this note, we derive a Liouville theorem for the complex Monge-Ampère equation. Our result states that if the global solution $u$ of the complex Monge-Ampère equation with constant right-hand side differs from a quadratic polynomial solution by $o(\abs{x}^2)$ at infinity, then $u$ is a quadratic polynomial.

New framework uses score-based priors to solve ill-conditioned polynomial equations, improving signal recovery from noisy data.

Solutions near infinity to special Lagrangian equations are asymptotic to quadratic polynomials with logarithmic terms.

In this paper, we prove that a quartic polynomial solution of the eikonal equation $|\nabla_x f|^2=16x^{6}$ in $\R{n}$ is either an isoparametric polynomial or congruent to a polynomial $f=(\sum_{i=1}^n x_i^2)^2-8(\sum_{i=1}^k x_i^2)(\sum_{i=k+1}^{n} x_i^2)$, $k=0,1,\...,[\frac{n}{2}]$.

Investigates polynomial solutions to minimal surface equation, proving constraints and structure theorems.

FNOs learn solution operators of dissipative equations efficiently via spectral methods.

Classifies special homogeneous curves with polynomial equations.

New knot polynomials derived from Nichols algebras and braided Hopf algebras.

We propose a method called ideal regression for approximating an arbitrary system of polynomial equations by a system of a particular type. Using techniques from approximate computational algebraic geometry, we show how we can solve ideal regression directly without resorting to numerical optimization. Ideal regression…

The paper extends a method to compute A-polynomials of 2-bridge knots.

Using the same method we provide negative answers to the following questions: Is it possible to find real equations for complex polynomials in two variables up to topological equivalence (Lee Rudolph) ? Can two topologically equivalent polynomials be connected by a continuous family of topologically equivalent polynomi…

Uniqueness found for elliptic equations with drift on manifolds.

Paper introduces equations to distinguish knots without using knot invariants.

We briefly review the current situation with various relations between knot/braid polynomials (Chern-Simons correlation functions), ordinary and extended, considered as functions of the representation and of the knot topology. These include linear skein relations, quadratic Plucker relations, as well as "differential" …

We study the uniqueness of a vortex equation involving an entire function on the complex plane. As geometric applications, we show that there is a unique harmonic map $u:\mathbb{C}\rightarrow \mathbb{H}^2$ satisfying $\partial u\neq 0$ with prescribed polynomial Hopf differential; there is a unique affine spherical imm…

New method learns low-dimensional models for systems with non-polynomial terms.

All polynomial invariants of links for two dimensional solutions of Yang-Baxter equation is constructed by employing Turaev's method. As a consequence, it is proved that the best invariant so constructed is the Jones polynomial and there exist three solutions connecting to the Alexander polynomial. Invariants for highe…

The study proves no smooth solutions for certain conformally invariant equations.

We introduce W-spin structures on a Riemann surface and give a precise definition to the corresponding W-spin equations for any quasi-homogeneous polynomial W. Then, we construct examples of nonzero solutions of spin equations in the presence of Ramond marked points. The main result of the paper is a compactness theore…

Representations of coherent state Lie algebras on coherent state manifolds as first order differential operators are presented. The explicit expressions of the differential action of the generators of semisimple Lie groups determine for linear Hamiltonians in the generators of the groups first order differential equati…

Complex manifold describes solvable Pell-Abel equations with fixed degrees.

We construct new knot polynomials. Let $V$ be the standard solid torus in 3-space and let $pr$ be its standard projection onto an annulus. Let $M$ be the space of all smooth oriented knots in $V$ such that the restriction of $pr$ is an immersion (e.g. regular diagrams of a classical knot in the complement of its meridi…

Classifies solutions of Toda equations near singularities.

We introduce a family of extremal polynomials associated with the prolongation of a stratified nilpotent Lie algebra. These polynomials are related to a new algebraic characterization of abnormal subriemannian geodesics in stratified nilpotent Lie groups. They satisfy a set of remarkable structure relations that are us…

The paper studies Fourier-Laplace transforms in polynomial OU volatility models for option pricing.

Efficiently finds sparse solutions to max-plus equations for convex regression.

Kashaev limits of quantum $A$-polynomials reveal classical action vanishing and hyperbolic volume deformation.

The paper studies parabolic representations of 2-bridge links using symplectic quandles.

This paper presents a new method for solving systems with polynomial stiffness.

We discuss relations between quantum BPS invariants defined in terms of a product decomposition of certain series, and difference equations (quantum A-polynomials) that annihilate such series. We construct combinatorial models whose structure is encoded in the form of such difference equations, and whose generating fun…

Study local perturbations of vector bundles with polynomial curvature solutions.

If a Lagrangian defining a variational problem has order $k$ then its Euler-Lagrange equations generically have order $2k$. This paper considers the case where the Euler-Lagrange equations have order strictly less than $2k$, and shows that in such a case the Lagrangian must be a polynomial in the highest-order derivati…

Given a polynomial map $ψ:S^m\to \mathbb{R}^k$ with components of degree $d$, we investigate the structure of the semialgebraic set $Z\subseteq S^m$ consisting of those points where $ψ$ and its derivatives satisfy a given list of polynomial equalities and inequalities (we call such a set a "singularity"). Concerning th…

New SDEs from affine and polynomial perspectives for path-dependent processes.