arXiv research

Stokes equations help uniquely identify manifold metrics from boundary data.

Algorithm describes Fourier transform of Stokes data at infinity.

Analyzes tt*-structures from $ADE$-type Stokes data.

Constructs positive energy representations from Toda equations Stokes data.

In "Isomonodromy aspects of the tt* equations of Cecotti and Vafa I. Stokes data" (arxiv:1209.2045) we described all smooth solutions of the two-function tt*-Toda equations in terms of asymptotic data, holomorphic data, and monodromy data. In this supplementary article we focus on the holomorphic data and its interpret…

Proves generic existence of spectral networks for many cases.

Stokes' theorem's boundary maximizes entropy.

Several intrinsic topological ways to encode connections on vector bundles on smooth complex algebraic curves will be described. In particular the notion of {\em Stokes decompositions} will be formalised, as a convenient intermediate category between the Stokes filtrations and the Stokes local systems/wild monodromy re…

Study well-posedness of generalized Stokes operator on cylindrical domains.

Lean 4 formalizes Stokes' theorem for smooth singular cubes.

Paper studies periodic solutions to Navier-Stokes equations on hyperbolic manifolds.

Study on Navier-Stokes equations on non-compact manifolds, proving existence and decay of solutions.

Explains quantum cohomology of Grassmannians using tt* equations.

Paper compares five surface Navier-Stokes derivations and finds some are equivalent.

Study submanifolds with boundaries in Heisenberg groups using Stokes' Theorem.

We compute Stokes matrices and monodromy for the quantum cohomology of projective spaces. We prove that the Stokes' matrix of the quantum cohomology coincides with the Gram matrix in the theory of derived categories of coherent sheaves.

Paper connects Stokes phenomena to quantum groups and Poisson-Lie groups.

Categorifies Stokes coefficients in Chern-Simons theory models.

We describe all smooth solutions of the two-function tt*-Toda equations (a version of the tt* equations, or equations for harmonic maps into SL(n,R)/SO(n)) in terms of (i) asymptotic data, (ii) holomorphic data, and (iii) monodromy data. This allows us to find all solutions with integral Stokes data. These include solu…

We consider the concept of Stokes-Dirac structures in boundary control theory proposed by van der Schaft and Maschke. We introduce Poisson reduction in this context and show how Stokes-Dirac structures can be derived through symmetry reduction from a canonical Dirac structure on the unreduced phase space. In this way, …

Framework predicts Navier-Stokes solutions on 2D domains using graph neural networks.

Stokes theorem holds for Lipschitz forms on a smooth manifold.

The paper validates Stokes' theorem for differential subcomplexes in positively graded Lie groups.

We propose a Lie-theoretic definition of the tt*-Toda equations for any complex simple Lie algebra $\mathfrak{g}$, based on the concept of topological-antitopological fusion which was introduced by Cecotti and Vafa. Our main result concerns the Stokes data of a certain meromorphic connection, whose isomonodromic deform…

A theory for the evolution of a metric $g$ driven by the equations of three-dimensional continuum mechanics is developed. This metric in turn allows for the local existence of an evolving three-dimensional Riemannian manifold immersed in the six-dimensional Euclidean space. The Nash-Kuiper theorem is then applied to th…

The author presents the generalized Stokes theorem for R-linear forms on Lie algebroids (which can be non-local). We apply the Stokes formula on forms to prove that two homotopic homomorphisms of Lie algebroids implies the existence of a chain operator joining their pullback operators.

Proves invertibility of layer potentials for generalized Stokes operators on smooth domains.

In this paper, we show the existence of real-analytic stationary Navier-Stokes flows with isotropic streamlines in all latitudes in some simply-connected flow region on a rotating round sphere. We also exclude the possibility of having a Poiseuille's flow profile to be one of these stationary Navier-Stokes flows with i…

Direct numerical simulation of Stokes flow through an impermeable, rigid body matrix by finite elements requires meshes fine enough to resolve the pore-size scale and is thus a computationally expensive task. The cost is significantly amplified when randomness in the pore microstructure is present and therefore multipl…

For a given bounded domain $Ω\subset {\Bbb R}^n$ with smooth boundary, we explicitly calculate the first two coefficients of the asymptotic expansion of the heat trace associated with the Stokes operator as $t\to 0^+$. These coefficients (i.e., heat invariants) provide precise information for the volume of the domain $…

We present type $θ$ Stokes' theorem for type $θ$ $k$-chains which extends the fundamental theorem of calculus in higher dimensions.

The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.

Study submanifolds with boundary in Heisenberg groups, proving Stokes' Theorem.

By proving graph theoretical versions of Green-Stokes, Gauss-Bonnet and Poincare-Hopf, core ideas of undergraduate mathematics can be illustrated in a simple graph theoretical setting. In this pedagogical exposition we present the main proofs on a single page and add illustrations. While discrete Stokes is is old, the …

The Navier-Stokes equations on certain manifolds can perform universal computation.

Stokes-Dirac structures are infinite-dimensional Dirac structures defined in terms of differential forms on a smooth manifold with boundary. These Dirac structures lay down a geometric framework for the formulation of Hamiltonian systems with a nonzero boundary energy flow. Simplicial triangulation of the underlaying m…

Model uses Navier-Stokes equations to assess liquidity and systemic risk.

In the first part of the paper, we solve the boundary and monodromy problems for the isomonodromy equation of the $n\times n$ meromorphic linear system of ordinary differential equations with Poncaré rank $1$. In particular, we derive an explicit expression of the Stokes matrices of the linear system, via the boundary …

Paper derives explicit expression of Alekseev-Meinrenken diffeomorphism.

This paper clarifies a global structure of Stokes-Dirac structures used for describing interconnected port-Hamiltonian systems defined on manifolds with non-trivial topology under consistent boundary condition.

The Wodzicki residue and the cut-off integral extend to classical symbol-valued forms. We show that they obey a Stokes' type property and that the extended Wodzicki residue can be interpreted as a complex residue like the ordinary one. In the case of cut-off integrals, Stokes' property (i.e. vanishing on exact forms) o…

What is the suitable Laplace operator on vector fields for the Navier-Stokes equation on a Riemannian manifold? In this note, by considering Nash embedding, we will try to elucidate different aspects of different Laplace operators such as de Rham-Hodge Laplacian as well as Ebin-Marsden's Laplacian. A probabilistic repr…

The purpose of this paper is to study the validity of Stokes' Theorem for singular submanifolds and differential forms with singularities in Euclidean space. The results are presented in the context of Lebesgue Integration, but their proofs involve techniques from gauge integration in the spirit of R.~Henstock, J.~Kurz…

For a projective algebraic variety $V$ with isolated singularities, endowed with a metric induced from an embedding, we consider the analysis of the natural partial differential operators on the regular part of $V$. We show that, in the complex case, the Laplacians of the de Rham and Dolbeault complexes are discrete op…

Numerical experiments support conjecture about opers and nonabelian Hodge.

Optimizes shapes in uncertain Navier-Stokes flow problems.

Study reveals how to determine area and curvature from fluid flow resonances.

New framework uses dynamics to justify Gaussian process for turbulent flows.