Proves summability of state integrals for specific hyperbolic knots.
problem Summability of perturbative series for hyperbolic knots.
method Algorithm to compute Borel-Laplace resummation as state integrals.
result Complete description of resurgent structure and explicit computations of Stokes constants.
Complex Chern-Simons theory reveals peacock patterns in perturbative series.
problem Understanding the structure of partition functions in complex Chern-Simons theory.
method Analyzing the partition function as a holomorphic function and using resurgence theory.
result Perturbative series are resurgent, with trans-series involving non-perturbative variables.
Study on Chern-Simons theory at generic levels, revealing universal resurgent structure.
problem Analyzing Chern-Simons theory at generic levels with small boundary holonomy.
method Examined resurgent structure of state integral models on knot complements with generic discrete level.
result Resurgent structure is universal, independent of the level k. Quantum modularity proved for a knot manifold.
problem Proving quantum modularity for a specific closed hyperbolic 3-manifold.
method Using factorization of state integrals and proving quantum modularity for functions and q-series. result Quantum modularity for the closed manifold provides a unification of volume conjecture and Witten's asymptotic expansion conjecture.
The trivial flat connection's Chern-Simons theory is resurgent, revealing its structure.
problem Understanding the resurgent structure of Chern-Simons theory at the trivial flat connection.
method Analyzing an extended square matrix of (x,q)-series to describe the resurgent structure and Stokes constants. result The resurgent structure and Stokes constants of the Chern-Simons series are completely described.
Stokes' theorem's boundary maximizes entropy.
problem Characterizing the boundary of a manifold using entropy.
method Maximizing entropy for codimension-1 submanifolds satisfying Stokes' theorem.
result The boundary of a manifold maximizes the entropy functional.
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…
Developed MF-PINNs to solve coupled Stokes-Darcy equations more accurately.
problem Solving coupled Stokes-Darcy equations with varying physical constants.
method Combining VP and SV forms with adjusted weights in MF-PINNs.
result Improved accuracy of streamline and pressure fields in numerical experiments.
Lean 4 formalizes Stokes' theorem for smooth singular cubes.
problem Formalizing Stokes' theorem for singular cubes in arbitrary dimensions.
method Using true differential-form pullback via Frechet derivative, bridging to mathlib4's extDeriv.
result d^2=0 for singular cubical chains, chain-level Stokes extended.
Paper studies periodic solutions to Navier-Stokes equations on hyperbolic manifolds.
problem Existence and uniqueness of asymptotically almost periodic solutions to Navier-Stokes equations on hyperbolic manifolds.
method Dispersive and smoothing estimates for the Stokes equation, Massera-type principle, fixed point argument.
result Existence and uniqueness of asymptotically almost periodic mild solutions in Lp(Γ(TM)) spaces. Study on Navier-Stokes equations on non-compact manifolds, proving existence and decay of solutions.
problem Existence and asymptotic behavior of solutions to Navier-Stokes equations on non-compact manifolds.
method Used Lp−Lq-dispersive and smoothing estimates of the Stokes semigroup, fixed point arguments, and Gronwall's inequality. result Established existence and exponential decay of almost periodic and asymptotically almost periodic mild solutions.
Paper compares five surface Navier-Stokes derivations and finds some are equivalent.
problem Modeling evolving fluidic surfaces using different principles and coordinate systems.
method Systematic comparison of five derivations using tangential and normal components.
result All derivations yield the same tangential surface Navier-Stokes equations.
Study submanifolds with boundaries in Heisenberg groups using Stokes' Theorem.
problem Understanding submanifolds with boundaries in Heisenberg groups.
method Introduce and study CH1-regular submanifolds with boundaries, prove Stokes' Theorem for them. result Prove a version of Stokes' Theorem for submanifolds with boundaries in Heisenberg groups.
Stokes equations help uniquely identify manifold metrics from boundary data.
problem Determining Riemannian metric from boundary Cauchy data.
method Proving uniqueness of metric from Stokes equations Cauchy data.
result Partial derivatives of all orders of the metric on the boundary are uniquely determined.
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.
Algorithm describes Fourier transform of Stokes data at infinity.
problem Understanding the Fourier transform of Stokes data at infinity.
method Topological description and algorithmic approach using recent results and language of Stokes local systems.
result Explicit isomorphisms between wild character varieties are induced.
Analyzes tt*-structures from ADE-type Stokes data.
problem Classifying tt*-structures over C∗. method Isomonodromic deformations with upper unitriangular real Stokes matrices.
result Establishes a direct analytic realization of the ADE classification. Paper connects Stokes phenomena to quantum groups and Poisson-Lie groups.
problem Analyse Stokes phenomena in Poisson-Lie groups and quantum groups.
method Use Ug-valued Stokes phenomena to construct quantum group U_hg and relate it to Poisson-Lie group G*.
result Show that Ug-valued Stokes phenomena can be obtained as a semiclassical limit of the KZ associator.
Categorifies Stokes coefficients in Chern-Simons theory models.
problem Stokes phenomenon in Chern-Simons theory around flat connections.
method Finite-dimensional model for analytically continued Chern-Simons theory, categorification of Stokes coefficients.
result Stokes coefficients can be promoted to graded vector spaces.
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, …
Stokes theorem holds for Lipschitz forms on a smooth manifold.
The paper validates Stokes' theorem for differential subcomplexes in positively graded Lie groups.
problem Validating Stokes' theorem for differential subcomplexes in positively graded Lie groups.
method Introducing geometric conditions and spectral complexes to recover Stokes' theorem on locally smooth intrinsic graphs.
result Stokes' theorem holds for Rumin complex and new spectral complexes on Carnot groups.
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.
Study well-posedness of generalized Stokes operator on cylindrical domains.
problem Analyzing the generalized Stokes operator on domains with cylindrical ends.
method Using layer potentials and developing algebra tools for limit and jump relations.
result Well-posedness results for the associated Stokes boundary value problem.
Proves invertibility of layer potentials for generalized Stokes operators on smooth domains.
problem Invertibility of layer potentials for generalized Stokes operators on smooth domains.
method Developed algebra toolkit to handle layer operators' limit and jump relations; proved Fredholm property and invertibility.
result 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…
For a given bounded domain Ω⊂Rn 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→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.
problem Eigenvalue problems and heat trace asymptotics for different operators.
method Establishes connections and inequalities for eigenvalues and heat traces.
result Eigenvalue inequalities and three-term asymptotic formulas for heat traces of various operators.
Study submanifolds with boundary in Heisenberg groups, proving Stokes' Theorem.
problem Understanding submanifolds with boundary in sub-Riemannian Heisenberg groups.
method Introduced examples and proved Stokes' Theorem involving Rumin's differential forms.
result Stokes' Theorem for submanifolds with boundary in Heisenberg groups.
Constructs positive energy representations from Toda equations Stokes data.
problem Creating positive energy representations of affine algebras.
method Using Stokes data of tt*-Toda equations to construct representations.
result Illustrates construction with examples in conformal field theory.
We prove the analyticity in time for solutions of two parabolic equations in the whole space, without any decaying or vanishing conditions. One of them involves solutions to the heat equation of exponential growth of order 2 on $\M$. Here $\M$ is Rd or a complete noncompact manifold with Ricci curvature bounded f…
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.
problem Computational universality in viscous fluids.
method Cosymplectic geometry and harmonic 1-forms.
result Stationary Navier-Stokes solutions exhibit Turing completeness.
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…
Paper derives explicit expression of Alekseev-Meinrenken diffeomorphism.
problem Understanding the Alekseev-Meinrenken diffeomorphism.
method Via the Stokes phenomenon of meromorphic linear systems of ODEs with Poncaré rank 1.
result Explicit expression of the 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…
Proves generic existence of spectral networks for many cases.
problem Existence of spectral networks for a broad range of spectral data.
method Generic existence proof for spectral networks.
result Proves existence for a large class of spectral 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…
Optimizes shapes in uncertain Navier-Stokes flow problems.
problem Optimizing shapes with geometric constraints and physical uncertainty.
method Multi-shape calculus and stochastic augmented Lagrangian method.
result Successfully optimized shapes in uncertain Navier-Stokes flow.
Study reveals how to determine area and curvature from fluid flow resonances.
problem Determining geometric properties from fluid flow data.
method Asymptotic expansion of heat kernel and Steklov spectral invariants.
result Area and total mean curvature can be inferred from Steklov eigenvalues.
Paper variates Navier-Stokes-Fourier system for thermodynamic consistency.
problem Modeling compressible fluid dynamics with thermodynamic constraints.
method Variational discretization with discrete exterior calculus.
result Derives a nonholonomic variational integrator for NSF system.
Framework predicts Navier-Stokes solutions on 2D domains using graph neural networks.
problem Predicting stationary Navier-Stokes solutions in non-parametrized 2D geometries.
method Graph-based multi-fidelity learning framework combining reduced-order models, Transformers, and Mamba architectures.
result Mamba architecture reduces computational cost while maintaining performance.
New equations reveal viscosity from boundary measurements.
problem Determine viscosity from boundary measurements for incompressible fluids.
method Equivalent new system of elliptic equations, Dirichlet-to-Neumann map analysis.
result Dirichlet-to-Neumann map uniquely determines viscosity and its derivatives on the boundary.