arXiv research

Stokes' theorem's boundary maximizes entropy.

Lean 4 formalizes Stokes' theorem for smooth singular cubes.

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

Stokes theorem holds for Lipschitz forms on a smooth manifold.

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

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

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

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.

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…

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…

We present some new Stokes' type theorems on complete non-compact manifolds that extend, in different directions, previous work by Gaffney and Karp and also the so called Kelvin-Nevanlinna-Royden criterion for (p-)parabolicity. Applications to comparison and uniqueness results involving the p-Laplacian are deduced.

Survey revisits vector calculus results using exterior derivative and provides a new formulation of 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 …

We present a new geometric unfolding of a prototype problem of optimal control theory, the Mayer problem. This approach is crucially based on the Stokes Theorem and yields to a necessary and sufficient condition that characterizes the optimal solutions, from which the classical Pontryagin Maximum Principle is derived i…

Constructs differential forms on $C^\infty$-ringed spaces.

New equations reveal viscosity from boundary measurements.

Graph theory connects automorphisms to cohomology.

Study differential and integral calculus on noncommutative C*-algebras.

Geometric theory of integration developed in SDG.

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…

Develops analysis of Hölder continuous mappings on Heisenberg groups.

For an integer $m\geq 1$, a combinatorial manifold $\widetilde{M}$ is defined to be a geometrical object $\widetilde{M}$ such that for $\forall p\in\widetilde{M}$, there is a local chart $(U_p,φ_p)$ enable $φ_p:U_p\to B^{n_{i_1}}\bigcup B^{n_{i_2}}\bigcup...\bigcup B^{n_{i_{s(p)}}}$ with $B^{n_{i_1}}\bigcap B^{n_{i_2}}…

The paper extends stability theorem for Navier-Stokes equations to negatively curved manifolds.

In the present report, by using the Stokes-Helmholtz decomposition theorem the 3-dimensional Navier-Stokes equation (NSE) is uncoupled and transformed into a scalar equation for the velocity potential when the flow field is toroidal. The dynamics of the velocity potential is independent of the vector potential. The red…

In this paper we prove that Dirac operators on non-compact complete orbifolds which are sufficiently regular at infinity, admit a unique extension. Additonally, we prove a generalized orbifold Stokes'/Divergence theorem.

This paper is motivated by recent developments of higher gauge theory. Different from its style of using higher category theory, we try to describe the concept of higher parallel transport within setting of classical principal bundle theory. From this perspective, we obtain a global geometric proof on a generalized 3-d…

Much of the vast literature on the integral during the last two centuries concerns extending the class of integrable functions. In contrast, our viewpoint is akin to that taken by Hassler Whitney [{\it Geometric integration theory}, Princeton Univ. Press, Princeton, NJ, 1957] and by geometric measure theorists because …

The generalization of the n-dimensional cube, an n-dimensional chain, the exterior derivative and the integral of a differential n-form on it are introduced and investigated. The analogue of Stokes theorem for the differential space is given.

We point out, and draw some consequences of, the fact that the Poisson Lie group G* dual to G=GL_n(C) (with its standard complex Poisson structure) may be identified with a certain moduli space of meromorphic connections on the unit disc having an irregular singularity at the origin. The Riemann-Hilbert map for such co…

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…

This paper extends de Rham theory of smooth manifolds to exploded manifolds. Included are versions of Stokes' theorem, De Rham cohomology, Poincare duality, and integration along the fiber. The resulting cohomology theory is used to define Gromov Witten invariants of exploded manifolds in a separate paper.

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.

We construct a 2-dimensional twisted nonabelian multiplicative integral. This is done in the context of a Lie crossed module (an object composed of two Lie groups interacting), and a pointed manifold. The integrand is a connection-curvature pair, that consists of a Lie algebra valued 1-form and a Lie algebra valued 2-f…

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

Stokes equations help uniquely identify manifold metrics from boundary data.

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.

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

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

Categorifies Stokes coefficients in Chern-Simons theory models.

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, …

We investigate the Berezin integral of non-compactly supported quantities. In the framework of supermanifolds with corners, we give a general, explicit and coordinate-free repesentation of the boundary terms introduced by an arbitrary change of variables. As a corollary, a general Stokes's theorem is derived - here, th…

This work is thought as an operative guide to discrete exterior calculus (DEC), but at the same time with a rigorous exposition. We present a version of (DEC) on cubic cell, defining it for discrete manifolds. An example of how it works, it is done on the discrete torus, where usual Gauss and Stokes theorems are recove…

For a Riemannian $G$-structure, we compute the divergence of the vector field induced by the intrinsic torsion. Applying the Stokes theorem, we obtain the integral formula on a closed oriented Riemannian manifold, which we interpret in certain cases. We focus on almost harmitian and almost contact metric structures.

Study of WKB asymptotics of Stokes matrices and spectral curves, proving rhombus inequalities.

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

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