A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Extends differential calculus to triole algebras.
New calculus framework for vector bundles with metrics.
The basic theorems of vector calculus are illuminated when we replace the original 3 stooges of vector calculus: Grad, Div, and Curl, with combinatorial substitutes. In addition to providing simple proofs of Green's theorem and the equivalence of the integral and derivative definitions of curl, we also provide a brief …
Develops global pseudo-differential calculus on homogeneous vector bundles.
Develops symmetric Cartan calculus linking to Patterson-Walker metric.
Defines vector fields and differential forms on local C-infinity-ringed spaces.
The study characterizes complex structures using calculus of variations.
New algebraic formalism for differential calculus in Diolic algebras.
We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but also discrete vector fields and the operators acting on these objects. This allow…
Derives new orthogonal coordinates for evolving surfaces and curves.
This article provides a pedagogically oriented introduction to geometric (Clifford) calculus on pseudo-Riemannian manifolds. Unlike usual approaches to the topic, which rely on embedding the geometric algebra either within a tensor algebra or within a vector manifold framework, here we define geometric calculus directl…
The aim of these notes is to relate covariant stochastic integration in a vector bundle (as in Norris \cite{Norris}) with the usual Stratonovich calculus via the connector $\K:TE \rightarrow E$ (cf. e.g. Paterson \cite{Paterson} or Poor \cite{Poor}) which carries the connection dependence.
We revisit the theory of Discrete Exterior Calculus (DEC) in 2D for general triangulations, relying only on Vector Calculus and Matrix Algebra. We present DEC numerical solutions of the Poisson equation and compare them against those found using the Finite Element Method with linear elements (FEML).
A new discrete calculus for bundle-valued forms is proposed and validated.
Survey revisits vector calculus results using exterior derivative and provides a new formulation of Stokes' theorem.
Extends calculus to topological manifolds using generalized functions.
We present a geometric interpretation of the integration-by-parts formula on an arbitrary vector bundle. As an application we give a new geometric formulation of higher-order variational calculus.
We extend the calculus of multiplicative vector fields and differential forms and their intrinsic derivatives from Lie groups to Lie groupoids; this generalization turns out to include also the classical process of complete lifting from arbitrary manifolds to tangent and cotangent bundles. Using this calculus we give a…
Solves inverse problem for Maxwell equations using vector fields.
New framework for diffusion geometry simplifies complex calculations.
Abstract: Tangent categories get a Cartan calculus with scalar multiplication by a commutative ring.
Develops calculus for tamed Dirichlet spaces using measure theory.
Reduces Lie (bi-)algebroids and Dirac manifolds using constraint vector bundles.
Computes Lie algebra structure constants using a graphical calculus.
In this article we present an intrinsec construction of foliated Brownian motion via stochastic calculus adapted to foliation. The stochastic approach together with a proposed foliated vector calculus provide a natural method to work on harmonic measures. Other results include a decomposition of the Laplacian in terms …
Let be a compact manifold with boundary. Suppose that the boundary is fibred, $φ:\pa X\longrightarrow Y,$ and let $x\in\CI(X)$ be a boundary defining function. This data fixes the space of `fibred cusp' vector fields, consisting of those vector fields on satisfying and which are tangent to the f…
Develops geometric Weyl calculus for curved spacetimes.
Teaches matrix calculus for machine learning and optimization.
Variational calculus on a vector bundle E equipped with a structure of a general algebroid is developed, together with the corresponding analogs of Euler-Lagrange equations. Constrained systems are introduced in the variational and in the geometrical setting. The constrained Euler-Lagrange equations are derived for ana…
We build a model using Gaussian processes to infer a spatio-temporal vector field from observed agent trajectories. Significant landmarks or influence points in agent surroundings are jointly derived through vector calculus operations that indicate presence of sources and sinks. We evaluate these influence points by us…
A spectral approach to building the exterior calculus in manifold learning problems is developed. The spectral approach is shown to converge to the true exterior calculus in the limit of large data. Simultaneously, the spectral approach decouples the memory requirements from the amount of data points and ambient space …
We explore the geometry that underlies the osculating nilpotent group structures of the Heisenberg calculus. For a smooth manifold with a distribution analysts use explicit (and rather complicated) coordinate formulas to define the nilpotent groups that are central to the calculus. Our aim in this p…
We present a self-contained proof of the Gauss-Bonnet theorem for two-dimensional surfaces embedded in using just classical vector calculus. The exposition should be accessible to advanced undergraduate and non-expert graduate students. It may be viewed as an illustration and exercise in multivariate calculus and…
Characterizes polyhomogeneous symbols and applies to Heisenberg calculus.
We define a simplicial differential calculus by generalizing divided differences from the case of curves to the case of general maps, defined on general topological vector spaces, or even on modules over a topological ring K. This calculus has the advantage that the number of evaluation points growths linearly with the…
Geometrically reformulates elasticity theory using exterior calculus.
It is shown that the new formula for the field theory Poisson brackets arise naturally in the extension of the formal variational calculus incorporating divergences. The linear spaces of local functionals, evolutionary vector fields, functional forms, multi-vectors and differential operators become graded with respect …
We consider differential operators between sections of arbitrary powers of the determinant line bundle over a contact manifold. We extend the standard notions of the Heisenberg calculus: noncommutative symbolic calculus, the principal symbol, and the contact order to such differential operators. Our first main result i…
Extends Hess-Schrader-Uhlenbrock inequality for 1-forms in tamed Dirichlet spaces.
Applying concepts and tools from classical tangent bundle geometry and using the apparatus of the calculus along the tangent bundle projection ('pull-back formalism'), first we enrich the known lists of the characterizations of affine vector fields on a spray manifold and conformal vector fields on a Finsler manifold. …
Develops a universal Hermitian projective calculus for complex hyperbolic two-space
Develops combinatorial theory of vector bundles on simplicial complexes.
Extends Lie bialgebroids for string and M theories with new calculus framework.
A conservative discretization of incompressible Navier-Stokes equations is developed based on discrete exterior calculus (DEC). A distinguishing feature of our method is the use of an algebraic discretization of the interior product operator and a combinatorial discretization of the wedge product. The governing equatio…
Extends elliptic operator regularity to maximally hypoelliptic operators.
A generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders. That is, a fractional exterior derivative is defined. This is found to generate new vector spaces of finite and infinite dimension, fractional differential form spaces. The def…
In this paper, we consider a generalization of variational calculus which allows us to consider in the same framework different cases of mechanical systems, for instance, Lagrangian mechanics, Hamiltonian mechanics, systems subjected to constraints, optimal control theory and so on. This generalized variational calculu…
Researchers calculate the second coefficient in the expansion of a Toeplitz operator.