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 algebraic formalism for differential calculus in Diolic algebras.
Introduces tractors for basic examples and modern differential calculus.
We explain that general differential calculus and Lie theory have a common foundation: Lie Calculus is differential calculus, seen from the point of view of Lie theory, by making use of the groupoid concept as link between them. Higher order theory naturally involves higher algebra (n-fold groupoids).(conceptual, topol…
Simplified calculus for manifold operators, proving index theorems.
In this paper we construct the Differential calculus on the Hopf Group Coalgebra introduced by Turaev [10]. We proved that the concepts introduced by S.L.Woronowicz in constructing Differential calculus on Hopf Compact Matrix Pseudogroups (Quantum Groups)[7] can be adapted to serve again in our construction.
Differential calculus on metric spaces is contained in the algebraic study of normed groupoids with -structures. Algebraic study of normed groups endowed with dilatation structures is contained in the differential calculus on metric spaces. Thus all algebraic properties of the small world of normed groups with dilat…
Secondary Calculus formalizes PDEs using cohomology, simplifying their study.
We introduce a noncommutative differential calculus on the two-parameter -superplane via a contraction of the (p,q)-superplane. We manifestly show that the differential calculus is covariant under transformations. We also give a two-parameter deformation of the (1+1)-dimensional phase space alge…
Here are considered some categorical aspects of "Differential calculus" archetype of local approximation of arbitrary morphisms by "linear" ones.
Teaches matrix calculus for machine learning and optimization.
New calculus framework for vector bundles with metrics.
Study differential and integral calculus on noncommutative C*-algebras.
A gauged bi-differential calculus over an associative (and not necessarily commutative) algebra A is an N-graded left A-module with two covariant derivatives acting on it which, as a consequence of certain (e.g., nonlinear differential) equations, are flat and anticommute. As a consequence, there is an iterative constr…
Following the programme set out in Part I of this work, we develop a conceptual higher order differential calculus. The '' local linear algebra '' defined in Part I is generalized by '' higher order local linear algebra ''. The underlying combinatorial object of such higher algebra is the natural n-dimensional hyper-cu…
Basic elements of integral calculus over algebras of iterated differential forms, are presented. In particular, defining complexes for modules of integral forms are described and the corresponding berezinians and complexes of integral forms are computed. Various applications and the integral calculus over the algebra $…
We construct a two-parameter covariant differential calculus on the quantum -exterior plane. We also give a deformation of the two-dimensional fermionic phase space.
Differential Calculus is a staple of the college mathematics major's diet. Eventually one becomes tired of the same routine, and wishes for a more diverse meal. The college math major may seek to generalize applications of the derivative that involve functions of more than one variable, and thus enjoy a course on Multi…
The study characterizes complex structures using calculus of variations.
Any discrete differential manifold (finite set endowed with an algebraic differential calculus) can be represented by appropriate polyhedron . This representation demonstrates the adequacy of the calculus of discrete differential manifolds and links this approach with that based on finitary substitutes…
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…
New integration theory on topological spaces, including fractals.
Extends exterior diff. sys. to Lie algebroids with examples.
Develops global pseudo-differential calculus on homogeneous vector bundles.
A new discrete calculus for bundle-valued forms is proposed and validated.
In this thesis, we study singular pseudo-differential operators defined by groupoids satisfying the Lauter-Nistor condition, by a method parallel to that of manifolds with boundary and edge differential operators. The example of the Bruhat sphere is studied in detail. In particular, we construct an extension to the cal…
We examine the N-Koszul calculus for the N-symmetric algebras. The case N=2 corresponds to the Elie Cartan calculus. We conjecture that, as in the case N=2, the N-Cartan calculus extends to manifolds when N>2, which would provide a new type of noncommutative differential geometry.
Survey revisits vector calculus results using exterior derivative and provides a new formulation of Stokes' theorem.
Defines vector fields and differential forms on local C-infinity-ringed spaces.
To give a Cartan calculus on the extended quantum 3d space, the noncommutative differential calculus on the extended quantum 3d space is extended by introducing inner derivations and Lie derivatives.
Global calculus for manifolds with boundary, solving evolution problems.
In 1974, Folland and Stein constructed an inhomogeneous pseudo-differential calculus based on analysis on the Heisenberg group. This Heisenberg calculus was generalized by several authors, to any subbundle of the tangent bundle. van Erp and Yuncken, following Debord and Skandalis showed that this calculus can be recove…
The concept of $\Zn$-supermanifold has been recently proposed as a natural generalization of classical ($\Zs$-graded) supergeometry, allowing for more complicated commutativity constraints. Here we continue the study of $\Zn$-supergeometry by developing the foundations of differential calculus on $\Zn$-supermanifolds.
New calculus on spacetimes for nonlinear differential equations.
Introduces non-regular spacetime geometry without smooth calculus.
A non-commutative differential calculus on the -superplane is presented via a contraction of the -superplane. An R-matrix which satisfies both ungraded and graded Yang-Baxter equations is obtained and a new deformation of the dimensional classical phase space (the super-Heisenberg algebra) is introduced.
Since the discovery of differential calculus by Newton and Leibniz and the subsequent continuous growth of its applications to physics, mechanics, geometry, etc, it was observed that partial derivatives in the study of various natural problems are (self-)organized in certain structures usually called geometric. Tensors…
Introduces geometric control theory for students.
New Spencer complexes for Lie groupoids developed.
Euler calculus is based on integrating simple functions with respect to the Euler characteristic. This paper makes the case for extending Euler calculus to continuous integrands by integrating with respect to (Gaussian) curvature. This requires a metric but is nevertheless defined within any O-minimal theory. It satisf…
Develops a new calculus for contact structures on manifolds.
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…
Study on stochastic mean curvature flow on networks using Ito calculus.
Rust library solves complex equations on abstract simplicial complexes.
In a 2006 article (\cite{A1}), Allouba gave his quadratic covariation differentiation theory for Itô's integral calculus. He defined the derivative of a semimartingale with respect to a Brownian motion as the time derivative of their quadratic covariation and a generalization thereof. He then obtained a systematic diff…
Differential calculus on the quantum quaternionic group GL(1,H) is introduced.
Global homotopies upgrade classical map in differential geometry.
Develops a mathematical model for automatic differentiation in machine learning.