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.
Study embedding calculus and link invariants using functor calculus.
Secondary Calculus formalizes PDEs using cohomology, simplifying their study.
This paper is concerned with pseudodifferential calculus on manifolds with fibred corners. Following work of Connes, Monthubert, Skandalis and Androulidakis, we associate to every manifold with fibred corners a longitudinally smooth groupoid which algebraic and differential structure is explicitely described. This grou…
A diagrammatic language for 3D manifolds with boundary.
Develops symmetric Cartan calculus linking to Patterson-Walker metric.
Simplified calculus for semimartingales makes complex transformations easier.
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…
This memoir deals with the hypoelliptic calculus on Heisenberg manifolds, or Heisenberg calculus. The Heisenberg manifolds generalize CR and contact manifolds and in this context the main differential operators at stake include the Hörmander's sum of squares, the Kohn Laplacian, the horizontal sublaplacian and its conf…
Cartan calculus applied to string topology homology.
New algebraic formalism for differential calculus in Diolic algebras.
We introduce and study graphic lambda calculus, a visual language which can be used for representing untyped lambda calculus, but it can also be used for computations in emergent algebras or for representing Reidemeister moves of locally planar tangle diagrams.
A new discrete calculus for bundle-valued forms is proposed and validated.
Simplified Khovanov-Rozansky calculus for bipartite knots.
New calculus solves boundary value problems for elliptic operators.
Study of embedding calculus using infinite operads.
We show that the existence of a Fredholm element of the zero calculus of pseudodifferential operators on a compact manifold with boundary with a given elliptic symbol is determined, up to stability, by the vanishing of the Atiyah-Bott obstruction. It follows that, up to small deformations and stability, the same symbol…
We give a description of local and global moves on a class of locally planar trivalent graphs and we show that it contains -Scale calculus, therefore in particular untyped lambda calculus. Surprisingly, the beta reduction rule comes from a local "sewing" transformation of trivalent locally planar graphs.
Embedding calculus proves convergence for surfaces.
We consider a standard optimal investment problem in a complete financial market driven by a Wiener process and derive an explicit formula for the optimal portfolio process in terms of the vertical derivative from functional It^o calculus. An advantage with this approach compared to the Malliavin calculus approach is t…
Study geometric characterization of asymptotic pseudodifferential calculus on spinor bundles.
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…
Study differential and integral calculus on noncommutative C*-algebras.
Teaches matrix calculus for machine learning and optimization.
In arXiv:1207.0332 [cs.LO] was proposed a graphic lambda calculus formalism, which has sectors corresponding to untyped lambda calculus and emergent algebras. Here we explore the sector covering knot diagrams, which are constructed as macros over the graphic lambda calculus.
A new approach to continuous-time universal portfolios using pathwise Itô calculus.
Extends differential calculus to triole algebras.
This study introduces computation of option sensitivities (Greeks) using the Malliavin calculus under the assumption that the underlying asset and interest rate both evolve from a stochastic volatility model and a stochastic interest rate model, respectively. Therefore, it integrates the recent developments in the Mall…
Introduces tractors for basic examples and modern differential calculus.
Unified Lie structures in homotopy and isotopy calculus.
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 paper connects calculus, gauge theory, and noncommutative worlds.
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.
In a previous paper ([1]), we associated a holonomy groupoid and a C*-algebra to any singular foliation (M,F). Using these, we construct the associated pseudodifferential calculus. This calculus gives meaning to a Laplace operator of any singular foliation F on a compact manifold M, and we show that it can be naturally…
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…
The thesis examines stochastic calculus in option pricing with logistic models and numerical methods.
Enhancing the Black-Scholes model with Lévy processes and Malliavin calculus
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…
The paper introduces a simple way of recording and manipulating general stochastic processes without explicit reference to a probability measure. In the new calculus, operations traditionally presented in a measure-specific way are instead captured by tracing the behaviour of jumps (also when no jumps are physically pr…
This is a short description of graphic lambda calculus, with special emphasis on a duality suggested by the two different appearances of knot diagrams, in lambda calculus and emergent algebra sectors of the graphic lambda calculus respectively. This duality leads to the introduction of the dual of the graphic beta move…
Develops a universal Hermitian projective calculus for complex hyperbolic two-space
New integration theory on topological spaces, including fractals.
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…
Develops a graphical calculus for stable curvature invariants.
In this paper we present a new theory of calculus over -dimensional domains in a smooth -manifold, unifying the discrete, exterior, and continuum theories. The calculus begins at a single point and is extended to chains of finitely many points by linearity, or superposition. It converges to the smooth continuum w…
Simplified calculus for manifold operators, proving index theorems.
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…
Derives optimal control conditions using calculus of variations.
The calculus correspondence has been known to exist between generic pedal evolutions and generic wave front evolutions. In this paper, we first extend the known results on the calculus correspondence to evolutions with multi-parameters, and then give applications of calculus correspondence. Moreover, we discuss the pos…