Introduces geometric control theory for students.
problem No specific problem addressed in the abstract.
method Expository presentation of geometric control theory.
result Suitable for advanced students with solid math background.
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48 results for “advanced calculus”
These are lecture notes on scale calculus and M-polyfolds written for a graduate course at UNICAMP March-June 2018 and an advanced mini-course given during the biannual meeting of Brazilian mathematicians, CBM-32, at IMPA in August 2019.
Using recent advances in integration theory, we give a proof of the fundamental theorem of geometric calculus. We assume only that the tangential derivative exists and is Lebesgue integrable. We also give sufficient conditions that exists.
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…
Enhancing the Black-Scholes Model for Option Valuation via Lévy Processes and Malliavin Calculusq-fin.MF
Enhancing the Black-Scholes model with Lévy processes and Malliavin calculus
problem Improving option valuation by incorporating stochastic volatility and jumps
method Deriving a pricing formula and exact implied volatility using multidimensional Itô calculus and Malliavin calculus
result Better capture of empirical features like volatility smiles
Phynanceq-fin.MF
These are the lecture notes for an advanced Ph.D. level course I taught in Spring'02 at the C.N. Yang Institute for Theoretical Physics at Stony Brook. The course primarily focused on an introduction to stochastic calculus and derivative pricing with various stochastic computations recast in the language of path integr…
New financial model with sandwiched volatility for option pricing.
problem Developing a new financial model for option pricing.
method Introducing a new model with stochastic volatility driven by a Gaussian Volterra process, ensuring the solution is sandwiched between two arbitrary Hölder continuous functions.
result Developed an algorithm for pricing options with discontinuous payoffs using Malliavin calculus.
Bringing errors into focusmath.PR
This lecture presents recent advances in the theory of errors propagation. We first explain in which cases the propagation of errors may be performed with a first order differential calculus or needs a second order differential calculus. Then we point out the link between error propagation and the concept of second ord…
We present a new class of solutions for the inverse problem in the calculus of variations in arbitrary dimension . This is the problem of determining the existence and uniqueness of Lagrangians for systems of second order ordinary differential equations. We also provide a number of new theorems concerning the in…
Optimal algorithm found for anytime regret with two experts.
problem Minimizing regret in prediction with two experts when time horizon is unknown.
method Designing a minimax optimal algorithm using ideas from stochastic calculus.
result Proved the optimal regret is γ√t / 2 for all time steps t.
Lecture Notes in Lie Groupsmath.DG
These lecture notes in Lie Groups are designed for a 1--semester third year or graduate course in mathematics, physics, engineering, chemistry or biology. This landmark theory of the 20th Century mathematics and physics gives a rigorous foundation to modern dynamics, as well as field and gauge theories in physics, engi…
Causal Effect Identification from Multiple Incomplete Data Sources: A General Search-based Approachstat.ML
Causal effect identification considers whether an interventional probability distribution can be uniquely determined without parametric assumptions from measured source distributions and structural knowledge on the generating system. While complete graphical criteria and procedures exist for many identification problem…
Innovative advances validate a conjecture on maximal hypoellipticity in sub-Riemannian geometry.
problem Characterizing maximal hypoellipticity in sub-Riemannian geometry.
method Generalization of Connes tangent groupoid, pseudodifferential calculus, and invertibility of principal symbol.
result Validation of Helffer and Nourrigat's conjecture on maximal hypoellipticity.
This chapter introduces quaternion machine learning for 3D rotations.
problem Lack of quaternion machine learning for 3D rotations.
method Augmented statistics, widely linear models, quaternion calculus, mean square estimation.
result Foundation for quaternion machine learning.
We explore the efficacy of using a novel activation function in Artificial Neural Networks (ANN) in characterizing exoplanets into different classes. We call this Saha-Bora Activation Function (SBAF) as the motivation is derived from long standing understanding of using advanced calculus in modeling habitability score …
New approach to score function in diffusion models using Malliavin calculus.
problem Estimating score function for complex data distributions.
method Combines Malliavin calculus with Bismut-type formula to derive exact score function expression.
result Derives exact, closed-form expression for score function in diffusion models.
Despite the major advances taken in causal modeling, causality is still an unfamiliar topic for many statisticians. In this paper, it is demonstrated from the beginning to the end how causal effects can be estimated from observational data assuming that the causal structure is known. To make the problem more challengin…
Morse theory in the 1990'smath.DG
This is a survey article on Morse theory based on lectures to graduate students and advanced undergraduates. After a brief review of standard material, mostly without proofs, the Morse theory of complex Grassmannian manifolds is worked out in detail. In contrast to standard treatments, gradient flow lines and their str…
Study embedding calculus and link invariants using functor calculus.
problem Detect Milnor invariants using embedding towers of string links.
method Use functor calculus and Goodwillie-Weiss embedding calculus.
result Embedding tower detects Milnor invariants.
A Causally Formulated Hazard Ratio Estimation through Backdoor Adjustment on Structural Causal Modelstat.ME
Novel approach to compute hazard ratios from observational studies using SCMs and backdoor adjustment.
problem Identifying causal relationships from observational data using hazard ratios.
method Backdoor adjustment through structural causal models (SCMs) and do-calculus.
result Novel approach for computing hazard ratios from observational studies.
Embedding calculus proves convergence for surfaces.
problem Proving convergence of embedding calculus for surfaces.
method Goodwillie-Weiss' embedding calculus for spaces of embeddings into a manifold of dimension at most two.
result Relates Johnson filtration of mapping class group to embedding calculus.
Study geometric characterization of asymptotic pseudodifferential calculus on spinor bundles.
problem Geometric characterization of asymptotic pseudodifferential calculus on spinor bundles.
method Groupoid approach to pseudodifferential calculus, rescaled bundle.
result Rescaled bundle provides geometric characterization to asymptotic pseudodifferential calculus on spinor bundles.
Lie Calculusmath.GR
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…
Secondary Calculus formalizes PDEs using cohomology, simplifying their study.
problem Formalizing and simplifying the study of partial differential equations (PDEs).
method Using cohomology of diffieties to formalize PDEs and their properties.
result Differential calculus on PDE solution spaces is homotopy calculus on horizontal De Rham algebras of diffieties.
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.
Extends differential calculus to triole algebras.
problem No specific problem stated; focuses on extending differential calculus.
method Generalizes diolic differential calculus to triole algebras with fiber metrics.
result Established a conceptual framework for calculus on bundles with vector-valued fiber metrics.
Notes on Tractor Calculusmath.DG
Introduces tractors for basic examples and modern differential calculus.
problem None explicitly stated, focuses on introduction.
method Classical examples and modern invariant differential calculus.
result Introduction to tractors and related modern differential calculus.
A diagrammatic language for 3D manifolds with boundary.
problem Representing and manipulating 3D manifolds with boundary.
method Diagrammatic calculus and local moves.
result Completeness of the diagrammatic calculus proved.
Unified Lie structures in homotopy and isotopy calculus.
problem Compatibility of Lie structures in homotopy and isotopy calculus.
method New technical tool: bracket on total homotopy fibres of collapsing cubes of wedge sums.
result Unified understanding of Lie structures in homotopy and isotopy calculus.
N-Cartan calculusmath.RT
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.
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…
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…
Simplified calculus for semimartingales makes complex transformations easier.
problem Complex transformations of semimartingales.
method Unified treatment of transformations for real and complex semimartingales.
result Unified calculus for semimartingales simplifies various transformations.
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 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 symmetric Cartan calculus linking to Patterson-Walker metric.
problem No specific problem stated; focuses on developing a new calculus.
method Symmetric Cartan calculus, using torsion-free affine connections.
result Symmetric Cartan calculus is a complete analogue of classical Cartan calculus.
New algebraic formalism for differential calculus in Diolic algebras.
problem Studying differential calculus in vector bundles.
method Introducing functors of differential calculus over arbitrary graded commutative algebras (DCGCA) and applying this to Diolic algebras.
result Recovery of well-known objects and notions from ordinary differential, symplectic, and Poisson geometry, with unique aspects.
New calculus solves boundary value problems for elliptic operators.
problem Boundary value problems for 0-elliptic operators.
method Developed a new calculus called symbolic 0-calculus to handle boundary value problems.
result Construct left and right parametrices for 0-elliptic operators with boundary conditions.
New integration theory on topological spaces, including fractals.
problem Developing a universal integration theory for arbitrary topological spaces.
method Introducing a new integration framework using unital magma valued functions and measures.
result Integration, differentiation, and orientation defined for arbitrary topological spaces.
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…
Cartan calculus applied to string topology homology.
problem Understanding the structure of free loop spaces.
method Introduced Cartan calculus on loop homology, linked to string topology operations.
result Loop product and bracket behavior under Hodge decomposition.
Develops a graphical calculus for stable curvature invariants.
problem Calculating stable curvature invariants of Riemannian manifolds.
method Graphical calculus based on trivalent graphs with colored edges.
result Derives a curvature identity for compact Einstein manifolds.
Coarse pseudo-differential calculus and index theory on manifolds with a tangent Lie structuremath.DG
Simplified calculus for manifold operators, proving index theorems.
problem Developing calculus for manifold operators and proving index theorems.
method Introducing a simplified pseudo-differential calculus for zero-order operators on manifolds with a tangent Lie structure.
result Proving index theorems for `h-elliptic' operators on manifolds with a tangent Lie structure.
Derives optimal control conditions using calculus of variations.
problem Optimizing Markov control in stochastic control problems.
method Calculus of variations approach to derive necessary conditions.
result Solves the Merton portfolio optimization problem.
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…
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.
Quantum calculus models stock liquidity issues.
problem Capturing illiquidity in stock price distributions.
method Quantum stochastic calculus applied to finance.
result Modeling the impact of widened bid-ask spreads.
Survey explores cohomology's roles in applied math and sciences.
problem Understanding cohomology's role in solving differential equations.
method Examining differential complexes and structure-preserving discretizations.
result Various fundamental concepts in mechanics are formulated using differential complexes.