Calculates spinor heat flow using Gaussian-Grassmann integrals.
problem Computing the spinor heat flow symbol.
method Getzler calculus and Gaussian-Grassmann integrals.
result Computed the spinor heat flow symbol.
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48 results for “Gaussian calculus”
Novel knot polynomials from Gaussian calculus show half vanish and determine Jones polynomials.
problem Understanding and characterizing knot polynomials from Gaussian calculus.
method Gaussian calculus of generating series for noncommutative algebras, connected sum of knots.
result Half of the polynomials vanish and three polynomials are explicitly given.
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…
Paper analyzes holdout cross-validation for large non-Gaussian covariance estimation.
problem Estimating large covariance matrices for non-Gaussian data.
method Use of Weingarten calculus and Ledoit-Péché formula for theoretical error derivation.
result Optimal train-test split ratio is proportional to square root of matrix dimension.
The paper compares and optimizes estimators for treatment effects with observed confounders and mediators.
problem Estimating treatment effects with observed confounders and mediators.
method Investigates the linear Gaussian causal model, compares and optimizes estimators, and combines datasets.
result An optimal estimator outperforms the backdoor and frontdoor estimators by an unbounded constant factor.
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…
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.
The paper provides formulas for volatility in various models, including rough volatility.
problem Calibrating SPX and VIX options with rough volatility models.
method Developed explicit formulae using Malliavin calculus for Gaussian processes.
result New insights on joint calibration of SPX and VIX options.
Method learns Dirichlet-to-Neumann maps on graphs using Gaussian processes.
problem Coupling multiphysics simulations on graphs with conservation constraints.
method Gaussian processes combined with discrete exterior calculus and maximum likelihood estimation.
result Data-driven predictions with uncertainty quantification on entire graph.
This paper is a step-by-step tutorial for fitting a mixture distribution to data. It merely assumes the reader has the background of calculus and linear algebra. Other required background is briefly reviewed before explaining the main algorithm. In explaining the main algorithm, first, fitting a mixture of two distribu…
New approach to quantum knot invariants using perturbed Gaussian generating functions.
problem Developing universal quantum knot invariants.
method Introducing generating functions of the form where is quadratic and is a perturbation, and developing a calculus for such functions.
result The rank one invariant dominates -colored Jones polynomials and relates to knot genus and Whitehead doubling.
The article derives a novel Gram-Charlier A (GCA) Series based Extended Rule-of-Thumb (ExROT) for bandwidth selection in Kernel Density Estimation (KDE). There are existing various bandwidth selection rules achieving minimization of the Asymptotic Mean Integrated Square Error (AMISE) between the estimated probability d…
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.
Three-candidate plurality voting is stable for small correlations.
problem Stability of plurality voting in small correlation scenarios.
method Calculus of variations and noise stability analysis.
result Proof of Plurality is Stablest Conjecture for 3 candidates.
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.
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…
A Malliavin-Gamma calculus approach to Score Based Diffusion Generative models for random fieldsmath.PR
Abstract: Generalizes SGMs to infinite-dimensional Hilbertian setting.
problem Difficulties in extending SGMs to infinite-dimensional settings.
method Uses Gamma and Malliavin Calculus, Dirichlet forms, Wiener chaoses, and time-reversal formula.
result Generalized SGMs to Hilbertian setting with finite-dimensional entropic convergence bounds.
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.
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.
CCHM algorithm learns BN structure with latent variables, improving causal effect measurement.
problem Latent variables cause spurious relationships in BN structure learning.
method Hybrid approach combining constraint-based and score-based learning, incorporating do-calculus.
result CCHM outperforms state-of-the-art in reconstructing true BN structure.
Develops calculus for tamed Dirichlet spaces using measure theory.
problem Defines calculus for measure spaces with Dirichlet forms.
method Introduces first and second order calculus on tamed Dirichlet spaces.
result Defines various geometric objects on tamed Dirichlet spaces.
Develops a smooth operator framework for analyzing neural network representations.
problem Analyzing the geometry of feedforward neural network representations.
method Introduces a smooth operator-theoretic approach based on diffusion Markov operators derived from feature clouds.
result Establishes a stable operator-geometric framework for tracking training, width, and perturbation stability.
Koszul duality for manifold modules proven.
problem Proving Koszul self duality of manifold modules.
method Using generalized Thom complexes and operads in Top.
result Koszul self duality of little disk modules proven.
The Accardi-Boukas quantum Black-Scholes framework, provides a means by which one can apply the Hudson-Parthasarathy quantum stochastic calculus to problems in finance. Solutions to these equations can be modelled using nonlocal diffusion processes, via a Kramers-Moyal expansion, and this provides useful tools to under…
Simplified Khovanov-Rozansky calculus for bipartite knots.
problem Complexity in calculating superpolynomials for knots.
method Bipartite calculus generalizes Khovanov-Rozansky calculus for a restricted class of knots.
result Simplification of Khovanov-Rozansky polynomials for bipartite knots.