Defines tangent spaces on causal sets using partial derivatives and metrics.
problem Defining geometric structures on causal sets.
method Using partial derivatives and metrics to define tangent spaces, connection, curvature, parallel transport, and geodesics.
result Approaches expected values for a flat spacetime as density increases.
Paper introduces efficient methods for estimating cross-partial derivatives and sensitivity indices.
problem Efficiently estimating cross-partial derivatives and sensitivity indices in complex models.
method Using randomized points and constraints, the paper develops estimators with optimal convergence rates and low bias.
result The estimators achieve optimal rates of convergence and do not suffer from the curse of dimensionality.
We introduce Hermite fractional financial markets, where market uncertainties are described by multidimensional Hermite motions. Hermite markets include as particular cases financial markets driven by multivariate fractional Brownian motion and multivariate Rosenblatt motion. Conditions for no-arbitrage and market comp…
Derives derivatives and geometric framework for functions with non-independent variables.
problem Characterizing functions with non-independent variables in probabilistic models.
method Derives actual and dependent partial derivatives, dependent Jacobian matrix, and tensor metric.
result Derives gradient, Hessian, and Taylor expansion for functions with non-independent variables.
Researchers found the Wigner derivative and its inverse are equal for spherical tetrahedra.
problem Computing the relationship between dihedral angles and edge lengths in tetrahedra.
method Computed the Wigner derivative and its inverse for spherical tetrahedra.
result The Wigner derivative and its inverse are equal for spherical tetrahedra.
Estimates smooth functions and their derivatives from noisy data.
problem Estimating smooth functions and their derivatives from noisy data.
method Least squares estimators and minimizers of smoothness subject to error bounds.
result Consistent estimators with convergence rates as n increases.
We consider various homological operations on homology of quandles. We introduce the notion of quandle partial derivatives, and extreme chains on which appropriate partial derivatives vanish. Extreme chains yield homological operations. We also consider the degree one homology operations created using elements of the q…
The paper derives risk measures for metalog distributions.
problem Deriving risk measures for metalog distributions.
method Closed-form expressions for Conditional Value at Risk and first-order partial moments.
result First-order partial moments are convex with respect to metalog parameters.
Most of the empirical studies on stochastic volatility dynamics favor the 3/2 specification over the square-root (CIR) process in the Heston model. In the context of option pricing, the 3/2 stochastic volatility model is reported to be able to capture the volatility skew evolution better than the Heston model. In this …
The L2-∂∂-Lemma is extended to complete Kähler manifolds with a gap in the spectrum.
problem Extending the L2-∂∂-Lemma to non-compact Kähler manifolds. method Proving the L2-∂∂-Lemma on complete Kähler manifolds with a gap in the spectrum. result The L2-∂∂-Lemma is generalized to complete Kähler manifolds. In this paper, we derive some ∂∂-Bochner formulas for holomorphic maps between Hermitian manifolds. As applications, we prove some Schwarz lemma type estimates, rigidity and degeneracy theorems. For instance, we show that there is no non-constant holomorphic map from a comapct Hermitian manif…
Study uniquely determines Riemannian metric derivatives from boundary data.
problem Determining Riemannian metric derivatives from boundary data.
method Computing the full symbol of the elastic Dirichlet-to-Neumann map.
result The elastic Dirichlet-to-Neumann map uniquely determines all partial derivatives of the Riemannian metric on the boundary.
Study partial derivatives on non-smooth metric measure structures.
problem Understanding partial derivatives in non-smooth settings.
method Extension of Schwarz's theorem and analysis of Sobolev regularity.
result Complete set of results relating properties of functions in non-smooth spaces.
Sharp bounds derived for the first two Steklov eigenvalues of exterior domains.
problem Finding bounds for the first two eigenvalues of Steklov eigenvalue problems on exterior domains.
method Sharp lower and upper bounds derived using the support function and distance function to the origin of the boundary.
result Sharp bounds for the first two eigenvalues of Steklov eigenvalue problems on exterior domains.
Abstract: Determines Lamé coefficients from boundary measurements.
problem Determining Lamé coefficients from elastic boundary measurements.
method Explicit symbol of elastic Dirichlet-to-Neumann map, partial derivatives determination.
result Elastic Dirichlet-to-Neumann map uniquely determines Lamé coefficients.
Optimizes material distribution on surfaces using topological derivatives.
problem Optimal distribution of two materials on smooth submanifolds in Rd. method Topological derivative approach for shape optimization constrained by PDEs.
result Numerical solution of topology optimization problem on surfaces.
The paper proves a logarithmic partial derivative lemma and applies it to several geometric problems.
problem Proving a logarithmic partial derivative lemma for compact Kähler manifolds.
method Developed a new ∂∂ˉ-type lemma for logarithmic differential forms. result Confirmed a conjecture by X. Wan and derived several geometric applications.
The paper characterizes when the ∂∂-lemma holds for twistor spaces.
problem Characterizing the ∂∂-lemma for twistor spaces. method Study Bott-Chern and Aeppli cohomologies of twistor spaces.
result Explicit computation of Dolbeault cohomology for flat torus twistor space.
Partial dependence curves (FPD) introduced by Friedman, are an important model interpretation tool, but are often not accessible to business analysts and scientists who typically lack the skills to choose, tune, and assess machine learning models. It is also common for the same partial dependence algorithm on the same …
Abstract: Determines thermoelastic coefficients from boundary data.
problem Determining coefficients of thermoelastic system from boundary information.
method Explicit expression for thermoelastic Dirichlet-to-Neumann map with variable coefficients.
result Thermoelastic Dirichlet-to-Neumann map uniquely determines coefficients on the manifold.
New findings on complex manifold properties under deformations.
problem Properties of Dolbeault and Bott-Chern formalities are not preserved under holomorphic deformations.
method Construction of a complex manifold to demonstrate non-preservation of properties.
result Existence of a manifold satisfying ∂∂-lemma but with non-vanishing Aeppli-Bott-Chern-Massey product. Derives gradient estimation for a specific heat equation on evolving manifolds.
problem Gradient estimation for a generalized heat equation on evolving weighted Riemannian manifolds.
method Derives gradient estimation for a specific heat equation on evolving weighted Riemannian manifolds.
result Derives a Harnack type inequality and a Liouville type theorem as applications of gradient estimation.
In this article, we combine replication pricing with expectation pricing for derivative trades that are partially collateralized by cash. The derivatives are replicated by underlying assets and cash, using repurchasing agreement (repo) and margining, which incur funding costs. We derive a partial differential equation …
New complex non-Kähler manifolds with specific properties are constructed.
problem Constructing complex non-Kähler manifolds with special properties.
method Using families of compact solvmanifolds and properties of the ∂∂ˉ-Lemma. result Provided families of compact (n+1)-dimensional complex non-Kähler manifolds with specific properties. The paper introduces and studies hedging for game (Israeli) style extension of swing options considered as multiple exercise derivatives. Assuming that the underlying security can be traded without restrictions we derive a formula for valuation of multiple exercise options via classical hedging arguments. Introducing t…
New quantum algorithm simplifies complex financial derivatives pricing.
problem Complex financial derivatives pricing with high dimensionality.
method Quantum-inspired variational algorithms combined with neural-network quantum states.
result Simplified pricing of European options with many correlated assets.
We give necessary conditions for certain real analytic tube generic submanifolds in C^n to be locally algebraizable. As an application, we exhibit families of real analytic non locally algebraizable tube generic submanifolds in C^n. During the proof, we show that the local CR automorphism group of a minimal, finitely n…
Method estimates sparse inverse covariance and partial correlation matrices efficiently.
problem Sparse high-dimensional inverse covariance and partial correlation matrix estimation.
method Two-stage estimation method using partial regression with positive semi-definiteness.
result Efficient estimation of inverse covariance and partial correlation matrices with derived non-asymptotic rates.
Let Q be a smooth compact orientable 3--manifold with smooth boundary ∂Q. Let B be the set of exact 2--forms B∈Ω2(Q) such that j∂Q∗B=0, where j∂Q:∂Q→Q is the inclusion map. The group D=Diff0(Q) of self-diffeomorphisms of Q isot…
New framework discovers PDEs from sparse, noisy data.
problem Discovering PDEs with high-order derivatives and heterogeneous parameters.
method Combines deep-learning and integral form to handle sparse and noisy data.
result More robust and accurate compared to existing methods.
New method for analyzing elliptic and parabolic equations.
problem Analyzing elliptic and parabolic equations.
method Level set version of partial uniform ellipticity.
result Effective approach to investigate equations.
Paper tackles distribution matching by partially matching distributions, achieving robust results.
problem Robustly aligning two probability distributions.
method Developed a partial Wasserstein adversarial network (PWAN) to efficiently approximate the partial Wasserstein-1 (PW) discrepancy.
result The PWAN effectively produces highly robust matching results, outperforming state-of-the-art methods.
A one-factor asset pricing model with an Ornstein--Uhlenbeck process as its state variable is studied under partial information: the mean-reverting level and the mean-reverting speed parameters are modeled as hidden/unobservable stochastic variables. No-arbitrage pricing formulas for derivative securities written on a …
Improved algorithm for partial recovery of tree-structured graphs with noisy data.
problem Learning Ising tree models with noisy observations.
method Symmetrized Geometric Averaging (SGA) algorithm with improved sample complexity.
result Significantly better sample complexity for partial tree recovery.
Study methods to recover unknown processes in PDEs from data.
problem Identifying unknown processes in time-dependent PDEs using observational data.
method Theoretical analysis and numerical approaches including Galerkin and collocation algorithms.
result The Galerkin algorithm is more suitable for practical situations with noisy data.
The paper uses belief propagation to analyze rankings and partial orders from partial information.
problem Analyzing rankings and partial orders from incomplete data.
method Continuous spin system and belief propagation algorithm.
result Computes marginal distribution and approximates number of linear extensions.
In common finance literature, Black-Scholes partial differential equation of option pricing is usually derived with no-arbitrage principle. Considering an asset market, Merton applied the Hamilton-Jacobi-Bellman techniques of his continuous-time consumption-portfolio problem, deriving general equilibrium relationships …
Derives PDEs from data using manifold learning and neural networks.
problem Identifying PDEs from unknown variables and dynamics.
method Combines manifold learning (Diffusion Maps) and neural networks.
result Emergent space identification connects with multiscale computation.
Neural nets replicate hedging payoffs for realistic discrete-time settings.
problem Hedging in realistic, discrete-time financial markets with transaction costs.
method Deep learning techniques to train neural networks to replicate modified payoff functions.
result Neural networks can better accommodate realistic hedging scenarios and transaction costs.
We study a market model in which the volatility of the stock may jump at a random time from a fixed value to another fixed value. This model was already described in the literature. We present a new approach to the problem, based on partial derivative equations, which gives a different perspective to the problem. Withi…
Survey on smooth function and form density in Riemannian Sobolev spaces.
problem Density of smooth functions and forms in Sobolev spaces on Riemannian manifolds.
method Careful examination of weak covariant derivatives and partial derivatives.
result Equivalence of weak covariant derivatives to weak partial derivatives.
Inference for normal and Monte Carlo distributions using minimum relative entropy.
problem Inference from partial information on expectations and covariances.
method Minimum relative entropy sub-manifolds, analytical formulas, Monte Carlo simulations.
result Improved numerical implementation for inference from partial information.
Harmonic basis vector fields on surfaces
problem Parameterizing surfaces with harmonic vector fields
method Introducing harmonic basis vector fields and deriving conditions for their existence
result Classifying parameterizations of surfaces with harmonic basis vector fields
Unified framework for selecting variables with uncertainty quantification.
problem Uncertainty in nonlinear variable selection for various models.
method Develops a unified framework using integrated partial derivatives for quantifying variable importance and uncertainty.
result The approach provides a principled method for quantifying variable selection uncertainty and is generalizable to non-differentiable models.
A general theory of rigid completely integrable analytic partial differential equations is endeavoured. The tube over the light cone in C^3 is shown to be the unique model (up to biholomorphisms) having CR automorphism group of maximal dimension equal to 10. Explicit formulas for the Lie prolongation of vector fields t…
For a convex domain D bounded by the hypersurface ∂D in a space of constant curvature we give sharp bounds on the width R−r of a spherical shell with radii R and r that can enclose ∂D, provided that normal curvatures of ∂D are pinched by two positive constants. Furthermore, in the …
Optimizes ranking of top-k players from partial comparison data.
problem Identifying the top-k players from incomplete pairwise comparisons.
method Maximum Likelihood Estimator (MLE) and Spectral Method.
result MLE achieves optimal partial and exact recovery, while Spectral Method is sub-optimal.
Study explores how scalar functionals evolve under Ricci flow.
problem Understanding the evolution of functionals involving scalar quantities under Ricci flow.
method Deriving explicit expressions for the time derivative of integrals of scalar functionals under extended Ricci flow.
result Explicit expressions for the time derivative of integrals involving scalar functionals under Ricci flow.