The paper extends first-order asymptotics for path-dependent derivatives in multiscale stochastic volatility.
problem Analyzing path-dependent derivatives in a multiscale stochastic volatility environment.
method First-order asymptotics analysis using Dupire's functional Ito calculus.
result Market parameters calibrated to vanilla options can price path-dependent derivatives to the same order.
Derives derivatives of risk measures for various types of portfolio losses.
problem Calculating precise risk measures for portfolio losses.
method Analyzes first and second order derivatives of risk measures for both continuous and discrete portfolio loss scenarios.
result Provides asymptotic results for conditional moments of heavy-tailed portfolio losses.
Derives Lagrangian for minimal surfaces, proving tangential variations vanish.
problem Variational calculus for minimal surfaces.
method Lagrangian formulation, pullback covariant derivative, geometric argument.
result Tangential variations vanish for minimal surfaces.
New method computes first Vassiliev derivative of Khovanov homology.
problem Computing Vassiliev derivatives of Khovanov homology.
method Developed a crux complex to compute the first derivative.
result Direct computation of the first derivative of Khovanov homology.
The article explores derivatives of shapes other than circles and spheres.
problem Understanding derivatives of various shapes.
method First-year calculus approach.
result Derivatives of shapes other than circles and spheres are explored.
We calculate the higher derivatives of length functions on Teichmuller space along earthquake deformations. This generalizes the cosine formula for the first derivative by Kerckhoff and Wolpert and the sine formula for second derivative by Wolpert.
Sharp bounds derived for eigenvalues on specific geometric spaces.
problem Eigenvalue problems on asymptotically hyperbolic manifolds and submanifolds.
method Sharp bounds derived for three types of eigenvalue problems: p-Dirichlet, polyharmonic, and weakly Poincaré-Einstein. result Sharp bounds and their implications for asymptotic sectional curvatures and mean curvature.
The first eigenfunction of a specific domain in hyperbolic space is log-concave.
problem Proving the log-concavity of the first eigenfunction on horoconvex domains in hyperbolic space.
method Proof by contradiction, using properties of Killing derivatives and nodal domains.
result The first eigenfunction is log-concave throughout the domain.
We formalize geometrically the idea that the (de Donder) Hamiltonian formulation of a higher derivative Lagrangian field theory can be constructed understanding the latter as a first derivative theory subjected to constraints.
We characterize the Lie derivative of spinor fields from a variational point of view by resorting to the theory of the Lie derivative of sections of gauge-natural bundles. Noether identities from the gauge-natural invariance of the first variational derivative of the Einstein(--Cartan)--Dirac Lagrangian provide restric…
We prove Bismut-type formulae for the first and second derivatives of a Feynman-Kac semigroup on a complete Riemannian manifold. We derive local estimates and give bounds on the logarithmic derivatives of the integral kernel. Stationary solutions are also considered. The arguments are based on local martingales, althou…
Schwarzian derivative connects to Euler-Lagrange equations in variational calculus.
problem Understanding the relationship between the Schwarzian derivative and variational equations.
method Analyzing the Schwarzian derivative as a first integral and Euler-Lagrange operator for specific variations.
result The Schwarzian derivative is both a first integral and the Euler-Lagrange operator for a certain class of variations.
New methods boost first-order optimization with faster rates.
problem Designing efficient first-order methods for convex problems.
method Shifted objective function with interpolation condition.
result New schemes achieve faster convergence rates.
We derive pointwise curvature estimates for graphical mean curvature flows in higher codimensions. To the best of our knowledge, this is the first such estimates without assuming smallness of first derivatives of the defining map. An immediate application is a convergence theorem of the mean curvature flow of the graph…
We consider the Lie algebra consisting of all derivations on the free associative algebra, generated by the first homology group of a closed oriented surface, which kill the symplectic class. We find the first non-trivial abelianization of this Lie algebra and discuss its relation to unstable cohomology classes of the …
Extends pricing theory for collateralized derivatives to include jumps and dividends.
problem Pricing collateralized derivatives with jumps and dividends.
method Extends No-Arbitrage theory to semimartingales, deriving pricing, dynamics, and forward prices.
result Derives pricing, dynamics, and forward prices of collateralized derivatives.
In the present paper a generalized Kählerian space G1KN of the first kind is considered, as a generalized Riemannian space GRN with almost complex structure Fih, that is covariantly constant with respect to the first kind of covariant derivative. Using the non-symmetr…
Computes derivatives of sections in vector bundles using Lie derivatives.
problem Computing time derivatives of sections in natural vector bundles.
method Extending a lemma to compute Lie derivatives of sections of natural vector bundles.
result Computed derivatives of sections in vector bundles using Lie derivatives.
The paper bounds eigenvalues of the Jacobi operator and derives rigidity results for CMC hypersurfaces.
problem Bounding the first eigenvalue of the Jacobi operator for CMC hypersurfaces.
method Geometric upper bounds for eigenvalues and rigidity results.
result New rigidity results for the area and length of CMC hypersurfaces.
A neural network derived from first principles using MaxEnt.
problem Developing a neural network from first principles.
method Derived a neural network using the principle of Maximum Entropy, with linear dimension-reducing transformations and conditional mean estimators.
result Unified theoretical justification for activation functions like sigmoid, softplus, and relu.
Closed-form relations and approximations for SE(3) derivatives for robust numerical simulations.
problem Deriving closed-form derivatives and approximations for SE(3) for robust numerical simulations.
method Avoiding block partitioning, deriving higher-order approximations for differential, first and second derivatives, Jacobian, and Hessian.
result Compact and numerically robust closed-form relations for SE(3) derivatives.
Paper derives formulas for surface variations in shell theory.
problem Deriving first variation formulas for surfaces in thin shell theory.
method Using strain-displacement relations from thin shell theory.
result Provides formulas for linear Weingarten surfaces as stationary points.
Tangent automates derivatives in Python, improving expressiveness and performance.
problem Efficiently calculating derivatives for complex models in Python.
method Source-code transformation for dynamically typed array programming.
result Demonstrates improved expressiveness and performance in automatic differentiation.
Study on ion travel time on curved surfaces.
problem Mean first passage time of ion on curved surfaces.
method Layer potential argument and microlocal analysis.
result Derivation of mean first passage time and spatial average.
New bounds on continuous random variables' right-tail probabilities.
problem Finding precise upper and lower limits for right-tail probabilities of continuous random variables.
method Developed new bounds based on PDF, first derivative, and two parameters.
result The new bounds are tight for various continuous random variables.
For a linear combination of random variables, fix some confidence level and consider the quantile of the combination at this level. We are interested in the partial derivatives of the quantile with respect to the weights of the random variables in the combination. It turns out that under suitable conditions on the join…
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.
Study shows a modified cobordism category's first derivative is equivalent to a Thom spectrum.
problem Analyzing the homotopy type of surface cobordism categories.
method Defined a new cobordism category over a base space, proving properties of induced functors and derivatives.
result The first derivative of the induced functor is equivalent to a Thom spectrum.
Derives a formula for the k-th covariant derivative of tensor fields.
problem Finding a formula for the k-th covariant derivative of tensor fields.
method Introducing symbols P and Q depending on Christoffel symbols, deriving a formula (3.1).
result Derives a formula for the k-th covariant derivative of tensor fields.
We show that a certain symmetry exists in the stable irreducible decomposition of the Lie algebra consisting of symplectic derivations of the free Lie algebra generated by the first homology group of compact oriented surfaces.
Geodesic coordinates derived for a specific metric in surface group representations.
problem Computing geodesic coordinates for a specific metric in surface group representations.
method Using thermodynamic formalism and gauge-theoretic formulas, computing first and second derivatives of the pressure metric.
result First derivatives of the pressure metric vanish at the Fuchsian locus.
Derives inequalities for eigenvalues and renormalized volume of Poincaré-Einstein manifolds.
problem Eigenvalues and renormalized volume of Poincaré-Einstein manifolds.
method Integral inequality and eigenvalue estimates.
result Sharp lower bound for first eigenvalue and new upper bound for renormalized volume.
Derives an inequality for submanifolds in spheres.
problem Characterize submanifolds in spheres.
method Derives an integral inequality.
result Characterizes spheres.
The author constructs height functions and studies their first derivatives on closed manifolds.
problem Understanding the first derivatives of height functions on closed manifolds.
method Constructing explicit maps and height functions, studying their global behaviors.
result Detailed study of the first derivatives of height functions on closed manifolds.
Method solves complex optimization problems with high probability bounds.
problem Nonlinear equality constrained stochastic optimization problems.
method Step-search sequential quadratic programming method.
result High-probability bound on iteration complexity for first-order stationarity.
Based on the work of Schoen-Yau, we derive an estimate of the first eigenvalue of a Schrödinger Operator (the Jaocbi operator of minimal surfaces in flat 3-spaces) on surfaces.
New bounds on manifold Betti numbers derived from semigroup norms.
problem Estimating the first Betti number of compact Riemannian manifolds.
method Birman-Schwinger principle and Schatten norm estimates for semigroup differences, without ultracontractivity assumptions.
result Explicit bounds on Betti numbers depend on Ricci tensor norms.
A method to estimate high order derivatives of data distributions from samples.
problem Estimating high order derivatives of data distributions efficiently and accurately.
method Generalizing denoising score matching via Tweedie's formula to estimate higher order derivatives.
result Models trained with the proposed method can approximate second order derivatives more efficiently and accurately than via automatic differentiation.
Formulae derived for survival and first passage times in stochastic processes.
problem Computing survival and first passage times for jump and diffusion processes.
method Recursive formula derivation for nextth survival and first passage time distributions. result General formulae for nextth survival and first passage times in multi-coordinate stochastic processes. The geometrical structure known as Tulczyjew triple has been used with success in analytical mechanics and first order field theory to describe a wide range of physical systems including Lagrangian/Hamiltonian systems with constraints and/or sources, or with singular Lagrangian. Starting from the first principles of th…
Derives a new first order differential equation for smooth surfaces.
problem Finding new equations to describe smooth surfaces.
method Derives a linear differential equation of the first order.
result Proves the maximum principle for Darboux rotation fields.
In this paper we present a new method to compute the first-order approximation of the price of derivatives on futures in the context of multiscale stochastic volatility of Fouque \textit{et al.} (2011, CUP). It provides an alternative method to the singular perturbation technique presented in Hikspoors and Jaimungal (2…
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 derives second order approximations for limit order books.
problem Modeling fluctuations in price and volume processes.
method Second order approximation of infinite dimensional limit order book dynamics.
result Confidence intervals for optimal portfolio liquidation models.
Abstract reviews recent Lagrangian analysis on immersions into higher dimensions.
problem Analyzing Lagrangians on immersions into higher dimensions.
method Reviews recent progress on Lagrangians on immersions with first and second fundamental forms and their derivatives.
result Recent progress in the analysis of Lagrangians on immersions into higher dimensions.
Study categorifies link invariants using Soergel bimodules.
problem Categorification of link invariants.
method Explicit computation of derived traces of Soergel bimodules.
result Derived annular Khovanov-Rozansky link invariant.
The virtual Betti number conjecture states that any hyperbolic three-manifold has a finite cover with positive first Betti number. We show that this would follow if it were known that the derived series of the fundamental group G of a hyperbolic three-manifold satisfies a certain stability property. The stability pro…
A regular normal parabolic geometry of type G/P on a manifold M gives rise to sequences Di of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative $\na^\om$ on the corresponding tractor bundle V, where $\om$ is…