arXiv research

Compactness proven for isospectral Birkhoff billiard tables.

We provide new exact Taylor's series with fixed coefficients and without the remainder. We demonstrate the usefulness of this contribution by using it to obtain very simple solutions to (non-linear) PDEs. We also apply the method to the portfolio model.

The conformal powers of the Laplacian of a Riemannian metric which are known as the GJMS-operators admit a combinatorial description in terms of the Taylor coefficients of a natural second-order one-parameter family $\H(r;g)$ of self-adjoint elliptic differential operators. $\H(r;g)$ is a non-Laplace-type perturbation …

We apply results of Malliavin-Thalmaier-Watanabe for strong and weak Taylor expansions of solutions of perturbed stochastic differential equations (SDEs). In particular, we work out weight expressions for the Taylor coefficients of the expansion. The results are applied to LIBOR market models in order to deal with the …

Defines natural tensors for submanifolds of pseudo-Riemannian manifolds.

Explicit Taylor series for the volume of tubes in Lie groups

Expanding the rough Heston model in $H$

Examines how central bank policies affect stock markets and asset prices.

The paper calculates Bachelier option prices using Taylor expansions and applies it as a variance reduction technique.

By analyzing the affine Taylor expansion of a non-degenerate plane curve, we obtain characterizations of classes of such curves via curvature properties of the gravity curve. The proof is based on an analysis of the degree parity and leading coefficients of polynomials occurring in the expansion.

This paper extends AD techniques to Monte Carlo processes for efficient derivative calculation.

This work continues the study of a homotopy-theoretic construction of the author inspired by the Bott-Taubes integrals. Bott and Taubes constructed knot invariants by integrating differential forms along the fiber of a bundle over the space of knots. Their techniques were later used by Cattaneo et al. to construct real…

We give upper bounds for the Bergman kernels associated to tensor powers of a smooth positive line bundle in terms of the rate of growth of the Taylor coefficients of the Kähler potential. As applications, we obtain improved off-diagonal rate of decay for the classes of analytic, quasi-analytic, and more generally Gevr…

We consider a general one-factor short rate model, in which the instantaneous interest rate is driven by a univariate diffusion with time independent drift and volatility. We construct recursive formula for the coefficients of the Taylor expansion of the bond price and its logarithm around $τ=0$, where $τ$ is time to m…

Study Bergman kernel metrics on degenerating hyperelliptic surfaces.

This study reveals statistical patterns in ERC20 token transactions on Ethereum blockchain.

Paper extracts features from time series to improve forecasting accuracy.

We will prove that Ruelle L-function for a cuspidal local system on an odd dimensional hyperbolic manifold with finite volume satisfies a functional equation and an analog of the Riemann hypothesis. We will also compute its Laurent expansion at the origin and will prove that the second coefficient coincides with a rati…

Kernelized Taylor diagram visualizes data populations with fewer assumptions.

Integrality of FJRW invariants for Lie algebras A_l, D_l, and E_6.

We derive asymptotic expansions for the prices of a variety of European and barrier-style claims in a general local-stochastic volatility setting. Our method combines Taylor series expansions of the diffusion coefficients with an expansion in the correlation parameter between the underlying asset and volatility process…

Neural networks learn higher-order derivatives for physics problems.

We study the counting function of topological Poincaré series associated with rational homology sphere plumbed 3-manifold with connected negative definite tree, interpreting as an alternating sum of coefficient functions associated with some Taylor expansions. It is motivated by a theorem of Szenes and Vergne which exp…

Develops AMITE for analyzing neural network nonlinearities.

Taylor expansions improve reinforcement learning policies.

Defines invariants for reflection groups and connects them to Frobenius structures.

We propose \emph{Taylorized training} as an initiative towards better understanding neural network training at finite width. Taylorized training involves training the $k$-th order Taylor expansion of the neural network at initialization, and is a principled extension of linearized training---a recently proposed theory …

Paper defines invariants for elliptic Weyl groups and connects them to Frobenius structures.

Study finds Deep Taylor Decomposition is unreliable for explaining neural networks.

We provide a general method to compute a Taylor expansion in time of implied volatility for stochastic volatility models, using a heat kernel expansion. Beyond the order 0 implied volatility which is already known, we compute the first order correction exactly at all strikes from the scalar coefficient of the heat kern…

The first aperiodic monotiling, introduced by Taylor, was based on a trapezoidal prototile equipped with 14 distinct decorations. A presentation of the closely related Taylor-Socolar aperiodic monotiling is based on a hexagonal prototile equipped with 7 decorations. This paper gives decoration-free algebraic descriptio…

The paper is concerned with non-linear Gaussian filtering and smoothing in continuous-discrete state-space models, where the dynamic model is formulated as an Itô stochastic differential equation (SDE), and the measurements are obtained at discrete time instants. We propose novel Taylor moment expansion (TME) Gaussian …

Proposes a Taylor framework to unify and analyze attribution methods.

NN2Poly converts deep neural networks into polynomial models for better understanding.

New approximations for Asian basket spread options using stochastic Taylor expansions.

Paper develops a new algorithm to find shortest paths on surfaces.

TaylorPODA uses Taylor expansions to improve feature attributions for opaque models.

Developed Taylor series for muscle-finger system analysis.

We study the finite horizon Merton portfolio optimization problem in a general local-stochastic volatility setting. Using model coefficient expansion techniques, we derive approximations for the both the value function and the optimal investment strategy. We also analyze the `implied Sharpe ratio' and derive a series a…

Modern convolutional networks, incorporating rectifiers and max-pooling, are neither smooth nor convex; standard guarantees therefore do not apply. Nevertheless, methods from convex optimization such as gradient descent and Adam are widely used as building blocks for deep learning algorithms. This paper provides the fi…

The Adomian decomposition method is shown to be equivalent to the Taylor series approach.

A mathematical framework connects neural networks and polynomial regression for better model understanding.

J.P. Levine showed that the Conway polynomial of a link is a product of two factors: one is the Conway polynomial of a knot which is obtained from the link by banding together the components; and the other is determined by the $\barμ$-invariants of a string link with the link as its closure. We give another description…

Proves a special case of the Gaussian kinematic formula using large sphere limits.

Unified framework for analyzing machine learning model attributions.

We introduce Taylor expansions that do not require the differentiability. We also provide new solutions to partial differential equations. We apply our methods to finance.

An expanding literature articulates the view that Taylor rules are helpful in predicting exchange rates. In a changing world however, Taylor rule parameters may be subject to structural instabilities, for example during the Global Financial Crisis. This paper forecasts exchange rates using such Taylor rules with Time V…

Unified framework connects two market-making models, revealing their underlying equivalence.