A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Analytic functions on specific domains are characterized by their smoothness and composites with polynomial curves.
Analytic networks with bounded coefficients can't outperform polynomial approximations.
The paper proves deep neural networks with analytic activation can approximate any function.
Any smooth geodesic flow is locally integrable with smooth integrals. We show that generically this fails if we require, in addition, that the integrals are polynomial (or, more generally, analytic) in momenta. Consequently we obtain that a generic real-analytic metric does not admit, even locally, a real-analytic inte…
Polynomials' roots count tied to surface umbilics.
The paper proves real-analyticity of superintegrable metrics and solves two conjectures.
New method detects essential tori in mixed singularity links.
We obtain a complete time expansion of the pull-back operator generated by a real analytic flow of real analytic automorphisms acting on analytic tensor sections of a manifold. Our expansion is given in terms of multiple Lie derivatives. Motivated by this expansion, we provide a rather simple and explicit estimate for …
Approximates discounted moments for financial products using polynomial expansions.
Study resolves polynomial germs, proving no mixed critical points and strict transform properties.
The paper generalizes polynomial functions on Lie groups and their properties.
To a knot in 3-space, one can associate a sequence of Laurent polynomials, whose th term is the th colored Jones polynomial. The Volume Conjecture for small angles states that the value of the -th colored Jones polynomial at $e^{\a/n}$ is a sequence of complex numbers that grows subexponentially, for a fixed s…
In this paper we study both analytic and numerical solutions of option pricing equations using systems of orthogonal polynomials. Using a Galerkin-based method, we solve the parabolic partial diferential equation for the Black-Scholes model using Hermite polynomials and for the Heston model using Hermite and Laguerre p…
Neural networks learn modular arithmetic but not all, extending known solutions to generalize.
Proves volume conjecture for twist knots using complex analysis.
The detrending moving average (DMA) algorithm is one of the best performing methods to quantify the long-term correlations in nonstationary time series. Many long-term correlated time series in real systems contain various trends. We investigate the effects of polynomial trends on the scaling behaviors and the performa…
Using Maple, we compute some analytical solutions of a modified Black-Scholes equation, recently proposed, in the case of the European put option. We show that the modified Black-Scholes equation with the European put option is exactly solvable in terms of associated Laguerre polynomials. We make some numerical experim…
This paper simplifies conditional Sobol' indices calculation using PCE bases.
This paper presents a new method for solving systems with polynomial stiffness.
New method learns low-dimensional models for systems with non-polynomial terms.
The paper contains a combinatorial theorem (the sequence of Newton polygons of a reccurent sequence of polynomials is quasi-linear) and two applications of it in classical and quantum topology, namely in the behavior of the -polynomial and a fixed quantum invariant (such as the Jones polynomial) under filling. Our c…
We derive analytic series representations for European option prices in polynomial stochastic volatility models. This includes the Jacobi, Heston, Stein-Stein, and Hull-White models, for which we provide numerical case studies. We find that our polynomial option price series expansion performs as efficiently and accura…
Extends machine learning models for analytic boundary conditions in differential equations.
Mathematical problems of digital terrain analysis include interpolation of digital elevation models (DEMs), DEM generalization and denoising, and computation of morphometric variables by calculation of partial derivatives of elevation. Traditionally, these procedures are based on numerical treatments of two-variable di…
Wide networks with polynomial activations have proven asymptotic behavior.
The Jones polynomial of a knot in 3-space is a Laurent polynomial in , with integer coefficients. Many people have pondered why is this so, and what is a proper generalization of the Jones polynomial for knots in other closed 3-manifolds. Our paper centers around this question. After reviewing several existing defin…
Study shows not all smooth paths are optimal in certain geometric structures.
We give two formulae which express the Alexander polynomial of several variables of a plane curve singularity in terms of the ring of germs of analytic functions on the curve. One of them expresses in terms of dimensions of some factorspaces corresponding to a (multi-indexed) filtration o…
Smooth knots with odd Conway polynomial terms have inscribed trefoils.
Algorithm finds real-analytic Legendrian representatives for every link type.
Algorithm creates polynomials for knotted surfaces, with bounds on degree.
A new formula approximates knot volume using Jones polynomial evaluations.
We generalize a theorem of Bismut-Zhang, which extends the Cheeger-Mueller theorem on Ray-Singer torsion and Reidemeister torsion, to the case where the flat vector bundle over a closed manifold carries a nondegenerate symmetric bilinear form. As a consequence, we prove the Burghelea-Haller conjecture which gives an an…
We give a purely complex geometric proof of the existence of the Bergman kernel expansion. Our method provides a sharper estimate, and in the case that the metrics are real analytic, we prove that the remainder decays faster than any polynomial.
This paper presents a new framework for manifold learning based on a sequence of principal polynomials that capture the possibly nonlinear nature of the data. The proposed Principal Polynomial Analysis (PPA) generalizes PCA by modeling the directions of maximal variance by means of curves, instead of straight lines. Co…
We derive a factorization of the Alexander polynomial of the 4-strand Turk's head knot using hypergeometric representations.
The paper explores how polynomial roots and operator eigenvalues change with parameters.
The paper describes relations between Liouville type theorems for solutions of a periodic elliptic equation (or a system) on an abelian cover of a compact Riemannian manifold and the structure of the dispersion relation for this equation at the edges of the spectrum. Here one says that the Liouville theorem holds if th…
We study complex analytic (possibly singular) projective connections on the plane. We characterize some of them in terms of their families of integral curves. We also give a beginning of classification of second order odes polynomial in the first and second derivatives, and with holomorphic coefficients.
OPAA estimates probability densities using functional analysis.
Geodesics in 3D space with 1-2 analytic obstacles, proving geodesic independence.
Recent proofs of classical theorems in polynomial algebra and functional analysis are discussed, which use tools from the topology of real manifolds. Simpler proofs were discovered in the new century, of the Hilbert Nullstellensatz, and the Gelfand-Mazur Theorem. We give a related proof that an irreducible real polynom…
The projective hull X^ of a subset X in complex projective space P^n is an analogue of the classical polynomial hull of a set in C^n. If X is contained in an affine chart C^n on P^n, then the affine part of X^ is the set of points x in C^n for which there exists a constant M=M_x so that |p(x)| < M^d sup{|p(y)| : y in X…
Proves solution uniqueness for biomembrane shape prediction.
The study optimizes polynomial regression for learning under Gaussian distributions.
Hermite polynomials improve private data generation by reducing feature count.
Resurgent analysis reveals full partition function for 3-manifold invariants.
Numerical discovery matches eta invariant on Berger spheres with conformal anomaly on round spheres.