arXiv research

Solutions near infinity to special Lagrangian equations are asymptotic to quadratic polynomials with logarithmic terms.

New Kelvin transform for anisotropic elliptic problems.

Hydrodynamics principles applied to finance, solving complex market models.

We study strict local martingales via h-transforms, a method which first appeared in Delbaen-Schachermayer. We show that strict local martingales arise whenever there is a consistent family of change of measures where the two measures are not equivalent to one another. Several old and new strict local martingales are i…

Soft cells fill space without gaps, derived from minimal surfaces and deformed using edge bending.

We present some new Stokes' type theorems on complete non-compact manifolds that extend, in different directions, previous work by Gaffney and Karp and also the so called Kelvin-Nevanlinna-Royden criterion for (p-)parabolicity. Applications to comparison and uniqueness results involving the p-Laplacian are deduced.

We consider the regularity of an interface between two incompressible and inviscid fluids flows in the presence of surface tension. We obtain local in time estimates on the interface in $H^{\frac32k +1}$ and the velocity fields in $H^{\frac32k}$. These estimates are obtained using geometric considerations which show th…

Let $u_1$ and $u_2$ be two different positive smooth solutions of the equation $Δu + n (n - 2) u^{{n + 2}\over {n - 2}} = 0$ in $R^n (n \ge 3).$ By a result of Gidas, Ni and Nirenberg, $u_1$ and $u_2$ are radially symmetric above the points $ξ_1$ and $ξ_2$, respectively. Let $u$ be a positive $C^2$-function on $R^n$ su…

The Lagrangian and Hamiltonian structures for an ideal gauge-charged fluid are determined. Using a Kaluza-Klein point of view, the equations of motion are obtained by Lagrangian and Poisson reductions associated to the automorphism group of a principal bundle. As a consequence of the Lagrangian approach, a Kelvin-Noeth…

We construct explicit solutions to continuous motion of discrete plane curves described by a semi-discrete potential modified KdV equation. Explicit formulas in terms the $τ$ function are presented. Bäcklund transformations of the discrete curves are also discussed. We finally consider the continuous limit of discrete …

Modified Hennings invariant defined using quantum groups and integrals.

Multidimensional scaling is an important dimension reduction tool in statistics and machine learning. Yet few theoretical results characterizing its statistical performance exist, not to mention any in high dimensions. By considering a unified framework that includes low, moderate and high dimensions, we study multidim…

Improved numerical solution for BSDEs with reduced boundary errors.

We present an explicit example of a fast decaying solution to the modified Novikov--Veselov equation with a one-point singularity in the space-time. It is constructed by using the geometrical interpretation of the Moutard transformation of solutions to this equation and the Enneper minimal surface.

A construction of blowing up solutions to the modified Novikov-Veselov equation is proposed. It is based on the Moutard transformation of two-dimensional Dirac operators and its geometrical interpretation via surface geometry. An explicit example of such a solution constructed by using the Enneper minimal surface is di…

Survey and new results link hydrodynamics, molecular physics, and financial engineering.

Paper introduces a modified Allen-Cahn equation for better energy equipartition.

We tackle the task of environmental event classification by drawing inspiration from the transformer neural network architecture used in machine translation. We modify this attention-based feedforward structure in such a way that allows the resulting model to use audio as well as video to compute sound event prediction…

Our goal is to extract meaningful transformations from raw images, such as varying the thickness of lines in handwriting or the lighting in a portrait. We propose an unsupervised approach to learn such transformations by attempting to reconstruct an image from a linear combination of transformations of its nearest neig…

This note re-addresses the Paris barrier options proposed by Yor and collaborators and their valuation using the Laplace transform approach. The notion of Paris barrier options, based on excursion theory and using the Brownian meander, is extended such that their valuation is now possible at any point during their life…

Modified Wasserstein metric for Gaussian distributions, invariant to isometries.

New Transformer architecture prevents rank degeneracy in deep attention models.

Study examines how Lorentz transformations affect foliations in spacetime.

Paper shows invertibility of tensor X-ray transform on certain manifolds.

This paper adresses the valuation of the Paris barrier options proposed by Yor, Jeanblanc-Picque, and Chesnay (Advances in Applied Probability, 29(1997), 165-184) using the Laplace transform approach. Based on suggestions by Pliska the notion of Paris options is extended such that their valuation is possible at any poi…

Miura-type transformations (MTs) are an essential tool in the theory of integrable nonlinear partial differential and difference equations. We present a geometric method to construct MTs for differential-difference (lattice) equations from Darboux-Lax representations (DLRs) of such equations. The method is applicable t…

We use algebraic Backlund transformations (BTs) to construct explicit solutions of the modified 2+1 chiral model from $T^2\times R$ to SU(n), where $T^2$ is a 2-torus. Algebraic BTs are parameterized by $z\in C$ (poles) and holomorphic maps $π$ from $T^2$ to Gr$(k,C^n)$. We apply Bäcklund transformations with carefully…

Study phase transitions in noisy transformer dynamics on spheres.

New theory explains signal propagation in normalization-free transformers.

Given a vector field on a manifold M, we define a globally conserved quantity to be a differential form whose Lie derivative is exact. Integrals of conserved quantities over suitable submanifolds are constant under time evolution, the Kelvin circulation theorem being a well-known special case. More generally, conserved…

In this thesis, we study a class of special Lagrangian submanifolds of toric Calabi-Yau manifolds and construct their mirrors using some techniques developed in the SYZ programme. We present a justification on the conjecture on the mirror construction of D- branes in Aganagic-Vafa [2]. We apply the techniques employed …

We present a theoretical analysis and empirical evaluations of a novel set of techniques for computational cost reduction of classifiers that are based on learned transform and soft-threshold. By modifying optimization procedures for dictionary and classifier training, as well as the resulting dictionary entries, our t…

Empirical time series of inter-event or waiting times are investigated using a modified Multifractal Detrended Fluctuation Analysis operating on fluctuations of mean detrended dynamics. The core of the extended multifractal analysis is the non-monotonic behavior of the generalized Hurst exponent $h(q)$ -- the fundament…

Improved DeepONet variants using Transformer cross-conditioning enhance PDE solution efficiency.

A spin model relating physical to financial variables is presented. This work is the first to introduce the concept of negative absolute temperature into stock market dynamics by establishing a rigorous formal analogy between physical and financial variables. Based on this model, an algorithm evaluating negative temper…

Study $p$-parabolicity on graphs using various energy functionals.

Equivariance is a nice property to have as it produces much more parameter efficient neural architectures and preserves the structure of the input through the feature mapping. Even though some combinations of transformations might never appear (e.g. an upright face with a horizontal nose), current equivariant architect…

It is demonstrated that the stationary Veselov-Novikov (VN) and the stationary modified Veselov-Novikov (mVN) equations describe one and the same class of surfaces in projective differential geometry: the so-called isothermally asymptotic surfaces, examples of which include arbitrary quadrics and cubics, quartics of Ku…

Extended LSTM improves language modeling performance.

New proof confirms noncompact locally conformally flat manifolds are compact.

Modified perturbation method removes non-smoothness in solving Black-Scholes equations.

In this work we propose a method for learning wavelet filters directly from data. We accomplish this by framing the discrete wavelet transform as a modified convolutional neural network. We introduce an autoencoder wavelet transform network that is trained using gradient descent. We show that the model is capable of le…

nGPT learns to transfer learning rates across model dimensions and token horizons.

This study proposes a trainable adaptive window switching (AWS) method and apply it to a deep-neural-network (DNN) for speech enhancement in the modified discrete cosine transform domain. Time-frequency (T-F) mask processing in the short-time Fourier transform (STFT)-domain is a typical speech enhancement method. To re…

Robust Transformer-Based One-Step Stock Index Forecasting via Shifted Data Augmentation

In some scientific fields, a scaling is able to modify the topology of an observed object. Our goal in the present work is to introduce a new formalism adapted to the mathematical representation of this kind of phenomenon. To this end, we introduce a new metric structure - the galactic spaces - which depends on an orde…

The paper introduces new methods for Asian option pricing using Laguerre quadrature.

Deep vanilla transformers trained without shortcuts achieve similar performance to standard models.