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.
New integrable matrix PDEs derived from Frölicher-Nijenhuis brackets.
Quaternionic differential geometry expands geometric concepts using quaternions.
Effective Gram matrix predicts deep network generalization.
We prove constrained trace, matrix and constrained matrix Harnack inequalities for the nonlinear heat equation on closed manifolds. We also derive a new interpolated Harnack inequality for the equation on closed surfaces under the -Ricci flow. Finally we prove…
Efficiently solves high-dimensional ODEs with probabilistic methods.
Traditional models of macroeconomic dynamics are fundamentally incorrect. The reason lies in a misunderstanding of peculiarities of the analysis of infinitesimal quantities. However, even those types of solutions that are envisaged by the above-mentioned models are nonrepresentative in the sense of the reflection of re…
Study on Langevin dynamics for recovering planted signals in spiked matrix models.
Representations of coherent state Lie algebras on coherent state manifolds as first order differential operators are presented. The explicit expressions of the differential action of the generators of semisimple Lie groups determine for linear Hamiltonians in the generators of the groups first order differential equati…
New method finds efficient low-rank neural networks during training.
New bounds for private matrix approximation using Gaussian noise and Dyson Brownian Motion.
Develops a new non-abelian framework for Riemann surfaces and differential equations.
Particles representing tokens cluster in Transformers, influenced by initial tokens and matrix spectrum.
A non-commutative differential calculus on the -superplane is presented via a contraction of the -superplane. An R-matrix which satisfies both ungraded and graded Yang-Baxter equations is obtained and a new deformation of the dimensional classical phase space (the super-Heisenberg algebra) is introduced.
Gradients of neural networks can be computed efficiently for any architecture, but some applications require differential operators with higher time complexity. We describe a family of restricted neural network architectures that allow efficient computation of a family of differential operators involving dimension-wise…
Study on neural network initialization with shaped infinite depth-and-width networks.
We determine the homogeneous Kähler diffeomorphism which expresses the Kähler two-form on the Siegel-Jacobi ball $\mc{D}^J_n=\C^n\times \mc{D}_n$ as the sum of the Kähler two-form on $\C^n$ and the one on the Siegel ball $\mc{D}_n$. The classical motion and quantum evolution on $\mc{D}^J_n$ determined by a hermiti…
We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for Weingarten's formula, the Ricci curvature and the Codazzi-Mainardi equations. For ma…
We propose a new, unified approach to solving jump-diffusion partial integro-differential equations (PIDEs) that often appear in mathematical finance. Our method consists of the following steps. First, a second-order operator splitting on financial processes (diffusion and jumps) is applied to these PIDEs. To solve the…
In this paper we study the heat equation (of Hodge-Laplacian) deformation of -forms on a Kähler manifold. After identifying the condition and establishing that the positivity of a -form solution is preserved under such an invariant condition we prove the sharp differential Harnack (in the sense of Li-Ya…
A new method estimates SDEs using occupation kernels.
The paper analyzes convergence of Langevin dynamics with time-dependent metrics.
A new neural approach for generating origin-destination matrices in ABMs.
Paper establishes convergence rates for learning elliptic pseudo-differential operators.
A new graph neural network framework captures long-range interactions efficiently.
Consider the problem of learning the drift coefficient of a stochastic differential equation from a sample path. In this paper, we assume that the drift is parametrized by a high dimensional vector. We address the question of how long the system needs to be observed in order to learn this vector of parameters. We prove…
These lecture notes provide a self-contained introduction to the mathematical methods required in a Bachelor degree programme in Business, Economics, or Management. In particular, the topics covered comprise real-valued vector and matrix algebra, systems of linear algebraic equations, Leontief's stationary input-output…
CoLA automates efficient numerical linear algebra for complex matrix structures.
Study solves DREs for trading strategies using signals and past prices.
A new algorithm solves high-dimensional nonlinear BSDEs efficiently.
Utilizing a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and vo…
We give necessary and sufficient local conditions for the simultaneous unitarizability of a set of analytic matrix maps from an analytic 1-manifold into SL_n(C) under conjugation by a single analytic matrix map. We apply this result to the monodromy arising from an integrable partial differential equation to construct …
A regression algorithm uses Green's function and covariance matrix for predictive distributions.
HTE improves PINNs for high-dimensional, high-order PDEs by reducing computational cost and memory usage.
Efficiently values and computes sensitivities of Bermudan options using Method of Lines.
We prove that the Ricci flow equation for left invariant metrics on Lie groups reduces to a first order ordinary differential equation for a map , where is the group of upper triangular matrices. We decompose the matrix of Ricci tensor coordinates with respect to an orthonormal frame fi…
Study on SGD dynamics and scaling laws for training quadratic neural networks in high dimensions.
Efficiently differentiate functions of large matrices using new adjoint systems.
If a curve in R^3 is closed, then the curvature and the torsion are periodic functions satisfying some additional constraints. We show that these constraints can be naturally formulated in terms of the spectral problem for a 2x2 matrix differential operator. This operator arose in the theory of the self-focusing Nonlin…
Consider a formally self-adjoint first order linear differential operator acting on pairs (2-columns) of complex-valued scalar fields over a 4-manifold without boundary. We examine the geometric content of such an operator and show that it implicitly contains a Lorentzian metric, Pauli matrices, connection coefficients…
The Davey Stewartson hierarchy will be developed based on a set of three matrix differential operators. These equations will act as evolution equations for different types of surface deformation in Euclidean four space. The Weierstrass representation for surfaces will be developed and its uniqueness up to gauge transfo…
We consider stochastic partial differential equations appearing as Markovian lifts of matrix valued (affine) Volterra type processes from the point of view of the generalized Feller property (see e.g., \cite{doetei:10}). We introduce in particular Volterra Wishart processes with fractional kernels and values in the con…
Model liquidity premia using a risk-sharing economy with quadratic costs.
These lecture notes are concerned with the solvability of the second boundary value problem of the prescribed affine mean curvature equation and related regularity theory of the Monge-Ampère and linearized Monge-Ampère equations. The prescribed affine mean curvature equation is a fully nonlinear, fourth order, geometri…
We study the linearization of the Dirichlet-to-Neumann map for Poincaré-Einstein metrics in even dimensions on an arbitrary compact manifold with boundary. By fixing a suitable gauge, we make the linearized Einstein equation elliptic. In this gauge the linearization of the Dirichlet-to-Neumann map appears as the scatte…
Study differential properties of matrix square roots in specific cases.
New Transformer architecture prevents rank degeneracy in deep attention models.
Research proves unique continuation for Einstein-vacuum equations on aAdS spacetimes.
Study real logarithms of semi-simple matrices, focusing on differential structure.