arXiv research

The reduction theorems for general linear and classical connections are generalized for operators with values in higher order gauge-natural bundles. We prove that natural operators depending on the $s_1$-jets of classical connections, on the $s_2$-jets of general linear connections and on the $r$-jets of tensor fields …

We generalize reduction theorems for classical connections to operators with values in $k$-th order natural bundles. Using the first reduction theorem in order two we classify all (0,2)-tensor fields on the cotangent bundle of a manifold with a linear (non-symmetric) connection.

We lift ambit fields as introduced by Barndorff-Nielsen and Schmiegel to a class of Hilbert space-valued volatility modulated Volterra processes. We name this class Hambit fields, and show that they can be expressed as a countable sum of weighted real-valued volatility modulated Volterra processes. Moreover, Hambit fie…

In this paper we study a continuous time equilibrium model of limit order book (LOB) in which the liquidity dynamics follows a non-local, reflected mean-field stochastic differential equation (SDE) with evolving intensity. Generalizing the basic idea of Ma et al. (2015), we argue that the frontier of the LOB (e.g., the…

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…

Theory developed for Hilbert geometry over valued fields, linking real and non-Archimedean geometries.

The perturbative approach to nonlinear Sigma models and the associated renormalization group flow are discussed within the framework of Euclidean algebraic quantum field theory and of the principle of general local covariance. In particular we show in an Euclidean setting how to define Wick ordered powers of the underl…

Paper develops a new method for solving IBVPs on star-shaped domains.

Generalizes beam models to include curvature and torsion.

In this note, we consider a fixed vector field $V$ on $S^2$ and study the distribution of points which lie on the nodal set (of a random spherical harmonic) where $V$ is also tangent. We show that the expected value of the corresponding counting function is asymptotic to the eigenvalue with a leading coefficient that i…

We develop a systematic method for renormalizing the AdS/CFT prescription for computing correlation functions. This involves regularizing the bulk on-shell supergravity action in a covariant way, computing all divergences, adding counterterms to cancel them and then removing the regulator. We explicitly work out the ca…

Unified field theory from higher-order Riemannian geometry.

New method for mesh denoising using TGV of normal vector field.

Proposes a normalization technique for manifold valued data.

High order discretization schemes of SDEs by using free Lie algebra valued random variables are introduced by Kusuoka, Lyons-Victoir, Ninomiya-Victoir and Ninomiya-Ninomiya. These schemes are called KLNV methods. They involve solving the flows of vector fields associated with SDEs and it is usually done by numerical me…

A framework for quantifying uncertainty in feature importance values.

The paper deals with a formally self-adjoint first order linear differential operator acting on m-columns of complex-valued half-densities over an n-manifold without boundary. We study the distribution of eigenvalues in the elliptic setting and the propagator in the hyperbolic setting, deriving two-term asymptotic form…

The paper explores how duality applies to reinforcement learning.

A Lie system is a system of first-order ordinary differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields: a so-called Vessiot-Guldberg Lie algebra. We suggest the definition of a particular class of Lie systems, the $k…

Study of Dirac fields on Kerr spacetimes using peeling method.

Contact Lie systems analyze integral curves of Hamiltonian vector fields.

Study on measure-valued CARMA processes in Banach spaces.

Study Transformer layers under cross-entropy training using mean field control.

Study improves $L^{\infty}$ estimates and extreme value behavior in stochastic differential games.

The differential system for minimal Lagrangian surfaces in a $2_{\mathbb{C}}$-dimensional, non-flat, complex space form is an elliptic system defined on the bundle of oriented Lagrangian planes. This is a 6-symmetric space associated with the Lie group SL(3,$\mathbb{C}$), and the minimal Lagrangian surfaces arise as th…

Let M be a manifold endowed with a symmetric affine connection $Γ.$ The aim of this paper is to describe a quantization map between the space of second-order polynomials on the cotangent bundle T^{*} M and the space of second-order linear differential operators, both viewed as modules over the group of diffeomorphisms …

Modeling price impacts and trading signals for optimal execution and speculation.

We are going to use the Euler's vector fields in order to show that for real quasi-homogeneous singularities with isolated critical value, the Milnor's fibration in a "thin" hollowed tube involving the zero level and the fibration in the complement of "link" in sphere are equivalents, since they exist. Moreover, in ord…

A new sampling scheme based on DOE improves Shapley value estimation accuracy and speed.

Infinitesimal variation of Action functional in classical (non-quantum) field theory with higher derivatives is presented in terms of well-defined intrinsic geometric objects independent of the particular field which varies. 'Integration by parts' procedure for this variation is then described in purely formal language…

Game theory models how agents trade in a risky asset considering price impact and a common signal.

The paper solves complex control problems using neural networks.

New construction reveals SU(2)-flavor fields in heterotic M5-brane model.

The Higgs field growth is studied on special geometric spaces, confirming a conjecture.

New vector fields integrate first-order ODEs.

We prove the existence and uniqueness of a *projectively equivariant symbol map*, which is an isomorphism between the space of bidifferential operators acting on tensor densities over $R^n$ and that of their symbols, when both are considered as modules over an imbedding of $sl(n+1,\R)$ into polynomial vector fields. Th…

Develops higher-order Euler-Poincaré field equations for principal G-bundles.

We continue the study of a general class of spaces of 0-cycles on a manifold defined and begun by Farb-Wolfson-Wood. Using work of Gadish on linear subspace arrangements, we obtain representation stability for the cohomology of the ordered version of these spaces. We establish subexponential bounds on the growth of uns…

An accurate assessment of the risk of extreme environmental events is of great importance for populations, authorities and the banking/insurance/reinsurance industry. Koch (2017) introduced a notion of spatial risk measure and a corresponding set of axioms which are well suited to analyze the risk due to events having …

We show injectivity of the X-ray transform and the $d$-plane Radon transform for distributions on the $n$-torus, lowering the regularity assumption in the recent work by Abouelaz and Rouvière. We also show solenoidal injectivity of the X-ray transform on the $n$-torus for tensor fields of any order, allowing the tensor…

Paper uses autoencoders for efficient reduced-order modeling of eigenvalue problems.

Geometric structures on $\mathbb N Q$-manifolds, i.e.~non-negatively graded manifolds with an homological vector field, encode non-graded geometric data on Lie algebroids and their higher analogues. A particularly relevant class of structures consists of vector bundle valued differential forms. Symplectic forms, contac…

We develop a multivalued theory for the stability operator of (a constant multiple of) a minimally immersed submanifold $Σ$ of a Riemannian manifold $\mathcal{M}$. We define the multiple valued counterpart of the classical Jacobi fields as the minimizers of the second variation functional defined on a Sobolev space of …

Constructs unique bases for CY varieties over valued fields.

A first-order formulation of gravity is developed in which the fundamental fields consist of an SL(2,C) connection and two spinor-valued 1-forms. It is shown that the first term of an expansion of the Einstein-Hilbert action leads to an action for these fields which consists of dynamic L2 inner products of their covari…

Study stabilizes second-order systems to first-order dynamics.

This paper presents a geometric-variational approach to continuous and discrete {\it second-order} field theories following the methodology of \cite{MPS}. Staying entirely in the Lagrangian framework and letting $Y$ denote the configuration fiber bundle, we show that both the multisymplectic structure on $J^3Y$ as well…

Generalizes Carathéodory form for higher-order field theories.