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.
Inverse problem solved for relativistic Boltzmann equation on spacetime.
A variational equation of the fourth order for the free relativistic top is developed starting from the Dixon's system of equations for the motion of the relativistic dipole. The obtained equation is then cast into the homogeneous space-time Hamiltonian form.
Black-Scholes equation, after a certain coordinate transformation, is equivalent to the heat equation. On the other hand the relativistic extension of the latter, the telegraphers equation, can be derived from the Euclidean version of the Dirac equation. Therefore the relativistic extension of the Black-Scholes model f…
Using a simple and well-motivated modification of the stress-energy tensor for a viscous fluid proposed by Lichnerowicz, we prove that Einstein's equations coupled to a relativistic version of the Navier-Stokes equations are well-posed in a suitable Gevrey class if the fluid is incompressible and irrotational. These la…
The paper decomposes spacelike hypersurface properties for general relativistic vacuum equations.
We study the problem of coupling Einstein's equations to a relativistic and physically well-motivated version of the Navier-Stokes equations. Under a natural evolution condition for the vorticity, we prove existence and uniqueness in a suitable Gevrey class if the fluid is incompressible, where this condition is given …
Mathisson's 'new mechanics' of a relativistic spinning particle is shown to follow, in the case of planar motion, from only general requirements of relativistic invariance and of the dependence on third order derivatives along with the 'variationality' feature. The Hamiltonian counterpart ultimately recovers the Dixon …
In this article we consider the motion of relativistic strings in the Minkowski space . Those surfaces are known as a timelike minimal surface, and described by a system with nonlinear wave equations of Born-Infeld type. By constructing a suitable Nash-Moser iteration scheme, we prove that the …
This text explains how fiber bundle structure is fundamental for classical physics.
The structure of a diffeomorphism invariant Lagrangians for an extended object W embedded in a bulk space M is discussed by following a close analogy with the relativistic particle in electromagnetic field as a system that is reparametrization-invariant. The current construction naturally contains, relativistic point p…
Local gauge freedom in relativistic quantum mechanics is derived from a measurement principle for space and time. For the Dirac equation, one obtains local U(2,2) gauge transformations acting on the spinor index of the wave functions. This local U(2,2) symmetry allows a unified description of electrodynamics and genera…
We obtain basic estimates for a Monge-Ampère equation introduced by Moncrief in the study of the Relativistic Teichmüller Theory. We then give another proof of the parametrization of the Teichmüller space obtained by Moncrief. Our approach provides yet another proof of the classical Teichmüller theorem that the Teichmü…
We develop a suitable generalization of Almgren's theory of varifolds in a lorentzian setting, focusing on area, first variation, rectifiability, compactness and closure issues. Motivated by the asymptotic behaviour of the scaled hyperbolic Ginzburg-Landau equations, and by the presence of singularities in lorentzian m…
In recent work, Boltzmann and Fokker-Planck equations were derived for the "Yard-Sale Model" of asset exchange. For the version of the model without redistribution, it was conjectured, based on numerical evidence, that the time-asymptotic state of the model was oligarchy -- complete concentration of wealth by a single …
The application of the Legendre transformation to a hyperregular Lagrangian system results in a Hamiltonian vector field generated by a Hamiltonian defined on the phase space of the mechanical system. The Legendre transformation in its usual interpretation can not be applied to homogeneous Lagrangians found in relativi…
Geometrically describes pseudo-gauge freedom in relativistic hydrodynamics.
In 1870s, L. Boltzmann proved the famous -theorem for the Boltzmann equation in the kinetic theory of gas and gave the statistical interpretation of the thermodynamic entropy. In 2002, G. Perelman introduced the notion of -entropy and proved the -entropy formula for the Ricci flow. This plays a crucial role in…
Employing the Klein-Gordon equation, we propose a generalized Black-Scholes equation. In addition, we found a limit where this generalized equation is invariant under conformal transformations, in particular invariant under scale transformations. In this limit, we show that the stock prices distribution is given by a C…
Starting from a relativistic phenomenology of anyons in crystals, we discuss the concept of relativistic interaction and the need to unify electromagnetism and gravitation within the Spencer cohomology of Lie equations. Then, from the sophisticated non-linear Spencer complex of the Poincaré and conformal Lie pseudogrou…
Some intrinsic tools from the formal theory of variational equations are being demonstrated at work in application to one concrete example of the third-order evolution equation of free relativistic top in three-dimensional space-time. The main goal is to introduce a combined approach consisting in the simultaneous util…
In this paper we describe the geometry of distributions by their symmetries, and present a simplified proof of the Frobenius theorem and some related corollaries. Then, we study the geometry of solutions of Gordon equation; A PDE which appears in differential geometry and relativistic field theory.
A new training method for normalizing flows without samples.
We briefly review the current situation with various relations between knot/braid polynomials (Chern-Simons correlation functions), ordinary and extended, considered as functions of the representation and of the knot topology. These include linear skein relations, quadratic Plucker relations, as well as "differential" …
We briefly review results on nonlinear kinetic equation of Boltzmann type which describe the evolution of wealth in a simple agents market. The mathematical structure of the underlying kinetic equations allows to use well-known techniques of wide use in kinetic theory of rarefied gases to obtain information on the proc…
In this paper we introduce kinetic equations for the evolution of the probability distribution of two goods among a huge population of agents. The leading idea is to describe the trading of these goods by means of some fundamental rules in price theory, in particular by using Cobb-Douglas utility functions for the bina…
We study scale invariant but not necessarily conformal invariant deformations of non-relativistic conformal field theories from the dual gravity viewpoint. We present the corresponding metric that solves the Einstein equation coupled with a massive vector field. We find that, within the class of metric we study, when w…
Drawing insights from the triumph of relativistic over classical mechanics when velocities approach the speed of light, we explore a similar improvement to the seminal Black-Scholes (Black and Scholes (1973)) option pricing formula by considering a relativist version of it, and then finding a respective solution. We sh…
In this paper we propose a geometrization of the non-relativistic quantum mechanics for mixed states. Our geometric approach makes use of the Uhlmann's principal fibre bundle to describe the space of mixed states and as a novelty tool, to define a dynamic-dependent metric tensor on the principal manifold, such that the…
Maximally hyperbolic solutions contain future neighborhoods of intersecting hypersurfaces.
The Ahlfors Laplacian is applied to solve geometric and relativistic problems.
In this thesis, we consider the suitability of using the charged cold fluid model in the description of ultra-relativistic beams. The method that we have used is the following. Firstly, the necessary notions of kinetic theory and differential geometry of second order differential equations are explained. Then an averag…
EDG generates Boltzmann samples from latent variables efficiently.
We describe a procedure naturally associating relativistic Klein-Gordon equations in static curved spacetimes to non-relativistic quantum motion on curved spaces in the presence of a potential. Our procedure is particularly attractive in application to (typically, superintegrable) problems whose energy spectrum is give…
The work focuses upon the relativistic and geometric properties of the space--time endowed tentatively with the metric function of the Berwald--Moor type. The zero curvature of indicatrix is a remarkable property of the approach. We demonstrate how the associated geodesic equations can be solved in a transparent way, t…
We provide a Hilbert manifold structure {à} la Bartnik for the space of asymptotically hyperbolic initial data for the vacuum constraint equations. The adaptation led us to prove new weighted Poincar{é} and Korn type inequalities for AH manifolds with inner boundary and weakly regular metric.
A new technique for the study of geodesic connectedness in a class of Lorentzian manifolds is introduced. It is based on arguments of Brouwer's topological degree for the solution of functional equations. It is shown to be very useful for multiwarped spacetimes, which include different types of relativistic spacetimes.
Paper explores Monge-Ampère in deep learning and quantum geometry.
The relativistic quantum mechanic approach is used to develop a stock market dynamics. The relativistic is conceptional here as the meaning of big external volatility or volatility shock on a financial market. We used a differential geometry approach with the parallel transport of the prices to obtain a direct shift of…
Paper uses queue theory to model financial signals with relativistic delay.
Using the one dimensional free particle symmetries, the quantum finance symmetries are obtained. Namely, it is shown that Black-Scholes equation is invariant under Schrödinger group. In order to do this, the one dimensional free non-relativistic particle and its symmetries are revisited. To get the Black-Scholes equati…
We propose a modification of the cost function of the Hopfield model whose salient features shine in its Taylor expansion and result in more than pairwise interactions with alternate signs, suggesting a unified framework for handling both with deep learning and network pruning. In our analysis, we heavily rely on the H…
The variational principle and the corresponding differential equation for geodesic circles in two dimensional (pseudo)-Riemannian space are being discovered. The relationship with the physical notion of uniformly accelerated relativistic particle is emphasized. The known form of spin-curvature interaction emerges due t…
Within the description of stochastic differential equations it is argued that the existence of Boltzmann-Gibbs type distribution in economy is independent of the time reversal symmetry in econodynamics. Both power law and exponential distributions can be accommodated by it. The demonstration is based on a mathematical …
Nonnegative Boltzmann machines (NNBMs) are recurrent probabilistic neural network models that can describe multi-modal nonnegative data. NNBMs form rectified Gaussian distributions that appear in biological neural network models, positive matrix factorization, nonnegative matrix factorization, and so on. In this paper,…
Geometric framework for Newton's equations on diffeomorphism groups.
Hamiltonian Monte Carlo (HMC) is a popular Markov chain Monte Carlo (MCMC) algorithm that generates proposals for a Metropolis-Hastings algorithm by simulating the dynamics of a Hamiltonian system. However, HMC is sensitive to large time discretizations and performs poorly if there is a mismatch between the spatial geo…
The recent book by T. Piketty (Capital in the Twenty-First Century) promoted the important issue of wealth inequality. In the last twenty years, physicists and mathematicians developed models to derive the wealth distribution using discrete and continuous stochastic processes (random exchange models) as well as related…
Arguably, the two most popular accelerated or momentum-based optimization methods in machine learning are Nesterov's accelerated gradient and Polyaks's heavy ball, both corresponding to different discretizations of a particular second order differential equation with friction. Such connections with continuous-time dyna…