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.
A machine learning model captures non-Newtonian fluid dynamics from molecular details.
Class of Newtonian dynamical systems admitting normal blow-up of points in Riemannian manifolds is considered. Geometric interpretation for weak normality condition, which arose earlier in the theory of dynamical systems admitting the normal shift of hypersurfaces, is found.
Newtonian, Lagrangian, and Hamiltonian dynamical systems are well formalized mathematically. They give rise to geometric structures describing motion of a point in smooth manifolds. Riemannian metric is a different geometric structure formalizing concepts of length and angle. The interplay of Riemannian metric and its …
Explicit description for arbitrary Newtonian dynamical system admitting the normal shift in Riemannian manifold of the dimension is found. On the base of this result the kinematics of normal shift of hypersurfaces along trajectories of such system is studied.
Normality equations describe Newtonian dynamical systems admitting normal shift of hypersurfaces. They were first derived in Euclidean geometry, then in Riemannian geometry. Recently they were rederived in more general case, when geometry of manifold is given by generalized Legendre transformation. As appears, in this …
The system of weak normality equations constitutes a part in the complete system of normality equations. Solutions of each of these two systems of equations are associated with some definite classes of Newtonian dynamical systems in Riemannian manifolds. In this paper for the case of simplest flat Riemannian manifold $…
Normality equations describe Newtonian dynamical systems admitting normal shift of hypersurfaces. These equations were first derived in Euclidean geometry. Then very soon they were rederived in Riemannian and in Finslerian geometry. Recently I have found that normality equations can be derived in geometry given by clas…
Formula for the force field of Newtonian dynamical systems admitting the normal shift of hypersurfaces in Riemannian manifolds is considered. Problem of globalization for geometric structures associated with this formula is studied.
Newtonian dynamical systems which accept the normal shift on an arbitrary Riemannian manifold are considered. For them the determinating equations making the weak normality condition are derived. The expansion for the algebra of tensor fields is constructed.
Study periodic solutions in N-body problem, revealing braids with complex dynamics.
Dyna optimizes momentum for stochastic optimization of neural networks.
We study the notion of -quasihomotopy in Newtonian classes of mappings and link it to questions concerning lifts of Newtonian maps, under the assumption that the target space is nonpositively curved. Using this connection we prove that every -quasihomotopy class of Newtonian maps contains a minimizer of the -e…
Examines quantum mechanics equivalence with Newtonian geometry.
This thesis discusses the Newtonian limit of General Relativity for static isolated systems with compactly supported matter. We call these systems "geometrostatic" to underline their geometric nature. We introduce new quasi-local notions of mass and center of mass that can be read off locally in the vicinity of the mat…
Theory of Newtonian dynamical systems admitting normal shift of hypersurfaces was first developed for the case of Riemannian manifolds. Recently it was generalized for manifolds geometric equipment of which is given by some regular Lagrangian or, equivalently, by some regular Hamiltonian dynamical system. In present pa…
Reviewing Newtonian mechanics in curved spaces.
We show that any bounded zero-angular momentum solution for the Newtonian three-body problem must suffer infinitely many eclipses, or collinearities, provided that it does not suffer a triple collision. Motivation for the result comes from the dream of building a symbolic dynamics for the three-body problem, one whose …
This text explains how fiber bundle structure is fundamental for classical physics.
We present a new geometric approach to the study of static isolated general relativistic systems for which we suggest the name geometrostatics. After describing the setup, we introduce localized formulas for the ADM-mass and ADM/CMC-center of mass of geometrostatic systems. We then explain the pseudo-Newtonian characte…
A frame independent formulation of analytical mechanics in the Newtonian space-time is presented. The differential geometry of affine values i.e., the differential geometry in which affine bundles replace vector bundles and sections of one dimensional affine bundles replace functions on manifolds, is used. Lagrangian a…
Symmetric Poisson structures linked to geodesic foliations and Jordan algebras.
The Schroedinger operator on the Newtonian space-time is defined in a way which is independent on the class of inertial observers. In this picture the Schroedinger operator acts not on functions on the space-time but on sections of certain one-dimensional complex vector bundle over space-time. This bundle, constructed …
The following three geometrical structures on a manifold are studied in detail: (1) Leibnizian: a non-vanishing 1-form plus a Riemannian metric $\h$ on its annhilator vector bundle. In particular, the possible dimensions of the automorphism group of a Leibnizian G-structure are characterized. (2) Galilean: Leibnizi…
High frequency limit for most of wave phenomena is known as quasiclassical limit or ray optics limit. Propagation of waves in this limit is described in terms of wave fronts and rays. Wave front is a surface of constant phase whose points are moving along rays. As it appears, their motion can be described by Hamilton e…
Study of timelike surfaces with time-minimizing rulings in Newtonian and relativistic spacetimes.
The center of mass in General Relativity is hard to define due to coordinate freedom.
The existence of K-instantons on a cylinder M^7 = R_tau x K/H over a homogeneous nearly K"ahler 6-manifold K/H requires a conformally parallel or a cocalibrated G_2-structure on M^7. The generalized anti-self-duality on M^7 implies a Chern-Simons flow on K/H which runs between instantons on the coset. For K-equivariant…
Enhances data-driven models with physics knowledge for better system dynamics.
Higher-order geometry modifies Newtonian dynamics and predicts anomalies in spacecraft motion.
This monograph describes a Riemannian geometric reduction approach to the three-body problem. The fundamental theorems are presented in the introductory part, whereas their proofs are provided in later chapters where specific topics are analyzed in more detail. The basic idea is to reduce the kinematic and dynamics of …
The Galilean group is the group of symmetries of Newtonian mechanics, with Lie a lgebra $\gal(n)$. We find algebraically independent generators for the center of the universal enveloping algebra of $\gal(n)$ using coadjoint orbits.
Main ideas of the differential geometry on affine bundles are presented. Affine counterparts of Lie algebroid and Poisson structures are introduced and discussed. The developed concepts are applied in a frame-independent formulation of the time-dependent and the Newtonian mechanics.
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…
New algorithm estimates partially-observed linear systems with better rates than previous methods.
The paper explores scaling symmetries in symplectic geometry and their applications to central configurations.
In accessibility tests for digital preservation, over time we experience drifts of localized and labelled content in statistical models of evolving semantics represented as a vector field. This articulates the need to detect, measure, interpret and model outcomes of knowledge dynamics. To this end we employ a high-perf…
In 1883, as an early result, Sophus Lie established an explicit necessary and sufficient condition for an analytic second order ordinary differential equation y_xx = F(x,y,y_x) to be equivalent, through a point transformation (x,y) --> (X(x,y), Y(x,y)), to the Newtonian free particle equation Y_XX = 0. This result, pre…
The Schroedinger operators on the Newtonian space-time are defined in a way which make them independent on the class of inertial observers. In this picture the Schroedinger operators act not on functions on the space-time but on sections of certain one-dimensional complex vector bundle -- the Schroedinger line bundle. …
In this paper we present a new theory of calculus over -dimensional domains in a smooth -manifold, unifying the discrete, exterior, and continuum theories. The calculus begins at a single point and is extended to chains of finitely many points by linearity, or superposition. It converges to the smooth continuum w…
NeuralMD accelerates protein-ligand binding simulations 1Kx faster.
This study compares different thermodynamic structure-informed neural networks for solving differential equations.
We suggest an alternative mathematical model for the massless neutrino. Consider an elastic continuum in 3-dimensional Euclidean space and assume that points of this continuum can experience no displacements, only rotations. This framework is a special case of the so-called Cosserat theory of elasticity. Rotations of p…
We perform an optimal localization of asymptotically flat initial data sets and construct data that have positive ADM mass but are exactly trivial outside a cone of arbitrarily small aperture. The gluing scheme that we develop allows to produce a new class of -body solutions for the Einstein equation, which patently…
Physicists believe, with some justification, that there should be a correspondence between familiar properties of Newtonian gravity and properties of solutions of the Einstein equations. The Positive Mass Theorem (PMT), first proved over twenty years ago \cite{SchoenYau79b,Witten81}, is a remarkable testament to this f…
This paper extends Lie algebra contractions to infinite-dimensional spaces for better understanding of group limits.
Study rectifying curves in 3D multiplicative Euclidean space.
A general, consistent and complete framework for geometrical formulation of mechanical systems is proposed, based on certain structures on affine bundles (affgebroids) that generalize Lie algebras and Lie algebroids. This scheme covers and unifies various geometrical approaches to mechanics in the Lagrangian and Hamilt…
The paper deals with the Weyl equation which is the massless Dirac equation. We study the Weyl equation in the stationary setting, i.e. when the spinor field oscillates harmonically in time. We suggest a new geometric interpretation of the stationary Weyl equation, one which does not require the use of spinors, Pauli m…