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Close insight into mathematical and conceptual structure of classical field theories shows serious inconsistencies in their common basis. In other words, we claim in this work to have come across two severe mathematical blunders in the very…
The analysis of contours of scalar fields plays an important role in visualization. For example the contour tree and contour statistics can be used as a means for interaction and filtering or as signatures. In the context of tensor field…
We develop a methodology to identify finite-time Lagrangian structures from data and models using an extension of the Koopman operator-theoretic methods developed for velocity fields with simple (periodic, quasi-periodic) time-dependence.…
We study the geometries generated by two-dimensional causal dynamical triangulations (CDT) coupled to $d$ massless scalar fields. Using methods similar to those used to study four-dimensional CDT we show that there exists a $c=1$ "barrier",…
This paper is devoted to the problem of classification, up to smooth isomorphisms or up to orbital equivalence, of smooth integrable vector fields on 2-dimensional surfaces, under some nondegeneracy conditions. The main continuous…
The Lagrangian description of mechanical systems and the Legendre Transformation (considered as a passage from the Lagrangian to the Hamiltonian formulation of the dynamics) for point-like objects, for which the infinitesimal configuration…
We consider a nonlinear control system with vector-valued measures as controls and with dynamics depending on time delayed states. First, we introduce a notion of discontinuous, bounded variation solution associated with this system and…
The main result is the identification of the orthogonal complement of the subalgebra of conformal vector field inside the algebra of all vector fields of a compact flat 2-manifold. As a fundamental tool, the complete Hodge decomposition for…
Signal processing and machine learning algorithms for data supported over graphs, require the knowledge of the graph topology. Unless this information is given by the physics of the problem (e.g., water supply networks, power grids), the…
The aim of the paper is to understand the local forms of conformal vector fields in the neighborhood of a singularity. We begin a general study in this direction, for any pseudo-Riemannian type, and give a complete answer in the Riemannian…
In this paper, we investigate vector fields on polyhedral complexes and their associated trajectories. We study vector fields which are analogue of the gradient vector field of a function in the smooth case. Our goal is to define a nice…
Traffic prediction has drawn increasing attention in AI research field due to the increasing availability of large-scale traffic data and its importance in the real world. For example, an accurate taxi demand prediction can assist taxi…
A new technique is proposed to classify a topological field in abelian lattice gauge theories. We perform the classification by regarding the topological field as a local composite field of the gauge field tensor instead of the vector…
We define the concepts of topological particles and topological radiation. These are nothing more than connected components of defects of a vector field. To each topological particle we assign an index which is an integer which is conserved…
We introduce topological methods for quantifying spatially heterogeneous dynamics, and use these tools to analyze particle-tracking data for a quasi-two-dimensional granular system of air-fluidized beads on approach to jamming. In…
Standard two-dimensional orientation-field based phase-field models rely on a continuous scalar field to represent crystallographic orientation. The corresponding order parameter space is the unit circle, which is not simply-connected. This…
We study the spatiotemporal dynamics, in one and two spatial dimensions, of two complex fields which are the two components of a vector field satisfying a vector form of the complex Ginzburg-Landau equation. We find synchronization and…
This paper proposes a flexible framework for inferring large-scale time-varying and time-lagged correlation networks from multivariate or high-dimensional non-stationary time series with piecewise smooth trends. Built on a novel and unified…
This paper analyzes the Lamb vector divergence, also called the hydrodynamic charge density, and its implications to the Navier-Stokes system. It is shown that the pressure field can be always chosen in a way that ensures regularity of the…
Four-dimensional CDT (causal dynamical triangulations) is a lattice theory of geometries which one might use in an attempt to define quantum gravity non-perturbatively, following the standard procedures of lattice field theory. Being a…