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This thesis proposes a combinatorial generalization of a nilpotent operator on a vector space. The resulting object is highly natural, with basic connections to a variety of fields in pure mathematics, engineering, and the sciences. For the…
Contemporary materials science research is heavily conducted in silico, involving massive simulations of the atomic-scale evolution of materials. Cataloging basic patterns in the atomic displacements is key to understanding and predicting…
A new computational technique based on the symbolic description utilizing kneading invariants is proposed and verified for explorations of dynamical and parametric chaos in a few exemplary systems with the Lorenz attractor. The technique…
For a theory with first and second class constraints, we propose a procedure for conversion of second class constraints based on deformation the structure of local symmetries of the Lagrangian formulation. It does not require extension or…
Physical theories grounded in mathematical symmetries are an essential component of our understanding of a wide range of properties of the universe. Similarly, in the domain of machine learning, an awareness of symmetries such as rotation…
Building accurate and predictive models of the underlying mechanisms of celestial motion has inspired fundamental developments in theoretical physics. Candidate theories seek to explain observations and predict future positions of planets,…
This work introduces a new set of orbital elements to fully represent the zonal harmonics problem around an oblate celestial body. This new set of orbital elements allows to obtain a complete linear system for the unperturbed problem and,…
In this paper we study group actions on hyperbolic $\Lambda$-metric spaces, where $\Lambda$ is an ordered abelian group. $\Lambda$-metric spaces were first introduced by Morgan and Shalen in their study of hyperbolic structures and then…
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…
New features of a previously introduced Group Approach to Quantization are presented. We show that the construction of the symmetry group associated with the system to be quantized (the "quantizing group") does not require, in general, the…
In this paper, a new parametrization of the relative motion between two satellites orbiting a central body is presented. The parametrization is based on the nodal elements: a set of angles describing the orbit geometry with respect to the…
We consider Smale spaces, a particular class of hyperbolic topological dynamical systems, which include the basic sets for Smale's Axiom A systems. We present a homology theory for such systems which is based on the dimension group in the…
Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomorphisms, a 'heteromorphism' which connects polyadic…
We describe a new method to formulate classical Lagrangian mechanics on a finite-dimensional Lie group. This new approach is much more pedagogical than many previous treatments of the subject, and it directly introduces students to…
In this paper we study the notion of configuration for group actions. It is proved that some properties concerning configuration of groups can be extended for the case of group actions. The relationship between configuration and different…
The present paper derives the post-Newtonian Lagrangian of translational motion of N arbitrary-structured bodies with all mass and spin multipoles in a scalar-tensor theory of gravity. The multipoles depend on time and evolve in accordance…
By engineering laser-atom interactions, both Hall ribbons and Hall cylinders as fundamental theoretical tools in condensed matter physics have recently been synthesized in laboratories. Here, we show that turning a synthetic Hall ribbon…
Several problems in number theory when reformulated in terms of homogenous dynamics involve study of limiting distributions of translates of algebraically defined measures on orbits of reductive groups. The general non-divergence and…
We present an application of a recently introduced variant of orbit space reduction for symmetric dynamical systems. This variant works with suitable localizations of the algebra of polynomial invariants of the group actions, and provides…
We review the observations and the basic laws describing the essential aspects of collective motion -- being one of the most common and spectacular manifestation of coordinated behavior. Our aim is to provide a balanced discussion of the…