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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…

In this thesis, we study the one parameter point transformations which leave invariant the differential equations. In particular we study the Lie and the Noether point symmetries of second order differential equations. We establish a new…

We study the classical and quantum oscillator in the context of a non-additive (deformed) displacement operator, associated with a position-dependent effective mass, by means of the supersymmetric formalism. From the supersymmetric partner…

It is known that there is no finitely presented group for which the Dehn function lies asymptotically strictly between linear and quadratic functions. This work presents an example of a semigroup that has Dehn function equivalent to $n \log…

We study the Lie point symmetries of Einstein's equations for the Friedmann-Roberstson-Walker Cosmology. They form either a two - dimensional or a three - dimensional solvable group depending on the form of the self interacting potential.…

A natural and very important development of constrained system theory is a detail study of the relation between the constraint structure in the Hamiltonian formulation with specific features of the theory in the Lagrangian formulation,…

Geometric mechanics models of locomotion have provided insight into how robots and animals use environmental interactions to convert internal shape changes into displacement through the world, encoding this relationship in a ``motility…

We study deterministic and quantum dynamics from a constructive "finite" point of view, since the introduction of a continuum, or other actual infinities in physics poses serious conceptual and technical difficulties, without any need for…

Dynamical similarities are non-standard symmetries found in a wide range of physical systems that identify solutions related by a change of scale. In this paper we will show through a series of examples how this symmetry extends to the…

This article presents an overview of the theory of integrable systems with symmetries, focusing on toric systems, semitoric systems, and their classifications via decorated polygons. We discuss certain one-parameter families of integrable…

Cooperative systems are systems in which the forces among agents are non-repulsive. The free evolution of such systems can tend to the formation of patterns, such as consensus or clustering, depending on the properties and intensity of the…

Lie derivatives of various geometrical and physical quantities define symmetries and conformal symmetries in general relativity. Thus we obtain motions, collineations, conformal motions and conformal collineations. These symmetries are used…

A symmetric function of $N$ variables can be given in terms of symmetric polynomials of these variables. We determine those symmetric polynomials in which the dual differential operators take the neatest form when expressed in terms of our…

Using semi-tensor product of matrices, the structures of several kinds of symmetric games are investigated via the linear representation of symmetric group in the structure vector of games as its representation space. First of all, the…

Nonholonomic systems are variational models commonly used for mechanical systems with ideal no-slip constraints. This note provides a differential-geometric derivation of the nonholonomic equations of motion for an arbitrary rigid body…

We split the generic conformal mechanical system into a "radial" and an "angular" part, where the latter is defined as the Hamiltonian system on the orbit of the conformal group, with the Casimir function in the role of the Hamiltonian. We…

We explore topological manipulations in one spatial dimension, which are defined for a system with a global symmetry and map the system to another one with a dual symmetry. In particular, we discuss fusion category symmetries enhanced by…

A finite-dimensional pseudo-unitary framework is set up for describing the dynamics of free elementary particles in a purely relativistic quantum mechanical way. States of any individual particles or antiparticles are defined as suitably…

Associated to classical semi-simple groups and their maximal parabolics are genuine zeta functions. Naturally related to Riemann's zeta and governed by symmetries, including that of Weyl, these zetas are expected to satisfy the Riemann…

This paper is devoted to the study of mechanical systems subjected to external forces in the framework of symplectic geometry. We obtain a Noether's theorem for Lagrangian systems with external forces, among other results regarding…