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In physics, two systems that radically differ at short scales can exhibit strikingly similar macroscopic behaviour: they are part of the same long-distance universality class. Here we apply this viewpoint to geometry and initiate a program…
In the recent past, binary similarity measures have been applied in solving biometric identification problems, including fingerprint, handwritten character detection, and in iris image recognition. The application of the relevant…
We study ray optics in the context of double mirror systems, in the limit as the two mirrors approach one another (thin films). This leads to a novel set of differential equations on a mirror surface which have interesting structure as seen…
We present a one-to-one correspondence between equivalence classes of embeddings of a manifold (into a larger manifold of the same dimension) and equivalence classes of certain distances on the manifold. This correspondence allows us to use…
The paper investigates a generalization of the classical Sitnikov problem, concentrating on the movement of a satellite along the Z-axis as it interacts with $n$ primary bodies in periodic motion. It establishes the existence of an infinite…
Many systems of interest to control engineering can be modeled by linear complementarity problems. We introduce a new notion of equivalence between linear complementarity problems that sets the basis to translate the powerful tools of…
We study conformal mappings in the Grushin plane and provide a number of their characterizations in terms of the Sobolev mappings and their geometry. Furthermore, we connect conformality on the Grushin plane with conformality on the complex…
Relativity is an integral part of positioning systems, and this is taken into account in today's practice by applying many "relativistic corrections" to computations performed using concepts borrowed from Galilean physics. A different,…
Using the properties of the angular momentum, we develop a new geometrical technique to study relative equilibria for a system of $3$--bodies with positive masses, moving on the two sphere under the influence of an attractive potential…
The intention of this article is to illustrate the use of methods from symplectic geometry for practical purposes. Our intended audience is scientists interested in orbits of Hamiltonian systems (e.g. the three-body problem). The main…
Using isoperimetry we obtain new symmetrization inequalities that allow us to provide a unified framework to study Sobolev inequalities in metric spaces. The applications include concentration inequalities, as well as metric versions of the…
We propose a class of semimetrics for preference relations any one of which is an alternative to the classical Kemeny-Snell-Bogart metric. (We take a fairly general viewpoint about what constitutes a preference relation, allowing for any…
Analytical methods are used to prove the existence of a periodic, symmetric solution with singularities in the planar 4-body problem. A numerical calculation and simulation are used to generate the orbit. The analytical method easily…
This work focuses on the generation of non-self-intersecting relative trajectories, and their applications to satellite constellation design, slotting architectures, and Space Traffic Management. To that end, this paper introduces two…
In this paper we are using the poset representation to describe the complex answers given by IR systems after a clustering and ranking processes. The answers considered may be given by cartographical representations or by thematic sub-lists…
Three-body systems of scalar bosons are described in the framework of relativistic constraint dynamics. With help of a change of variables followed by a change of wave function, two redundant degrees of freedom get eliminated and the…
Central configurations are fundamental equilibrium solutions of the Newtonian $n$-body problem and play a key role in understanding the structure and dynamics of gravitational systems. However, the classification and enumeration of such…
This paper discusses some of the physical properties of plane symmetric self-similar solutions of the first kind (i.e., homothetic solutions). We are interested in calculating the expansion, the acceleration, the rotation, the shear tensor,…
Lipschitz equivalence of self-similar sets is an important area in the study of fractal geometry. It is known that two dust-like self-similar sets with the same contraction ratios are always Lipschitz equivalent. However, when self-similar…
Employing techniques from scattering amplitudes and effective field theory, we model the dynamics of hierarchical triples, which are three-body systems composed of two bodies separated by a distance $r$ and a third body a distance $\rho$…