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The $L_\infty$-algebra is an algebraic structure suitable for describing deformation problems. In this paper we construct one $L_\infty$-algebra, which turns out to be a differential graded Lie algebra, to control the deformations of Lie…
Reflective imaging systems form an important part of photonic devices such as spectrometers, telescopes, augmented and virtual reality headsets or lithography platforms. Reflective optics provide unparalleled spectral performance and can be…
Lie symmetry analysis is applied to study the nonlinear rotating shallow water equations. The 9-dimensional Lie algebra of point symmetries admitted by the model is found. It is shown that the rotating shallow water equations are related…
This paper aims at reviewing and analysing the method of reflections. The latter is an iterative procedure designed to linear boundary value problems set in multiply connected domains. Being based on a decomposition of the domain boundary,…
The notion of quantum algebras is merged with that of Lie systems in order to establish a new formalism called Poisson-Hopf algebra deformations of Lie systems. The procedure can be naturally applied to Lie systems endowed with a symplectic…
The optimization of an optical system benefits greatly from a study of its aberrations and an identification of each of its elements' contribution to the overall aberration figures. The matrix formalism developed by one of the authors was…
Traditional lens design is a numerical and forward process based on ray tracing and aberration theory. This method has limitations because the initial configuration of the lens has to be specified and the aberrations of the lenses have to…
We use ring-theoretic methods and methods from the theory of skew braces to produce set-theoretic solutions to the reflection equation. We also use set-theoretic solutions to construct solutions to the parameter-dependent reflection…
When a plane electromagnetic wave impinges upon a diffraction grating or other periodic structures, reflected and transmitted waves propagate away from the structure in different radiation channels. A diffraction anomaly occurs when the…
We consider the geometric optics problem of finding a system of two reflectors that transform a spherical wavefront into a beam of parallel rays with prescribed intensity distribution. Using techniques from optimal transportation theory, it…
It is pointed out that affine Lie algebras appear to be the natural mathematical structure underlying the notion of integrability for two-dimensional systems. Their role in the construction and classification of 2D integrable systems is…
The goal of this paper is to make the vertex operator algebra approach to mirror symmetry accessible to algebraic geometers. Compared to better-known approaches using moduli spaces of stable maps and special Lagrangian fibrations, this…
We derive a method for finding Lie Symmetries for third-order difference equations. We use these symmetries to reduce the order of the difference equations and hence obtain the solutions of some third-order difference equations. We also…
This paper examines the use of Lie group and Lie Algebra theory to construct the geometry of pairwise comparisons matrices. The Hadamard product (also known as coordinatewise, coordinate-wise, elementwise, or element-wise product) is…
This paper proposes a new extrinsic calibration of kaleidoscopic imaging system by estimating normals and distances of the mirrors. The problem to be solved in this paper is a simultaneous estimation of all mirror parameters consistent…
This is a survey work on Lie algebras with ad-invariant metrics. We summarize main features, notions and constructions, in the aim of bringing into consideration the main research on the topic. We also give some list of examples in low…
We propose a new method for calculating reflection and transmission coefficients for an arbitrarily polarized electromagnetic plane wave incident on a one-dimensional dielectric medium of finite thickness and with dielectric permittivity…
The leading order monochromatic aberrations are investigated for the optical systems, which obey single-plane symmetry and translational invariance. These aberrations are classified from the symmetry principles for the wave aberration…
Symmetry is a powerful tool for finding analytical solutions to differential equations, both partial and ordinary, via the similarity variables or via the invariance of the equation under group transformations. It is the largest group of…
We will show the usefulness of the tools of Symplectic and Presymplectic Geometry and the corresponding Lie algebraic methods in different problems in Geometric Optics.