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Numerous authors have considered the problem of determining the Lebesgue space mapping properties of the operator $\mathcal{A}$ given by convolution with affine arc-length measure on some polynomial curve in Euclidean space. Essentially,…
In Euclidean geometry, the Pythagorean theorem is presented as an equation involving three squares. This paper explores how analogous expressions may be identified in spherical and hyperbolic geometries.
Weierstrass elliptic and related functions have been recently shown to enable analytical explicit solutions to classical problems in astrodynamics. These include the constant radial acceleration problem, the Stark problem and the two-fixed…
An important problem in analytic and geometric combinatorics is estimating the number of lattice points in a compact convex set in a Euclidean space. Such estimates have numerous applications throughout mathematics. In this note, we exhibit…
This is a study of a problem in geodesy with methods from complex algebraic geometry: for a fixed number of measure points and target points at unknown position in the Euclidean plane, we study the problem of determining their relative…
The contour argument was introduced by Peierls for two dimensional Ising model. Peierls benefited from the particular symmetries of the Ising model. For non-symmetric models the argument was developed by Pirogov and Sinai. It is very…
The synchronization between two dynamical systems is one of the most appealing phenomena occurring in Nature. Already observed by Huygens in the case of two pendula, it is a current area of research in the case of chaotic systems, with…
Time-series data augmentation mitigates the issue of insufficient training data for deep learning models. Yet, existing augmentation methods are mainly designed for classification, where class labels can be preserved even if augmentation…
We describe a method to classify crystallographic tilings of the Euclidean and hyperbolic planes by tiles whose stabiliser group contains translation isometries or whose topology is not that of a closed disk. We tackle this problem from two…
Gaia Data Release 2 (Gaia DR2) provides high accuracy and precision astrometric parameters (position, parallax, and proper motion) for more than 1 billion sources and is revolutionizing astrometry. For a fast-moving target such as an…
Orbital solutions for binary or multiple stellar systems that combine astrometry (e.g., position angles and angular separations) with spectroscopy (radial velocities) have important advantages over astrometric-only or spectroscopic-only…
Although the differential calculus was invented by Newton, Kepler established his famous laws 70 years earlier by using the same idea, namely to find a path in a nonuniform field of force by small steps. It is generally not known that…
E. C. Zeeman [1] has criticized the fact that in all articles and books until that moment (1967) the topology employed to work with the Minkowski space was the Euclidean one. He has proposed a new topology, which was generalized for more…
We summarize the first exploratory investigation into whether Machine Learning techniques can augment science strategic planning. We find that an approach based on Latent Dirichlet Allocation using abstracts drawn from high impact astronomy…
We introduce the Primary Gasing Triangle, a right triangle with a hypotenuse of 1 unit, to define the primary trigonometric functions: sine and cosine. This triangle serves as the foundational element in a new approach to learning…
We show that the space of Euclid's parameters for Pythagorean triples is endowed with a natural symplectic structure and that it emerges as a spinor space of the Clifford algebra $\mathbb{R}_{2,1}$, whose minimal version may be…
The measurement of the positions, distances, motions and luminosities of stars represents the foundations of modern astronomical knowledge. Launched at the end of the eighties, the ESA Hipparcos satellite was the first space mission…
Topological transforms have been very useful in statistical analysis of shapes or surfaces without restrictions that the shapes are diffeomorphic and requiring the estimation of correspondence maps. In this paper we introduce two…
We present a method designed to correct for Lutz-Kelker effects in distance-limited samples. The method allows for the calculation of distances to individual objects and, at the same time, provides a fit to a parameterized, self-consistent…
Many numerical methods for multiscale differential equations require a scale separation between the larger and the smaller scales to achieve accuracy and computational efficiency. In the area of multiscale dynamical systems, so-called,…