Related papers: The Mystery of the Shape Parameter IV
Most genuine multi-sided surface representations depend on a 2D domain that enables a mapping between local parameters and global coordinates. The shape of this domain ranges from regular polygons to curved configurations, but the simple…
We study the geometry and codes of quartic surfaces with many cusps. We apply Gr\"obner bases to find examples of various configurations of cusps on quartics.
We investigate the isomorphism problem in the setting of definable sets (equivalent to sets with atoms): given two definable relational structures, are they related by a definable isomorphism? Under mild assumptions on the underlying…
The numerical phenomenon of $\pi$ appearing at parameters $c = 1/4$, $c=-3/4$ and $c=-5/4$ in the Mandelbrot set $\mathcal{M}$ has been known for over 30 years. In 2001, the first proof was provided by Aaron Klebanoff for the parameter…
The use of quadratic residues to construct matrices with specific determinant values is a familiar problem with connections to many areas of mathematics and statistics. Our research has focused on using cubic residues to construct matrices…
Based on geometric intuition, in this paper we are trying to give an idea and visualize the meaning of the determinants for the cubic-matrix. In this paper we have analyzed the possibilities of developing the concept of determinant of…
The amount of information in the form of features and variables avail- able to machine learning algorithms is ever increasing. This can lead to classifiers that are prone to overfitting in high dimensions, high di- mensional models do not…
In this paper, we characterize the polynomiality of surfaces of revolution by means of the polynomiality of an associated plane curve. In addition, if the surface of revolution is polynomial, we provide formulas for computing a polynomial…
We introduce the theoretical framework we use to study the bewildering variety of phases in condensed--matter physics. We emphasize the importance of the breaking of symmetries, and develop the idea of an order parameter through several…
We find a class of open-system models in which individual quantum trajectories may depend on parameters that are undetermined by the full open-system evolution. This dependence is imprinted in the geometric phase associated with such…
Given a family of rational curves depending on a real parameter, defined by its parametric equations, we provide an algorithm to compute a finite partition of the parameter space (${\Bbb R}$, in general) so that the shape of the family…
In this paper, we provide a criterion for determining whether multiple shells support a $t$-design. We construct as a corollary an infinite series of $2$-designs using power residue codes.
Shape metrics for objects in high dimensions remain sparse. Those that do exist, such as hyper-volume, remain limited to objects that are better understood such as Platonic solids and $n$-Cubes. Further, understanding objects of ill-defined…
We describe here the general mathematical approach to constructing likelihoods for fitting observed spectra in one or more dimensions with multiple sources, including the effects of systematic uncertainties represented as nuisance…
The interest in a system often resides in the interplay among different parameters governing its evolution. It is thus often required to access many of them at once for a complete description. Assessing how quantum enhancement in such…
Why are materials with specific characteristics more abundant than others? This is a fundamental question in materials science and one that is traditionally difficult to tackle, given the vastness of compositional and configurational space.…
One of the molecular properties most intuitive to the human perception is the geometrical shape. However, when exploring a large chemical space the determination of shape needs to be automated. We present a fast and simple approach to…
In this paper, we give a Zariski triple of the arrangements for a smooth quartic and its four bitangents. A key criterion to distinguish the topology of such curves is given by a matrix related to the height pairing of rational points…
Problems related to the application of hidden color components in multi-quark systems are discussed in this report.
Many standard structural quantities, such as order parameters and correlation functions, exist for common condensed matter systems, such as spherical and rod-like particles. However, these structural quantities are often insufficient for…