Search Results (9)

Recently, Yu et al. derived a factorization procedure for detecting and computing the potential self-intersection of 3D integral Bézier cubics, claiming that their proposal distinctly outperforms existing methodologies. First, we recall that in the 2D case, explicit formulas already exist for the parameter values at the self-intersection (the singularity called crunode in algebraic geometry). Such values are the solutions of a quadratic equation, and affine invariants depend only on the curve hodograph. Also, the factorization procedure for cubics is well known. Second, we note that only planar Bézier cubics can display a self-intersection, so there is no need to address the problem in the more involved 3D setting. Finally, we elucidate the connections with the previous literature and provide a geometric interpretation, in terms of the affine classification of cubics, of the algebraic conditions necessary for the existence of a self-intersection. Cubics with a self-intersection are affine versions of the celebrated Tschirnhausen cubic. Full article

The interactions between topology and algebraic geometry expose various interesting properties. This paper proposes the deformations of topological n-manifolds over the automorphic polynomial ring maps and associated isomorphic imbedding of locally flat submanifolds within the n-manifolds. The manifold deformations include topologically homeomorphic bending of submanifolds at multiple directions under algebraic operations. This paper introduces the concept of a topological equivalence class of manifolds and the associated equivalent class of polynomials in a real ring. The concepts of algebraic compositions in a real polynomial ring and the resulting topological properties (homeomorphism, isomorphism and deformation) of manifolds under algebraic compositions are introduced. It is shown that a set of ideals in a polynomial ring generates manifolds retaining topological isomorphism under algebraic compositions. The numerical simulations are presented in this paper to illustrate the interplay of topological properties and the respective real algebraic sets generated by polynomials in a ring within affine 3-spaces. It is shown that the coefficients of polynomials generated by a periodic smooth function can induce mirror symmetry in manifolds. The proposed formulations do not consider the simplectic class of manifolds and associated quantizable deformations. However, the proposed formulations preserve the properties of Nash representations of real algebraic manifolds including Nash isomorphism. Full article

The formulations of polynomials over a topological simplex combine the elements of topology and algebraic geometry. This paper proposes the formulation of simplicial polynomials and the properties of resulting topological manifolds in two classes, non-degenerate forms and degenerate forms, without imposing the conditions of affine topological spaces. The non-degenerate class maintains the degree preservation principle of the atoms of the polynomials of a topological simplex, which is relaxed in the degenerate class. The concept of hybrid decomposition of a simplicial polynomial in the non-degenerate class is introduced. The decompositions of simplicial polynomial for a large set of simplex vertices generate ideal components from the radical, and the components preserve the topologically isolated origin in all cases within the topological manifolds. Interestingly, the topological manifolds generated by a non-degenerate class of simplicial polynomials do not retain the homeomorphism property under polynomial extension by atom addition if the simplicial condition is violated. However, the topological manifolds generated by the degenerate class always preserve isomorphism with varying rotational orientations. The hybrid decompositions of the non-degenerate class of simplicial polynomials give rise to the formation of simplicial chains. The proposed formulations do not impose strict positivity on simplicial polynomials as a precondition. Full article

In teleparallel geometries, the coframe and corresponding spin connection are the principal geometric objects and, consequently, the appropriate definition of a symmetry is that of an affine symmetry. The set of invariant coframes and their corresponding spin connections that respect the full six dimensional Lie algebra of Robertson–Walker affine symmetries are displayed and discussed. We will refer to such geometries as teleparallel Robertson–Walker (TRW) geometries, where the corresponding derived metric is of Robertson–Walker form and is characterized by the parameter $k=(-1,0,1)$. The field equations are explicitly presented for the $F\left(T\right)$ class of teleparallel TRW spacetimes. We are primarily interested in investigating the $k\ne 0$ TRW models. After first studying the $k=0$ models and, in particular, writing their governing field equations in an appropriate form, we then study their late time stability with respect to perturbations in k in both the cases of a vanishing and non-vanishing effective cosmological constant term. As an illustration, we consider both quadratic $F\left(T\right)$ theories and power-law solutions. Full article

We propose a new paradigm for modelling and calibrating laser scanners with rotation symmetry, as is the case for lidars or for galvanometric laser systems with one or two rotating mirrors. Instead of bothering about the intrinsic parameters of a physical model, we use the geometric properties of the device to model it as a specific configuration of lines, which can be recovered by a line-data-driven procedure. Compared to universal data-driven methods that train general line models, our algebraic-geometric approach only requires a few measurements. We elaborate the case of a galvanometric laser scanner with two mirrors, that we model as a grid of hyperboloids represented by a grid of $3\times 3$ lines. This provides a new type of look-up table, containing not more than nine elements, lines rather than points, where we replace the approximating interpolation with exact affine combinations of lines. The proposed method is validated in a realistic virtual setting. As a collateral contribution, we present a robust algorithm for fitting ruled surfaces of revolution on noisy line measurements. Full article

This article deals with Lie algebra $\mathfrak{G}$ of all infinitesimal affine transformations of the manifold $\mathcal{M}$ with an affine connection, its stationary subalgebra $ℌ\subset \mathfrak{G}$, the Lie group $\mathcal{G}$ corresponding to the algebra $\mathfrak{G}$, and its subgroup $\mathcal{H}\subset \mathcal{G}$ corresponding to the subalgebra $ℌ\subset \mathfrak{G}$. We consider the center $ℨ\subset \mathfrak{G}$ and the commutant [$\mathfrak{G},\mathfrak{G}]$ of algebra $\mathfrak{G}$. The following condition for the closedness of the subgroup $\mathcal{H}$ in the group $\mathcal{G}$ is proved. If $ℌ\cap \left(ℨ+\left[\mathfrak{G};\mathfrak{G}\right]\right)=ℌ\cap [\mathfrak{G};\mathfrak{G}$], then $\mathcal{H}$ is closed in $\mathcal{G}$. To prove it, an arbitrary group $\mathcal{G}$ is considered as a group of transformations of the set of left cosets $\mathcal{G}/\mathcal{H}$, where $\mathcal{H}$ is an arbitrary subgroup that does not contain normal subgroups of the group $\mathcal{G}$. Among these transformations, we consider right multiplications. The group of right multiplications coincides with the center of the group$\mathcal{G}$. However, it can contain the right multiplication by element $\overline{\mathcal{𝒽}}$, belonging to normalizator of subgroup $\mathcal{H}$ and not belonging to the center of a group $\mathcal{G}$. In the case when $\mathcal{G}$ is in the Lie group, corresponding to the algebra $\mathfrak{G}$ of all infinitesimal affine transformations of the affine space $\mathcal{M}$ and its subgroup $\mathcal{H}$ corresponding to its stationary subalgebra $ℌ\subset \mathfrak{G}$, we prove that such element $\overline{{\displaystyle \mathcal{𝒽}}}$ exists if subgroup $\mathcal{H}$ is not closed in $\mathcal{G}$. Moreover $\overline{\mathcal{𝒽}}$ belongs to the closures $\overline{\mathcal{H}}$ of subgroup $\mathcal{H}$ in $\mathcal{G}$ and does not belong to commutant $\left(\mathcal{G},\mathcal{G}\right)$ of group $\mathcal{G}$. It is also proved that $\mathcal{H}$ is closed in $\mathcal{G}$ if $\left(\mathfrak{P}+ℨ\right)\cap ℌ=\mathfrak{P}\cap ℌ$ for any semisimple algebra $\mathfrak{P}\in \mathfrak{G}$ for which $\mathfrak{P}+\Re =\mathfrak{G}$. Full article

The affine Hecke algebra of type A has two parameters $\left(q,,,t\right)$ and acts on polynomials in N variables. There are two important pairwise commuting sets of elements in the algebra: the Cherednik operators and the Jucys–Murphy elements whose simultaneous eigenfunctions are the nonsymmetric Macdonald polynomials, and basis vectors of irreducible modules of the Hecke algebra, respectively. For certain parameter values, it is possible for special polynomials to be simultaneous eigenfunctions with equal corresponding eigenvalues of both sets of operators. These are called singular polynomials. The possible parameter values are of the form $q^m=t^{-n}$ with $2\le n\le N.$ For a fixed parameter, the singular polynomials span an irreducible module of the Hecke algebra. Colmenarejo and the author (SIGMA 16 (2020), 010) showed that there exist singular polynomials for each of these parameter values, they coincide with specializations of nonsymmetric Macdonald polynomials, and the isotype (a partition of N) of the Hecke algebra module is $\left(d,n,-,1,,,n,-,1,,,\dots ,,,n,-,1,,,r\right)$ for some $d\ge 1$. In the present paper, it is shown that there are no other singular polynomials. Full article

With the particular interest of being implemented in cryptography, the recognition and analysis of image patterns based on Latin squares has recently arisen as an efficient new approach for classifying partial Latin squares into isomorphism classes. This paper shows how the use of a Computer Algebra System (CAS) becomes necessary to delve into this aspect. Thus, the recognition and analysis of image patterns based on these combinatorial structures benefits from the use of computational algebraic geometry to determine whether two given partial Latin squares describe the same affine algebraic set. This paper delves into this topic by focusing on the use of a CAS to characterize when two partial Latin squares are either partial transpose or partial isotopic. Full article

We introduce the symplectic structure of information geometry based on Souriau’s Lie group thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physical observables like energy, heat, and moment as pure geometrical objects. Using geometric Planck temperature of Souriau model and symplectic cocycle notion, the Fisher metric is identified as a Souriau geometric heat capacity. The Souriau model is based on affine representation of Lie group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. In the framework of Lie group thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincaré-Cartan-Souriau integral. The Souriau-Fisher metric is linked to KKS (Kostant–Kirillov–Souriau) 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We apply this model in the framework of information geometry for the action of an affine group for exponential families, and provide some illustrations of use cases for multivariate gaussian densities. Information geometry is presented in the context of the seminal work of Fréchet and his Clairaut-Legendre equation. The Souriau model of statistical physics is validated as compatible with the Balian gauge model of thermodynamics. We recall the precursor work of Casalis on affine group invariance for natural exponential families. Full article