Real Projective Plane Research Articles

The covariant functoriality of graph algebras

AbstractIn the standard category of directed graphs, graph morphisms map edges to edges. By allowing graph morphisms to map edges to finite paths (path homomorphisms of graphs), we obtain an ambient category in which we determine subcategories enjoying covariant functors to categories of algebras given by constructions of path algebras, Cohn path algebras, and Leavitt path algebras, respectively. Thus, we obtain new tools to unravel homomorphisms between Leavitt path algebras and between graph C*‐algebras. In particular, a graph‐algebraic presentation of the inclusion of the C*‐algebra of a quantum real projective plane into the Toeplitz algebra allows us to determine a quantum CW‐complex structure of the former. It comes as a mixed‐pullback theorem where two ‐homomorphisms are covariantly induced from path homomorphisms of graphs and the remaining two are contravariantly induced by admissible inclusions of graphs. As a main result and an application of new covariant‐induction tools, we prove such a mixed‐pullback theorem for arbitrary graphs whose all vertex‐simple loops have exits, which substantially enlarges the scope of examples coming from noncommutative topology.

The Turán number of surfaces

AbstractWe show that there is a constant such that any 3‐uniform hypergraph with vertices and at least edges contains a triangulation of the real projective plane as a subgraph. This resolves a conjecture of Kupavskii, Polyanskii, Tomon and Zakharov. Furthermore, our work, combined with prior results, asymptotically determines the Turán number of all surfaces.

Birational involutions of the real projective plane

We classify birational involutions of the real projective plane up to conjugation. In contrast with an analogous classification over the complex numbers (due to E. Bertini, G. Castelnuovo, F. Enriques, L. Bayle and A. Beauville), which includes four different classes of involutions, we discover 12 different classes over the reals, and provide many examples when the fixed curve of an involution does not determine its conjugacy class in the real plane Cremona group.

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AbstractFor , let be a two‐dimensional algebraically defined graph, that is, a bipartite graph where each partite set is a copy of and two vertices and are adjacent if and only if . It is known that has girth 6 and can be extended to the point‐line incidence graph of the classical real projective plane. However, it was unknown whether there exists such that has girth 6 and is nonisomorphic to . This paper answers this question affirmatively and thus provides a construction of a nonclassical real projective plane. This paper also studies the diameter and girth of for families of bivariate functions .

Non‐Euclidean Sliced Optimal Transport Sampling

AbstractIn machine learning and computer graphics, a fundamental task is the approximation of a probability density function through a well‐dispersed collection of samples. Providing a formal metric for measuring the distance between probability measures on general spaces, Optimal Transport (OT) emerges as a pivotal theoretical framework within this context. However, the associated computational burden is prohibitive in most real‐world scenarios. Leveraging the simple structure of OT in 1D, Sliced Optimal Transport (SOT) has appeared as an efficient alternative to generate samples in Euclidean spaces. This paper pushes the boundaries of SOT utilization in computational geometry problems by extending its application to sample densities residing on more diverse mathematical domains, including the spherical space 𝕊d, the hyperbolic plane ℍd, and the real projective plane ℙd. Moreover, it ensures the quality of these samples by achieving a blue noise characteristic, regardless of the dimensionality involved. The robustness of our approach is highlighted through its application to various geometry processing tasks, such as the intrinsic blue noise sampling of meshes, as well as the sampling of directions and rotations. These applications collectively underscore the efficacy of our methodology.

Three Conics Determine a Cubic

Given a cubic K in the real projective plane. Then for each point P there is a conic CP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{{P}}$$\\end{document} associated to P. The conic CP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{{P}}$$\\end{document} is called the polar conic of K with respect to the pole P. We investigate the situation when three conics C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_1$$\\end{document}, C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_2$$\\end{document}, and C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_3$$\\end{document} are polar conics of K with respect to the poles P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_1$$\\end{document}, P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_2$$\\end{document}, and P3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_3$$\\end{document}, respectively. In particular, we give an elementary proof—without using any results from algebraic geometry—that any three conics C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_1$$\\end{document}, C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_2$$\\end{document}, C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_3$$\\end{document} in general position, satisfying only a non-degeneracy condition, determine a unique cubic K and three points P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_1$$\\end{document}, P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_2$$\\end{document}, P3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_3$$\\end{document}, such that C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_1$$\\end{document}, C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_2$$\\end{document}, C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_3$$\\end{document} are polar conics of K with respect to the three poles P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_1$$\\end{document}, P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_2$$\\end{document}, P3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_3$$\\end{document}. This can be seen as a higher degree variant of von Staudt’s Theorem.

Fractal interpolation on the real projective plane

Fractal interpolation on the real projective plane

Fractal Dimension of Fractal Functions on the Real Projective Plane

In this article, we consider an iterated functions system on the non-Euclidean real projective plane which has a linear structure. Then, we study the fractal dimension of the associated curve as a subset of the projective space and like a set of the Euclidean space. At the end, we initiate a dual real projective iterated function system and pose an open problem.

Low-energy points on the sphere and the real projective plane

We present a generalization of a family of points on S2, the Diamond ensemble, containing collections of N points on S2 with very small logarithmic energy for all N∈N. We extend this construction to the real projective plane RP2 and we obtain upper and lower bounds with explicit constants for the Green and logarithmic energy on this last space.

On Dimension of Configurations

The combinatorial (or abstract) configuration is an incidence structure, which can often be represented in Euclidean or real projective plane by a system of points and lines. B. Grünbaum introduced the concept of the dimension of a configuration as the largest dimensional space into which the configuration can be embedded to preserve the incidence. In this paper we examine the dimensions of some configurations and introduce the concept of the natural d-configuration.

Partitioning the projective plane and the dunce hat

We show that any finite triangulation of the real projective plane or the dunce hat is partitionable. To prove this, we introduce simple yet useful gluing tools that allow us to reduce partitionability of a given complex to that of smaller constituent relative subcomplexes such as the disk or the open Möbius strip. The gluing process generates partitioning schemes with a distinctive shelling/constructible flavor. We also give a tool to lift partitionability of relative simplicial complexes to that of the preimages of certain simplicial quotient maps.

Entanglement of valence-bond-solid state models on topological surfaces

We investigated the scaling behaviors of entanglements on the two-leg ladder lattice structure implemented with Affleck–Kennedy–Lieb–Tasaki (AKLT) model on six types of topological surfaces, namely, torus, sphere, real projective plane, Klein bottle, cylinder and Möbius strip. All six topological surfaces on which our lattices were embedded are classified according to whether their boundaries are closed or open. Lattice structures on these topological surfaces are formed either by a trivalent or four-valent site to fulfill a complete spin-32 or spin-2 system, respectively. Their ground states, which are the valence-bond-solid states, are derived by the tensor-product method. The von Neumann entropies for entanglement of the bipartite region on all six topological surfaces are calculated and all results show that the entanglement entropies per area are bounded from above or below when approaching the thermodynamic limit. For the spin-2 and the spin-32 systems on all surfaces, except on the sphere, numerical results between even and odd cases of the number of lattice sites should be considered separately for scrutinizing their scaling behaviors. However, their entanglement entropies still satisfy the area law, but with a correction term.

Twistor Theory of Dancing Paths

Given a path geometry on a surface $\mathcal{U}$, we construct a causal structure on a four-manifold which is the configuration space of non-incident pairs (point, path) on $\mathcal{U}$. This causal structure corresponds to a conformal structure if and only if $\mathcal{U}$ is a real projective plane, and the paths are lines. We give the example of the causal structure given by a symmetric sextic, which corresponds on an ${\rm SL}(2,{\mathbb R})$-invariant projective structure where the paths are ellipses of area $\pi$ centred at the origin. We shall also discuss a causal structure on a seven-dimensional manifold corresponding to non-incident pairs (point, conic) on a projective plane.

Dynamics of the Secant map near infinity

We investigate the root finding algorithm given by the Secant method applied to a real polynomial p of degree k as a discrete dynamical system defined on . We extend the Secant map to the real projective plane . The line at infinity is invariant, and there is one (if k is odd) or two (if k is even) fixed points at . We show that these are of saddle type, and this allows us to better understand the dynamics of the Secant map near infinity.

On foliations of the real projective plane defined by decomposable pencils of cubics

On foliations of the real projective plane defined by decomposable pencils of cubics

A note on the minimum number of red lines needed to pierce the intersections of blue lines

Let L be a set of n non-concurrent blue lines and let R be a set of m red lines in the real projective plane. In this note, using elementary geometric arguments, we show that if L∩R=∅ and there is a line from R through every intersection point of lines in L, then m≥411(n−1311). This lower bound improves the previous ones whenever n≥4. Also we give an application of this result.

On the disposition of cubic and pair of conics in a real projective plane. II

On the disposition of cubic and pair of conics in a real projective plane. II

A Triple of Projective Billiards

A projective billiard is a polygon in the real projective plane with a circumconic and an inconic. Similar to the classical billiards in conics, the intersection points between the extended sides of a projective billiard are located on a family of conics which form the associated Poncelet grid. We extend the projective billiard by the inner and outer billiard and disclose various relations between the associated grids and the diagonals, in particular other triples of projective billiards.

Duality Notions in Real Projective Plane

Summary In this article, we check with the Mizar system [1], [2], the converse of Desargues’ theorem and the converse of Pappus’ theorem of the real projective plane. It is well known that in the projective plane, the notions of points and lines are dual [11], [9], [15], [8]. Some results (analytical, synthetic, combinatorial) of projective geometry are already present in some libraries Lean/Hott [5], Isabelle/Hol [3], Coq [13], [14], [4], Agda [6], . . . . Proofs of dual statements by proof assistants have already been proposed, using an axiomatic method (for example see in [13] - the section on duality: “[...] For every theorem we prove, we can easily derive its dual using our function swap [...]2”). In our formalisation, we use an analytical rather than a synthetic approach using the definitions of Leończuk and Prażmowski of the projective plane [12]. Our motivation is to show that it is possible by developing dual definitions to find proofs of dual theorems in a few lines of code. In the first part, rather technical, we introduce definitions that allow us to construct the duality between the points of the real projective plane and the lines associated to this projective plane. In the second part, we give a natural definition of line concurrency and prove that this definition is dual to the definition of alignment. Finally, we apply these results to find, in a few lines, the dual properties and theorems of those defined in the article [12] (transitive, Vebleian, at_least_3rank, Fanoian, Desarguesian, 2-dimensional). We hope that this methodology will allow us to continued more quickly the proof started in [7] that the Beltrami-Klein plane is a model satisfying the axioms of the hyperbolic plane (in the sense of Tarski geometry [10]).

Lefschetz fibrations on nonorientable 4-manifolds

Let $W$ be a nonorientable $4$-dimensional handlebody without $3$- and $4$-handles. We show that $W$ admits a Lefschetz fibration over the $2$-disk, whose regular fiber is a nonorientable surface with nonempty boundary. This is an analogue of a result of Harer obtained in the orientable case. As a corollary, we obtain a $4$-dimensional proof of the fact that every nonorientable closed $3$-manifold admits an open book decomposition, which was first proved by Berstein and Edmonds using branched coverings. Moreover, the monodromy of the open book we obtain for a given $3$-manifold belongs to the twist subgroup of the mapping class group of the page. In particular, we construct an explicit minimal open book for the connected sum of arbitrarily many copies of the product of the circle with the real projective plane. We also obtain a relative trisection diagram for $W$, based on the nonorientable Lefschetz fibration we construct, similar to the orientable case first studied by Castro. As a corollary, we get trisection diagrams for some closed $4$-manifolds, e.g. the product of the $2$-sphere with the real projective plane, by doubling $W$. Moreover, if $X$ is a closed nonorientable $4$-manifold which admits a Lefschetz fibration over the $2$-sphere, equipped with a section of square $\pm 1$, then we construct a trisection diagram of $X$, which is determined by the vanishing cycles of the Lefschetz fibration. Finally, we include some simple observations about low-genus Lefschetz fibrations on closed nonorientable $4$-manifolds.