Homography in ℝℙ

It is proved that for a 3-dimensional compact metrizable space X the infinite real projective space is an absolute extensor of X if and only if the real projective plane is an absolute extensor of X.

Homotopy type theory is a version of Martin-Lof type theory taking advantage of its homotopical models. In particular, we can use and construct objects of homotopy theory and reason about them using higher inductive types. In this article, we construct the real projective spaces, key players in homotopy theory, as certain higher inductive types in homotopy type theory. The classical definition of RPn, as the quotient space identifying antipodal points of the n-sphere, does not translate directly to homotopy type theory. Instead, we define RPn by induction on n simultaneously with its tautological bundle of 2-element sets. As the base case, we take RP-1 to be the empty type. In the inductive step, we take RPn+1 to be the mapping cone of the projection map of the tautological bundle of RPn, and we use its universal property and the univalence axiom to define the tautological bundle on RPn+1.By showing that the total space of the tautological bundle of RPn is the n-sphere Sn, we retrieve the classical description of RPn+1 as RPn with an (n + 1)-disk attached to it. The infinite dimensional real projective space RP∞, defined as the sequential colimit of RPn with the canonical inclusion maps, is equivalent to the Eilenberg-MacLane space K(Z/2Z, 1), which here arises as the subtype of the universe consisting of 2-element types. Indeed, the infinite dimensional projective space classifies the 0-sphere bundles, which one can think of as synthetic line bundles.These constructions in homotopy type theory further illustrate the utility of homotopy type theory, including the interplay of type theoretic and homotopy theoretic ideas.

Summary We continue in the Mizar system [2] the formalization of fuzzy implications according to the book of Baczyński and Jayaram “Fuzzy Implications” [1]. In this article we define fuzzy negations and show their connections with previously defined fuzzy implications [4] and [5] and triangular norms and conorms [6]. This can be seen as a step towards building a formal framework of fuzzy connectives [10]. We introduce formally Sugeno negation, boundary negations and show how these operators are pointwise ordered. This work is a continuation of the development of fuzzy sets [12], [3] in Mizar [7] started in [11] and partially described in [8]. This submission can be treated also as a part of a formal comparison of fuzzy and rough approaches to incomplete or uncertain information within the Mizar Mathematical Library [9].

The Mizar system is one of the pioneering systems aimed at supporting mathematical proof development on a computer that have laid the groundwork for and eventually have evolved into modern interactive proof assistants. We claim that an important milestone in the development of these systems was the creation of organized libraries accumulating all previously available formalized knowledge in such a way that new works could effectively re-use all previously collected notions. In the case of Mizar, the turning point of its development was the decision to start building the Mizar Mathematical Library as a centrally-managed knowledge base maintained together with the formalization language and the verification system. In this paper we show the process of forming this library, the evolution of its design principles, and also present some data showing its current use with the modern version of the Mizar proof checker, but also as a rich corpus of semantically linked mathematical data in various areas including web-based and natural language proof presentation, maths education, and machine learning based automated theorem proving.

Summary The main aim of this article is to introduce formally ternary Boolean algebras (TBAs) in terms of an abstract ternary operation, and to show their connection with the ordinary notion of a Boolean algebra, already present in the Mizar Mathematical Library [2]. Essentially, the core of this Mizar [1] formalization is based on the paper of A.A. Grau “Ternary Boolean Algebras” [7]. The main result is the single axiom for this class of lattices [12]. This is the continuation of the articles devoted to various equivalent axiomatizations of Boolean algebras: following Huntington [8] in terms of the binary sum and the complementation useful in the formalization of the Robbins problem [5], in terms of Sheffer stroke [9]. The classical definition ([6], [3]) can be found in [15] and its formalization is described in [4]. Similarly as in the case of recent formalizations of WA-lattices [14] and quasilattices [10], some of the results were proven in the Mizar system with the help of Prover9 [13], [11] proof assistant, so proofs are quite lengthy. They can be subject for subsequent revisions to make them more compact.

We examine graphs that contain a non-trivial link in every embedding into real projective space, using a weaker notion of unlink than was used by Flapan, et al. We call such graphs intrinsically linked in projective space. We fully characterize such graphs with connectivity 0,1 and 2. We also show that only one Petersen-family graph is intrinsically linked in projective space and prove that K7 minus any two edges is also minor-minimal intrinsically linked. In all, 594 graphs are shown to be minor-minimal intrinsically linked in projective space.

Welschinger’s invariant bounds from below the number of real rational curves through a given generic collection of real points in the real projective plane. We estimate this invariant using Mikhalkin’s approach which deals with a corresponding count of tropical curves. In particular, our estimate implies that, for any positive integer d, there exists a real rational curve of degree d through any collection of 3d − 1 real points in the projective plane, and, moreover, asymptotically in the logarithmic scale at least one third of the complex plane rational curves through a generic point collection are real. We also obtain similar results for curves on other toric Del Pezzo surfaces.

For any sequence of properly convex domains in the real projective plane such that the zeros of Pick differentials have bounded multiplicity and get further and further apart, we determine all Hausdorff limit domains that one can obtain after normalizing each member of the sequence by a projective transformation. We then show that the result can be applied to convex domains generated by projective triangular reflection groups.

Summary In this article, we formalize in Mizar [1], [2] the 3-fold product space of real normed spaces for usefulness in application fields such as engineering, although the formalization of the 2-fold product space of real normed spaces has been stored in the Mizar Mathematical Library [3]. First, we prove some theorems about the 3-variable function and 3-fold Cartesian product for preparation. Then we formalize the definition of 3-fold product space of real linear spaces. Finally, we formulate the definition of 3-fold product space of real normed spaces. We referred to [7] and [6] in the formalization.

Dual quaternions and dual projective spaces

A basis is presented for the RP" classes in y\n(BO).This basis is used to prove that every smooth involution (M, T) on a closed manifold bounds if its fixed point set is a disjoint union of odd-dimensional projective spaces of constant dimension.Given any real vector bundle f -* RP" , let (RP" , <*) 6 Wn(BO) denote the stable bordism class determined by the map RP" -> BO(c) <- BO classifying .With n fixed, the classes (RP" ,,) as t, ranges over all vector bundles span a Z2-vector subspace of yi"(BO), where the sum (RP", <*) + (RP" , n) represents the disjoint union.We will exhibit a basis for this subspace.Letting A - RP" denote the canonical line bundle, the KO-theory of real vector bundles tells us that any -* RP" is stably isomorphic to kX for some (possibly large) k , so that (RP", ) = (RP" , kX).The main result of this paper is that up to bordism one need only consider those k < n .Theorem.A basis for the RP" classes in y\n(BO) is {(RP",;A)|0 <;<} ifn = 2m, {(RP" , (2) + 1)A)|0 < j < m} if n = 2m + 1.Thus, given a bundle kX - RP" with k > n , the theorem tells us that (rp" , kX) = Yi"1 > M j with each j < n .As an application we include the following Proposition.Given a closed manifold with differentiate involution (M, T) with fixed point set any disjoint union F = |JRP2"+I (n fixed), then (M, ) bounds.Capobianco [1] has shown this is true when = Rp2m+1.Royster [4] proved it when = RP2"+1 u RP2m+1 .No such result is possible for = IJRP2" Indeed, Stong [5] characterized all involutions fixing RP2".

The purpose of this paper is to study extendibility and stable extendibility of vector bundles over real projective spaces. We determine a necessary and sufficient condition that a vector bundle ζ over the real projective n-space RPn is extendible (or stably extendible) to RPm for every m>n in the case where ζ is the complexification of the tangent bundle of RPn and in the case where ζ is the normal bundle associated to an immersion of RPn in the Euclidean (n+k)-space Rn+k or its complexification, and give examples of the normal bundle which is extendible to RPN but is not stably extendible to RPN+1.

The s-th higher topological complexity TC s ⁡ ( X ) {\operatorname{TC}_{s}(X)} of a space X can be estimated from above by homotopical methods, and from below by homological methods. We give a thorough analysis of the gap between such estimates when X = ℝ ⁢ P m {X=\operatorname{\mathbb{R}P}^{m}} , the real projective space of dimension m. In particular, we describe a number r ⁢ ( m ) {r(m)} , which depends on the structure of zeros and ones in the binary expansion of m, and with the property that 0 ≤ s ⁢ m - TC s ⁡ ( ℝ ⁢ P m ) ≤ δ s ⁢ ( m ) {0\leq sm-\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})\leq\delta_{s}(% m)} for s ≥ r ⁢ ( m ) {s\geq r(m)} , where δ s ⁢ ( m ) = ( 0 , 1 , 0 ) {\delta_{s}(m)=(0,1,0)} for m ≡ ( 0 , 1 , 2 ) mod 4 {m\equiv(0,1,2)\bmod 4} . Such an estimation for TC s ⁡ ( ℝ ⁢ P m ) {\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})} appears to be closely related to the determination of the Euclidean immersion dimension of ℝ ⁢ P m {\operatorname{\mathbb{R}P}^{m}} . We illustrate the phenomenon in the case m = 3 ⋅ 2 a {m=3\cdot 2^{a}} . In addition, we show that, for large enough s and even m, TC s ⁡ ( ℝ ⁢ P m ) {\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})} is characterized as the smallest positive integer t = t ⁢ ( m , s ) {t=t(m,s)} for which there is a suitable equivariant map from Davis’ projective product space P 𝐦 s {\mathrm{P}_{\mathbf{m}_{s}}} to the ( t + 1 ) {(t+1)} -st join-power ( ( ℤ 2 ) s - 1 ) ∗ ( t + 1 ) {((\mathbb{Z}_{2})^{s-1})^{\ast(t+1)}} . This is a (partial, but conjecturally complete) generalization of the work of Farber, Tabachnikov and Yuzvinsky relating TC 2 {\operatorname{TC}_{2}} to the immersion dimension of real projective spaces.

Let $l_{1},$ $l_{2},$ $\\ldots,$ $l_{7}$ be mutually different seven lines on the real projective plane. We consider two conditions;(A) No three of $l_{1},$ $l_{2},$ $\\ldots,$ $l_{7}$ intersect at a point. (B) There is no conic tangent to any six of $l_{1},$ $l_{2}$ , . . . , $l_{7}$ . Cummings [3] and White [16] showed that there are eleven non-equivalent classes of systems of seven lines with condition (A)(cf. [7], Chap. 18). The purposes of this article is to give an interpretation of the classification of Cummings and White in terms of the root system of type $E_{7}$ . To accomplish this, it is better to add condition (B) for systems of seven lines. Moreover we need the notion of tetrahedral sets which consist of ten roots modulo slgns in the root system of type $E_{7}$ and which plays an important role in our study.

This thesis focuses on the interplay of extremal combinatorics and finite geometry. Combinatorics is concerned with discrete (and usually finite) objects. Extremal combinatorics studies how large or how small a collection of finite objects can be under certain restrictions. Those objects can be sets, graphs, vectors, etc. These questions are often motivated by problems in information theory and computer science. Another branch of combinatorics is finite geometry over finite fields of order q. Although there is no field of order 1, certain substructures in finite geometry can be interpreted as versions of projective spaces for q = 1. A triangle in a projective plane is a good example of this phenomenon. Following an introductory chapter, Chapter 2 gives an overview of some classical problems and their q-analogues in extremal combinatorics. The most recent results for the q-analogues of Sperner's theorem, the Erdös-Ko-Rado theorem and several versions of Bollobás's theorem are described. For the q-analogue of the Hilton-Milner theorem we give some new results. We also give a new bound for the minimum size of the q-analogue of small maximal cliques. We conclude that sometimes results for a q-analogue can be obtained by using the same technique as in the original problem. In some cases the answer to the problem is even identical to that of the original problem. In other cases techniques used for the q-analogue could be used for improving bounds for the original problem. The third chapter describes the known results for the chromatic number of the Kneser graphs and gives new bounds for the chromatic number of the q-analogue. The vertices of a Kneser graph are subsets (of a fixed size) of a set, whereas two vertices are adjacent if they are disjoint in their subset representation. In 1955 M. Kneser conjectured the chromatic number of those graphs. In 1978 this was proven correct by L. Lóvasz. Two small cases in the projective space q-analogue were solved in 2001 by J. Eisfeld, L. Storme and P. Sziklai and in 2006 by A. Chowdhury, C. Godsil and G. Royle. We describe an asymptotic result for all cases (except for one parameter family, where we give a partial proof) using the bounds from the q-analogue of the Hilton-Milner theorem. Chapters 4, 5 and 6 describe other q-analogues of the Kneser graphs. In Chapter 4 we define a family of Kneser type graphs over pairs of incident points and hyperplanes of a projective space. We describe large maximal cocliques in these graphs and prove that for small dimensions these are the largest possible cocliques. We conjecture that this is the case for all dimensions. In Chapter 5 we extend the Kneser graphs to the case of finite polar spaces instead of finite projective spaces and in some cases give descriptions and chromatic numbers of these graphs. Chapter 6 describes the generalization of Kneser graphs over coset spaces of Chevalley groups (parameterized by q) with respect to parabolic subgroups. This encompasses all previous cases and extends to some interesting new cases. First we describe these graphs when q = 1 and give chromatic numbers for some families. Then we consider the graphs defined over coset spaces of Chevalley groups for general q with respect to parabolic subgroups and study what results of the q = 1 case can be translated to the general case. In this thesis we found different possibilities for the connection between the bounds for the q = 1 case and the case for general q. In some cases the bounds for the general case are identical to the bounds for q = 1. In other cases we obtain the bounds for q = 1 by taking the limit for q¿1 in the general bound. Other cases show a completely different bound in both cases.