Real Locus Research Articles

Polynomial Partitioning on Varieties of Codimension Two and Point-Hypersurface Incidences in Four Dimensions

We present a polynomial partitioning theorem for finite sets of points in the real locus of an irreducible complex algebraic variety of codimension at most two. This result generalizes the polynomial partitioning theorem on the Euclidean space of Guth and Katz, and its extension to hypersurfaces by Zahl and by Kaplan, Matousek, Sharir and Safernova. We also present a bound for the number of incidences between points and hypersurfaces in the four-dimensional Euclidean space. It is an application of our partitioning theorem together with the refined bounds for the number of connected components of a semi-algebraic set by Barone and Basu.

Lower estimates for the expected Betti numbers of random real hypersurfaces

We estimate from below the expected Betti numbers of real hypersurfaces taken at random in a smooth real projective n-dimensional manifold. These random hypersurfaces are chosen in the linear system of a large dth power of a real ample line bundle equipped with a Hermitian metric of positive curvature. As for the upper bounds that we recently established, these lower bounds read as a product of a factor which only depends on the dimension n of the manifold with the Kähler volume of its real locus R X and d n . Actually, any closed affine real algebraic hypersurface appears with positive probability as part of such random real hypersurfaces in any ball of R X of radius O ( 1 / d ) .

EXPECTED TOPOLOGY OF RANDOM REAL ALGEBRAIC SUBMANIFOLDS

Let$X$be a smooth complex projective manifold of dimension$n$equipped with an ample line bundle$L$and a rank$k$holomorphic vector bundle$E$. We assume that$1\leqslant k\leqslant n$, that$X$,$E$and$L$are defined over the reals and denote by$\mathbb{R}X$the real locus of$X$. Then, we estimate from above and below the expected Betti numbers of the vanishing loci in$\mathbb{R}X$of holomorphic real sections of$E\otimes L^{d}$, where$d$is a large enough integer. Moreover, given any closed connected codimension$k$submanifold${\it\Sigma}$of$\mathbb{R}^{n}$with trivial normal bundle, we prove that a real section of$E\otimes L^{d}$has a positive probability, independent of$d$, of containing around$\sqrt{d}^{n}$connected components diffeomorphic to${\it\Sigma}$in its vanishing locus.

CRIANÇAS DESATENTAS OU PRÁTICAS PEDAGÓGICAS SEM SENTIDO? RELAÇÕES ENTRE MOTIVO, SENTIDO PESSOAL E ATENÇÃO

O objetivo deste trabalho é contribuir para a análise do papel da atividade pedagógica na formação da atenção. Primeiramente, apresentaremos o conceito do Transtorno de Déficit de Atenção e Hiperatividade (TDAH) analisando os critérios utilizados no seu diagnóstico. Analisaremos, também, o processo de constituição da atenção como função psicológica superior examinando sua relação com a estrutura da atividade pedagógica. Finalmente focaremos as relações entre os motivos da atividade, tendo como referência a estrutura da atividade proposta por Leontiev, e a formação da atenção, apresentando o conceito de ações geradoras de motivos de aprendizagem como uma ferramenta teórica que pode colocar o professor no papel ativo de produtor de motivos que levem os alunos a desenvolverem funções psicológicas superiores, entre elas a atenção. Desse modo nos contrapomos ao processo de medicalização trazendo o desafio de enfrentamento do desenvolvimento da atenção para o seu lócus verdadeiro, a escola.http://dx.doi.org/10.14572/nuances.v25i1.2735

Developing Shared Product Platforms During a Merger: The Fiat-Chrysler Case

This paper, building on R&D integration and improvisation literature, explores how firms organize to jointly develop common product platforms if their integration process is still in progress. In order to address our exploratory research question, we draw on a unique set of empirical data gathered during the Fiat-Chrysler R&D integration process and during the development of their first shared product platform. We show that, how due to fierce competition on time in the industry, the two companies did not have time to complete the planned R&D integration. As a consequence, the first shared platform development project represented the real locus of technological, knowledge, and organizational integration between the two firms. In line with the R&D integration literature, this paper identifies a set of planned integration mechanisms: a centralized R&D area, two teams that mirror each other, integrator roles and shared technical norms. This organization was designed to help Fiat and Chrysler exploiting their complementarities. Furthermore, building on the improvisational literature, we show that the planned organization did not suffice: interstices between the two firms exist and planning and improvisation are complementary. Improvisation should be built upon a minimum structure and firms willing to accelerate their integration process can rely upon NPD activities.

A Note About Pruning and Hénon Maps

We prove that there exists a \(C^1\)-open set \(\mathcal{U }\) of diffeomorphisms of the plane that consists of pruning diffeomorphisms of the horseshoe (up to a conjugacy), with the property that the \(C^0\)-closure of \(\mathcal{U }\) contains all Henon map in the boundary of the real horseshoe locus.

Pencils on real curves

AbstractWe consider coverings of real algebraic curves to real rational algebraic curves. We show the existence of such coverings having prescribed topological degree on the real locus. From those existence results we prove some results on Brill‐Noether Theory for pencils on real curves. For coverings having topological degree \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\underline{0}$\end{document} we introduce the covering number k and we prove the existence of coverings of degree 4 with prescribed covering number.

Betti numbers of random real hypersurfaces and determinants of random symmetric matrices

We asymptotically estimate from above the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds. Our upper bounds grow as the square root of the degree of the hypersurfaces as the latter grows to infinity, with a coefficient involving the Kählerian volume of the real locus of the manifold as well as the expected determinant of random real symmetric matrices of given index. In particular, for large dimensions, these coefficients get exponentially small away from mid-dimensional Betti numbers. In order to get these results, we first establish the equidistribution of the critical points of a given Morse function restricted to the random real hypersurfaces.

The separating gonality of a separating real curve

Let X be a smooth real curve of genus g such that the real locus has s connected components. We say X is separating if the complement of the real locus is disconnected. In case there exists a morphism f from X to \({\mathbb{P}^1}\) such that the inverse image of the real locus of \({\mathbb{P}^1}\) is equal to the real locus of X then X is separating and such morphism is called separating. The separating gonality of a separating real curve X is the minimal degree of a separating morphism from X to \({\mathbb{P}^1}\). It is proved by Gabard that this separating gonality is between s and (g + s + 1)/2. In this paper we prove that all values between s and (g + s + 1)/2 do occur.

Feynman Integrals and Motives of Configuration Spaces

We formulate the problem of renormalization of Feynman integrals and its relation to periods of motives in configuration space instead of momentum space. The algebro-geometric setting is provided by the wonderful compactifications of arrangements of subvarieties associated to the subgraphs of a Feynman graph and a (quasi)projective variety. The motive and the class in the Grothendieck ring are computed explicitly for these wonderful compactifications, in terms of the motive of the variety and the combinatorics of the Feynman graph, using recent results of Li Li. The pullback to the wonderful compactification of the form defined by the unrenormalized Feynman amplitude has singularities along a hypersurface, whose real locus is contained in the exceptional divisors of the iterated blowup that gives the wonderful compactification. A regularization of the Feynman integrals can be obtained by modifying the cycle of integration, by replacing the divergent locus with a Leray coboundary. The ambiguities are then defined by Poincare' residues. While these residues give mixed Tate periods associated to the cohomology of the exceptional divisors and their intersections, the regularized integrals give rise to periods of the hypersurface complement in the wonderful compactification, which can be motivically more complicated.

HERMITIAN–EINSTEIN CONNECTIONS ON PRINCIPAL BUNDLES OVER FLAT AFFINE MANIFOLDS

Let M be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric g and a covariant constant volume form. Let G be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat principal G-bundle EG over M admits a Hermitian–Einstein structure if and only if EG is polystable. A polystable flat principal G-bundle over M admits a unique Hermitian–Einstein connection. We also prove the existence and uniqueness of a Harder–Narasimhan filtration for flat vector bundles over M. We prove a Bogomolov type inequality for semistable vector bundles under the assumption that the Gauduchon metric g is astheno-Kähler.

Do Uniruled Six-manifolds Contain Sol Lagrangian Submanifolds?

We prove using symplectic field theory combined with some Gromov–Witten theory that if the suspension of a hyperbolic diffeomorphism of the two-torus Lagrangian embeds in a closed uniruled symplectic six-manifold, then its first rational homology is generated by the boundary of a symplectic disk. This prevents any Sol manifold from being in the real locus of an orientable real Del Pezzo fibration over a curve, confirming an expectation of Kollár. It also constrains Hamiltonian diffeomorphisms of uniruled symplectic four-manifolds.

Hyperellipticity and systoles of Klein surfaces

Given a hyperelliptic Klein surface, we construct companion Klein bottles, extending our technique of companion tori already exploited by the authors in the genus 2 case. Bavard’s short loops on such companion surfaces are studied in relation to the original surface so to improve a systolic inequality of Gromov’s. A basic idea is to use length bounds for loops on a companion Klein bottle, and then analyze how curves transplant to the original non-orientable surface. We exploit the real structure on the orientable double cover by applying the coarea inequality to the distance function from the real locus. Of particular interest is the case of Dyck’s surface. We also exploit an optimal systolic bound for the Möbius band, due to Blatter.

In Defense of Reading: Or, Why Reading Still Matters in a Contextualist Age

Suspicion of reading as a lived experience is a consequence of the rhetorical success of a few key arguments that together have defined a critical landscape dominated by various forms of contextualism. Where the contextualist consensus prevails, reading is tacitly or explicitly regarded as an epiphenomenon, inasmuch as the real locus of meaning-creation is elsewhere. The essay analyzes three core contextualist doctrines (about consciousness, history, and the status of the subject) and argues that they need not delegitimate the experience of reading. Rather, in each case the defining assumptions and beliefs of contextualism require attention to reading in order to do their interpretive work. Giving reading its due may also have a corrective function to the extent that contradictions caused by its neglect have thwarted an understanding of issues such as the relation of form and history, the status of the aesthetic, and the disciplinary purpose of the lettered humanities. Recognizing reading as the hidden ground of our critical and theoretical activity can help get us past various conundrums, impasses, and dead ends that haunt our profession.

On meromorphic parameterizations of real algebraic curves

A singular flat geometry may be canonically assigned to a real algebraic curve Γ; namely, via analytic continuation of unit speed parameterization of the real locus \({\Gamma_\mathbb{R}}\) . Globally, the metric \({\rho=|Q|=|q(z)|dzd\bar{z}}\) is given by the meromorphic quadratic differential Q on Γ induced by the standard complex form dx2 + dy2 on \({\mathbb{C}^2=\{(x,y)\}}\) . By considering basic properties of Q, we show that the condition for local arc length parameterization along \({\Gamma_\mathbb{R}}\) to extend meromorphically to the complex plane is quite restrictive: For curves of degree at most four, only lines, circles and Bernoulli lemniscates have such meromorphic parameterizations.

On the real nerve of the moduli space of complex algebraic curves of even genus

The real locus in the moduli space of complex algebraic curves of given genus consists of curves having real forms, that is, Riemann surfaces admitting a symmetry (anticonformal involution). The real locus is covered by subsets, each formed by curves having a given topological type determined by the number of connected components and the separability type of the real models of the curves. In this paper, we study the structure of the nerve corresponding to this covering, called the real nerve of complex algebraic curves, for even genera. We find its geometrical and homological dimension, and we obtain some results concerning its global geometrical properties. In the proofs, we use the equivalent language of Riemann surfaces and their symmetries.

2D LOCUS CONFIGURATIONS AND THE TRIGONOMETRIC CALOGERO–MOSER SYSTEM

A central hyperplane arrangement in ℂ2 with multiplicity is called a "locus configuration" if it satisfies a series of "locus equations" on each hyperplane. Following [4], we demonstrate that the first locus equation for each hyperplane corresponds to a force-balancing equation on a related interacting particle system on ℂ*: the charged trigonometric Calogero-Moser system. When the particles lie on S1 ⊂ ℂ*, there is a unique equilibrium for this system. For certain classes of particle weight, this is enough to show that all the locus equations are satisfied, producing explicit examples of real locus configurations. This in turn produces new examples of Schrödinger operators with Baker–Akhiezer functions.

Minitwistor spaces, Severi varieties, and Einstein–Weyl structure

In this article, we show that the space of nodal rational curves, which is so called a Severi variety (of rational curves), on any non-singular projective surface is always equipped with a natural Einstein–Weyl structure, if the space is 3-dimensional. This is a generalization of the Einstein–Weyl structure on the space of smooth rational curves on a complex surface, given by Hitchin. As geometric objects naturally associated to Einstein–Weyl structure, we investigate null surfaces and geodesics on the Severi varieties. Also, we see that if the projective surface has an appropriate real structure, then the real locus of the Severi variety becomes a positive definite Einstein–Weyl manifold. Moreover, we construct various explicit examples of rational surfaces having 3-dimensional Severi varieties of rational curves.

The Optimization Design of the Pressure Foot Mechanism in a Single-Sheet Feeder

Based on the analysis of the working principle of the pressure foot (the cam- linkage compound mechanism) in a single-sheet feeder, using VB to develop the design of cam mechanism, a mathematical model of optimization design is set-up in which the object function is the minimum deviation between the real locus of the pressure foot and the required locus. The resolving method for this model is also studied with one real example. It is practically valuable.

Simplicity made difficult

The real locus of dispute, though, must be Tl. After all, analytic relativists tend to accept T3, T4, and T5.1 T2 is a technical claim; whether it is correct is a matter of what sorts of abstract objects it is most useful to assign to embedded occurrences of sentences in a compositional semantic theory. There are relativists and nonrelativists on both sides of the issue, and Cappelen and Hawthorne concede that much of