On Huisman’s conjectures about unramified real curves

Let X ⊂ ℙ n be an unramified real curve with X(ℝ) ≠ 0. If n ≥ 3 is odd, Huisman [9] conjectured that X is an M-curve and that every branch of X(ℝ) is a pseudo-line. If n ≥ 4 is even, he conjectures that X is a rational normal curve or a twisted form of such a curve. Recently, a family of unramified M-curves in ℙ3 providing counterexamples to the first conjecture was constructed in [11]. In this note we construct another family of counterexamples that are not even M-curves. We remark that the second conjecture follows for generic curves of odd degree from the de Jonquières formula.

Linear General Position (i.e. Arcs) for Zero-Dimensional Schemes Over a Finite Field

We extend some of the usual notions of projective geometry over a finite field (arcs and caps) to the case of zero-dimensional schemes defined over a finite field Fq. In particular we prove that for our type of zero-dimensional arcs the maximum degree in any r-dimensional projective space is r(q + 1) and (if either r = 2 or q is odd) all the maximal cases are projectively equivalent and come from a rational normal curve.

Normal Curves in 4-Dimensional Galilean Space G4

In this article, first, we give the definition of normal curves in 4-dimensional Galilean space G4. Second, we state the necessary condition for a curve of curvatures τ(s) and σ(s) to be a normal curve in 4-dimensional Galilean space G4. Finally, we give some characterizations of normal curves with constant curvatures in G4.

Geometric Invariants of Normal Curves under Conformal Transformation in $\mathbb{E}^3$

In this paper, we investigate the geometric invariant properties of a normal curve on a smooth immersed surface under conformal transformation. We obtain an invariant-sufficient condition for the conformal image of a normal curve. We also find the deviations of normal and tangential components of the normal curve under the same motion.

Hyperbolic worldsheets and worldlines of null Cartan curves in de Sitter 3-space

The hyperbolic worldsheets and the hyperbolic worldline generated by null Cartan curves are defined and their geometric properties are investigated. As applications of singularity theory, the singularities of the hyperbolic worldsheets and the hyperbolic worldline are classified by using the approach of the unfolding theory in singularity theory. It is shown that under appropriate conditions, the hyperbolic worldsheet is diffeomorphic to cuspidal edge or swallowtail type of singularity and the hyperbolic worldline is diffeomorphic to cusp. An important geometric invariant which has a close relation with the singularities of the hyperbolic worldsheets and worldlines is found such that the singularities of the hyperbolic worldsheets and worldlines can be characterized by the invariant. Meanwhile, the contact of the spacelike normal curve of a null Cartan curve with hyperbolic quadric or world hypersheet is discussed in detail. In addition, the dual relationships between the spacelike normal curve of a null Cartan curve and the hyperbolic worldsheet are described. Moreover, it is demonstrated that the spacelike normal curve of a null Cartan curve and the hyperbolic worldsheet are [Formula: see text]-dual each other.

Conhecimentos Didático-Matemáticos para Abordagem da Curva Normal no Ensino Médio

ResumoInvestigações que tratem do conhecimento do professor de Matemática são sempre pertinentes e necessárias porque podem contribuir para a ressignificação e o melhoramento da prática docente. No presente texto, são discutidos os resultados de um estudo dissertativo que teve o objetivo de investigar os conhecimentos didático-matemáticos de professores de Matemática do Ensino Médio para abordagem da inter-relação entre a Estatística e a Probabilidade por meio da Curva Normal. Esse estudo está fundamentado no modelo teórico de Conhecimentos e Competências Didático-Matemáticos do professor – CCDM, desenvolvido no âmbito do Enfoque Ontossemiótico do Conhecimento e da Instrução Matemática – EOS. Em termos metodológicos, essa pesquisa possui abordagem qualitativa, seu universo de participantes foi composto por 12 professores brasileiros de Matemática do Ensino Médio e seu desenrolar é contemplado por um estudo diagnóstico e um encontro formativo. Os resultados apontam que, inicialmente, os professores apresentaram lacunas nos conhecimentos didático-matemáticos sobre o tema. No entanto, por meio da realização do encontro formativo, no qual foi vivenciada a proposta de ensino, eles conseguiram avançar na construção, ressignificação e ampliação de seus conhecimentos didático-matemáticos sobre a inter-relação entre a Estatística e a Probabilidade por meio da Curva Normal. Concluímos que é pertinente o investimento nas formações acadêmicas e em formações continuadas de estudos semelhantes que possibilitem a apropriação e ampliação de conhecimentos didático-matemáticos de professores de Matemática sobre a Estatística e a Probabilidade. Palavras-chave: Estatística. Probabilidade. Curva Normal. Conhecimentos e Competências Didático-Matemáticos. Formação de Professores. AbstractInvestigations that deal with the mathematics teacher's knowledge are always relevant and necessary because they can contribute to the reframing and improvement of teaching practice. In the present text, the results of a dissertation study that aimed to investigate the didactic-mathematical knowledge of high school mathematics teachers to approach the interrelationship between Statistics and Probability through the Normal Curve are discussed. This study is based on the theoretical model of Teacher's Competences and Didactic-Mathematical Knowledge - CCDM, developed within the scope of the Ontosemiotic focus to Knowledge and Mathematical Instruction - EOS. In methodological terms, this research has a qualitative approach, its universe of participants was composed of 12 Brazilian high school mathematics teachers and its course is contemplated by a diagnostic study and a formative meeting. The results show that, initially, teachers had gaps in didactic-mathematical knowledge on the topic. However, through the formative meeting, in which the teaching proposal was experienced, they managed to advance in the construction, reframing and expansion of their didactic-mathematical knowledge about the interrelationship between Statistics and Probability through the Curve Normal. We conclude that it is pertinent to invest in academic training and in continuing training in similar studies that enable the appropriation and expansion of didactic-mathematical knowledge of mathematics teachers on statistics and probability. Keywords: Statistic. Probability. Normal Curve. Knowledge and Skills Didactic-Mathematics. Teacher Training.

The Charitable Continuum

There are powerful fairness and efficiency arguments for making charitable donations to soup kitchens 100% deductible. These arguments have no purchase for donations to fund opulent church organs, yet these too are 100% deductible under the current tax code. This stark dichotomy is only the tip of the iceberg. Looking at a wider sampling of charitable gifts reveals a charitable continuum. Based on sliding scales for efficiency, multiple theories of fairness, pluralism, institutional competence and social welfare dictate that charitable deductions should in most cases be fractions between zero and one. Moreover, the Central Limit Theorem strongly suggests that combining this welter of largely independent criteria with the wide variety of charitable gifts results in a classic bell-shaped normal curve of optimal deductions, with a peak at some central value and quickly decaying to zero at the extremes of 0% and 100%. Given that those are the only two options under the current tax code, the current charitable deduction regime inevitably makes large errors in most cases. Actually calculating a precise optimal percentage for each type of charitable donation is of course impractical. This Article suggests, however, that we can do much better than the systematically erroneous current charitable deduction. Granting a 100% deduction only for donations to the desperately poor, along with 50%, 25%, and 0% for gifts yielding progressively fewer efficiency, fairness, pluralism, and institutional competence benefits, promises to deliver a socially more desirable charitable deduction.