Geometry of singularities of a Pinchuk's map

We describe a singular variety associated to a Pinchuk's map and calculate its homology intersection. The result provides geometries of singularities of this Pinchuk's map

An algorithm to classify the asymptotic set associated to a polynomial mapping

We provide an algorithm to classify the asymptotic sets of the dominant polynomial mappings F: \C^3 \to \C^3 of degree 2, using the definition of the so-called "{\it fa\c{c}ons}" in \cite{Thuy}. We obtain a classification theorem for the asymptotic sets of dominant polynomial mappings F: \C^3 \to \C^3 of degree 2. This algorithm can be generalized for the dominant polynomial mappings F: \C^n \to \C^n of degree $d$, with any $(n, d) \in {(\N^*)}^2$

A remark on a polynomial mapping from \C^n to \C^n

We provide relations of the results obtained in the articles \cite{ThuyCidinha} and \cite{VuiThang}. Moreover, we provides some examples to illustrate these relations, using the software {\it Maple} to complete the complicate calculations of the examples. We give some discussions on these relations.Comment: arXiv admin note: text overlap with arXiv:1503.0807

The 2-dimensional Complex Jacobian Conjecture under the viewpoint of "pertinent variables"

Let F = (f,g): \C^2 \to \C^2 be a polynomial map. The 2-dimensional Complex Jacobian Conjecture, which is still open, can be expressed as follows: "if $F$ satisfies the Non-Zero Condition \det (JF(x,y)) = {\rm constant} \neq 0, \forall (x,y) \in \C^2, then $F$ is non-proper". A significant approach for the study of the Jacobian Conjecture is to remove the most possible polynomial maps that do not satisfy the Non-Zero Condition and work on the complementary set in the ring of polynomial maps. We define firstly in this paper good polynomial maps which satisfy the following property: if $F$ satisfies the Non-Zero Condition, then $F$ is a good polynomial map. Then, with the hypothesis "$F$ is non-proper", we define new variables called pertinent variables and we treat $F$ under these variables. That allows us to define a class $\mathcal{C}_1$ of good, non-proper polynomial maps F = (f,g): \C^2 \to \C^2 that are significant for the study of the Jacobian Conjecture. We continue to restrict the class $\mathcal{C}_1$ by expliciting a subclass $\mathcal{C}_2 \subset \mathcal{C}_1$ of polynomial maps which do not satisfy the Non-Zero Condition. Then, on the one side, we get a model of a counter-example for the Conjecture if there exists. On the other side, we provide a criterion for verifying the 2-dimensional Complex Jacobian Conjecture for the class of good, non-proper maps. Moreover, for verifying the 2-dimensional Complex Jacobian Conjecture, it is enough to verify it for the complementary set of the set ${\mathcal C}_2$ in the set of good maps. The second part of this paper is to provide a criterion to verify the dominancy of a polynomial map in general. Applying this criterion, we prove that a polynomial map of the class $\mathcal{C}_1 \setminus \mathcal{C}_2$ is dominant and we describe its asymptotic set

An elementary proof of Euler formula using Cauchy's method

The use of Cauchy's method to prove Euler's well-known formula is an object of many controversies. The purpose of this paper is to prove that Cauchy's method applies for convex polyhedra and not only for them, but also for surfaces such as the torus, the projective plane, the Klein bottle and the pinched torus

On a singular variety associated to a polynomial mapping from \C^n to \C^{n-1}

We construct a singular variety ${\mathcal{V}}_G$ associated to a polynomial mapping G : \C^{n} \to \C^{n - 1} where $n \geq 2$. We prove that in the case G : \C^{3} \to \C^{2}, if $G$ is a local submersion but is not a fibration, then the homology and the intersection homology with total perversity (with compact supports or closed supports) in dimension two of the variety ${\mathcal{V}}_G$ is not trivial. In the case of a local submersion G : \C^{n} \to \C^{n - 1} where $n \geq 4$, the result is still true with an additional condition

On a singular variety associated to a polynomial mapping

In the paper "Geometry of polynomial mapping at infinity via intersection homology" the second and third authors associated to a given polynomial mapping F : \C^2 \to \C^2 with nonvanishing jacobian a variety whose homology or intersection homology describes the geometry of singularities at infinity of the mapping. We generalize this result.Comment: 1 figur

THE EFFECTS OF QUESTION-ANSWER RELATIONSHIP STRATEGY ON EFL HIGH SCHOOL STUDENTS’ READING COMPREHENSION

This paper reports the effects of the question-answer relationship strategy (QARS) on English as a foreign language (EFL) high school students’ reading comprehension and their perceptions of implementing this reading strategy. An experimental study was conducted with 50 tenth-grade students in a district high school in Can Tho City. The data were collected from the pretest and posttest, questionnaire, and interviews. The findings show that the QARS had positive effects on students’ reading comprehension and that students had positive perceptions of implementing this reading strategy. The findings of this research add to the contemporary literature the value of using the QARS to promote student learning reading.  Article visualizations

EFFECTS OF ADDITIVES, PIGMENT AND FILLER ON PHYSICO- MECHANICAL PROPERTIES AND WEATHER RESISTANCE OF POLYUREA COATINGS

Polyurea coatings are known to provide high hardness, stable color, excellent weather durability and good scratch resistance. Being cured, the coatings can also express heat resistance up to 150 oC, good resistance to acids, oils, as well as other chemicals, high flexibility and strain, excellent adhesion to various substrate materials such as concrete, steel, ceramic, glass and others. In this paper, a polyurea coating has been developed on basis of polyaspartic acid esters and polyisocyanate curing agent, using 3 % nanoclay additive. The physical, mechanical, and chemical properties of the coatings were examined by current technical specifications. The results for developed coating were recorded as follow: 76 in.lb impact strength, 7.5 mm elongation, and 0.7 relative hardness; no blistering was observed after 30 days of immersion in  5 % HCl and 5 % NaOH solutions. Weather resistance of the coatings was also evaluated with full absence of peeling, blistering and high gloss retention after 500 h accelerated UV testing