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

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

The tumor suppressing roles of tissue structure in cervical cancer development

Indiana University-Purdue University Indianapolis (IUPUI)Cervical cancer is caused by the persistent infection of human papilloma virus (HPV) in the cervix epithelium. Although effective preventative care is available, the widespread nature of infection and the variety of HPV strains unprotected by HPV vaccines necessitate a better understanding of the disease for development of new therapies. A major tumor suppressing mechanism is the inhibition of cell division by tissue structure; however, the underlining molecular circuitry for this regulation remains unclear. Recently, the Yap transcriptional co-activator has emerged as a key growth promoter that mediates contact growth arrest and limits organ size. Thus, we aimed to uncover upstream signals that connect tissue organization to Yap regulation in the inhibition of cervical cancer. Two events that disrupt tissue structure were examined including the loss of the tumor suppressor LKB1 and the expression of the viral oncogene HPV16-E6. We identified that Yap mediates cell growth regulation downstream of both LKB1 and E6. Restoration of LKB1 expression in HeLa cervical cancer cells, which lack this tumor suppressor, or shRNA knockdown of LKB1 in NTERT immortalized normal human dermal keratinocytes, demonstrated that LKB1 promotes Yap phosphorylation, nuclear exclusion, and proteasomal degradation. The ability of phosphorylation-defective Yap mutants to rescue LKB1 phenotypes, such as reduced cell proliferation and cell size, suggest that Yap inhibition contributes to LKB1 tumor suppressor function(s). Interestingly, LKB1’s suppression of Yap activity required neither the canonical Yap kinases, Lats1/2, nor metabolic downstream targets of LKB1, AMPK and mTORC1. Instead, the scaffolding protein NF2 was required for LKB1 to induce a specific actin cytoskeleton structure that associates with Yap suppression. Meanwhile, HPV16-E6 promoted Yap activation in all stages of keratinocyte differentiation. E6 activated the Rap1 small GTPase, which in turn promoted Yap activity. Since Rap1 does not mediate differentiation inhibition caused by E6, E6 may play a role in promoting cell growth through Rap1-Yap activation rather than preventing growth arrest through the disruption of differentiation. Altogether, the LKB1-NF2-Yap and E6-Rap1-Yap pathways represent two examples of a novel phenomenon, whereby the structure of a cell directly influences its gene expression and proliferation