A Generic Position Based Method for Real Root Isolation of Zero-Dimensional Polynomial Systems

We improve the local generic position method for isolating the real roots of a zero-dimensional bivariate polynomial system with two polynomials and extend the method to general zero-dimensional polynomial systems. The method mainly involves resultant computation and real root isolation of univariate polynomial equations. The roots of the system have a linear univariate representation. The complexity of the method is $\tilde{O}_B(N^{10})$ for the bivariate case, where $N=\max(d,\tau)$, $d$ resp., $\tau$ is an upper bound on the degree, resp., the maximal coefficient bitsize of the input polynomials. The algorithm is certified with probability 1 in the multivariate case. The implementation shows that the method is efficient, especially for bivariate polynomial systems.Comment: 24 pages, 5 figure

Multiplicity Preserving Triangular Set Decomposition of Two Polynomials

In this paper, a multiplicity preserving triangular set decomposition algorithm is proposed for a system of two polynomials. The algorithm decomposes the variety defined by the polynomial system into unmixed components represented by triangular sets, which may have negative multiplicities. In the bivariate case, we give a complete algorithm to decompose the system into multiplicity preserving triangular sets with positive multiplicities. We also analyze the complexity of the algorithm in the bivariate case. We implement our algorithm and show the effectiveness of the method with extensive experiments.Comment: 18 page

Root Isolation of Zero-dimensional Polynomial Systems with Linear Univariate Representation

In this paper, a linear univariate representation for the roots of a zero-dimensional polynomial equation system is presented, where the roots of the equation system are represented as linear combinations of roots of several univariate polynomial equations. The main advantage of this representation is that the precision of the roots can be easily controlled. In fact, based on the linear univariate representation, we can give the exact precisions needed for roots of the univariate equations in order to obtain the roots of the equation system to a given precision. As a consequence, a root isolation algorithm for a zero-dimensional polynomial equation system can be easily derived from its linear univariate representation.Comment: 19 pages,2 figures; MM-Preprint of KLMM, Vol. 29, 92-111, Aug. 201

Fibroblast Growth Factor 10 in Pancreas Development and Pancreatic Cancer

The tenacious prevalence of human pancreatic diseases such as diabetes mellitus and adenocarcinoma has prompted huge research interest in better understanding of pancreatic organogenesis. The plethora of signaling pathways involved in pancreas development is activated in a highly coordinated manner to assure unmitigated development and morphogenesis in vertebrates. Therefore, a complex mesenchymal–epithelial signaling network has been implicated to play a pivotal role in organogenesis through its interactions with other germ layers, specifically the endoderm. The Fibroblast Growth Factor Receptor FGFR2-IIIb splicing isoform (FGFR2b) and its high affinity ligand Fibroblast Growth Factor 10 (FGF10) are expressed in the epithelium and mesenchyme, respectively, and therefore are well positioned to transmit mesenchymal to epithelial signaling. FGF10 is a typical paracrine FGF and chiefly mediates biological responses by activating FGFR2b with heparin/heparan sulfate (HS) as cofactor. A substantial number of studies using genetically engineered mouse models have demonstrated an essential role of FGF10 in the development of many organs and tissues including the pancreas. During mouse embryonic development, FGF10 signaling is crucial for epithelial cell proliferation, maintenance of progenitor cell fate and branching morphogenesis in the pancreas. FGF10 is also implicated in pancreatic cancer, and that overexpression of FGFR2b is associated with metastatic invasion. A thorough understanding of FGF10 signaling machinery and its crosstalk with other pathways in development and pathological states may provide novel opportunities for pancreatic cancer targeted therapy and regenerative medicine

Root Isolation of Zero-dimensional Polynomial Systems with Linear Univariate Representation

Abstract In this paper, a linear univariate representation for the roots of a zero-dimensional polynomial equation system is presented, where the roots of the equation system are represented as linear combinations of roots of several univariate polynomial equations. The main advantage of this representation is that the precision of the roots can be easily controlled. In fact, based on the linear univariate representation, we can give the exact precisions needed for isolating the roots of the univariate equations in order to obtain the roots of the equation system to a given precision. As a consequence, a root isolation algorithm for a zero-dimensional polynomial equation system can be easily derived from its linear univariate representation

An Improved Complexity Bound for Computing the Topology of a Real Algebraic Space Curve

We propose a new algorithm to compute the topology of a real algebraic space curve. The novelties of this algorithm are a new technique to achieve the lifting step which recovers points of the space curve in each plane fiber from several projections and a weaken notion of generic position. As opposed to previous work, our sweep generic position does not require that x-critical points have different x-coordinates. The complexity of achieving this sweep generic position is thus no longer a bottleneck in term of complexity. The bit complexity of our algorithm is O(d^18 + d ^17 t) where d and t bound the degree and the bitsize of the integer coefficients of the defining polynomials of the curve and polylogarithmic factors are ignored. To the best of our knowledge, this improves upon the best currently known results at least by a factor of d 2

Are metal-free pristine carbon nanotubes electrocatalytically active?

Metal-free (i.e., residual metallic impurities-blocked) carbon nanotubes (CNTs) do show electrocatalytic activity for H2 evolution, O2 evolution and O2 reduction reactions (HER, OER & ORR) in alkaline solutions, but their activities strongly depend on the number of walls or inner tubes with a maximum for CNTs with 2–3 walls