Juana Sendra

Approximate parametrization of plane algebraic curves by linear systems of curves

It is well known that an irreducible algebraic curve is rational (i.e. parametric) if and only if its genus is zero. In this paper, given a tolerance @e>0 and an @e-irreducible algebraic affine plane curve C of proper degree d, we introduce the notion of @e-rationality, and we provide an algorithm to parametrize approximately affine @e-rational plane curves by means of linear systems of (d-2)-degree curves. The algorithm outputs a rational parametrization of a rational curve C@? of degree d which has the same points at infinity as C. Moreover, although we do not provide a theoretical analysis, our empirical analysis shows that C@? and C are close in practice.

An algorithm to parametrize approximately space curves

This is the author’s version of a work that was accepted for publication in Journal of Symbolic Computation. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Symbolic Computation vol. 56 pp. 80-106 (2013). DOI: 10.1016/j.jsc.2013.04.002

A Novel Topology Control Approach to Maintain the Node Degree in Dynamic Wireless Sensor Networks

Topology control is an important technique to improve the connectivity and the reliability of Wireless Sensor Networks (WSNs) by means of adjusting the communication range of wireless sensor nodes. In this paper, a novel Fuzzy-logic Topology Control (FTC) is proposed to achieve any desired average node degree by adaptively changing communication range, thus improving the network connectivity, which is the main target of FTC. FTC is a fully localized control algorithm, and does not rely on location information of neighbors. Instead of designing membership functions and if-then rules for fuzzy-logic controller, FTC is constructed from the training data set to facilitate the design process. FTC is proved to be accurate, stable and has short settling time. In order to compare it with other representative localized algorithms (NONE, FLSS, k-Neighbor and LTRT), FTC is evaluated through extensive simulations. The simulation results show that: firstly, similar to k-Neighbor algorithm, FTC is the best to achieve the desired average node degree as node density varies; secondly, FTC is comparable to FLSS and k-Neighbor in terms of energy-efficiency, but is better than LTRT and NONE; thirdly, FTC has the lowest average maximum communication range than other algorithms, which indicates that the most energy-consuming node in the network consumes the lowest power.

The Influence of Communication Range on Connectivity for Resilient Wireless Sensor Networks Using a Probabilistic Approach

Wireless sensor networks (WSNs) consist of thousands of nodes that need to communicate with each other. However, it is possible that some nodes are isolated from other nodes due to limited communication range. This paper focuses on the influence of communication range on the probability that all nodes are connected under two conditions, respectively: (1) all nodes have the same communication range, and (2) communication range of each node is a random variable. In the former case, this work proves that, for 0 < ε < e − 1 , if the probability of the network being connected is 0.36 ε , by means of increasing communication range by constant C ( ε ) , the probability of network being connected is at least 1 − ε . Explicit function C ( ε ) is given. It turns out that, once the network is connected, it also makes the WSNs resilient against nodes failure. In the latter case, this paper proposes that the network connection probability is modeled as Cox process. The change of network connection probability with respect to distribution parameters and resilience performance is presented. Finally, a method to decide the distribution parameters of node communication range in order to satisfy a given network connection probability is developed.

Conchoid surfaces of rational ruled surfaces

The conchoid surface G of a given surface F with respect to a point O is roughly speaking the surface obtained by increasing the radius function of F with respect to O by a constant d. This paper studies real rational ruled surfaces in this context and proves that their conchoid surfaces possess real rational parameterizations, independently of the position of O. Thus any rational ruled surface F admits a rational radius function r(u,v) with respect to any point in space. Besides the general skew ruled surfaces and examples of low algebraic degree we study ruled surfaces generated by rational motions.

A FIRST APPROACH TOWARDS NORMAL PARAMETRIZATIONS OF ALGEBRAIC SURFACES

In this paper we analyze the problem of deciding the normality (i.e. the surjectivity) of a rational parametrization of a surface . The problem can be approached by means of elimination theory techniques, providing a proper close subset where surjectivity needs to be analyzed. In general, these direct approaches are unfeasible because is very complicated and its elements computationally hard to manipulate. Motivated by this fact, we study ad hoc computational alternative methods that simplifies . For this goal, we introduce the notion of pseudo-normality, a concept that provides necessary conditions for a parametrization for being normal. Also, we provide an algorithm for deciding the pseudo-normality. Finally, we state necessary and sufficient conditions on a pseudo-normal parametrization to be normal. As a consequence, certain types of parametrizations are shown to be always normal. For instance, pseudo-normal polynomial parametrizations are normal. Moreover, for certain class of parametrizations, we derive an algorithm for deciding the normality.

Localized and Energy-Efficient Topology Control in Wireless Sensor Networks Using Fuzzy-Logic Control Approaches

The sensor nodes in the Wireless Sensor Networks (WSNs) are prone to failures due to many reasons, for example, running out of battery or harsh environment deployment; therefore, the WSNs are expected to be able to maintain network connectivity and tolerate certain amount of node failures. By applying fuzzy-logic approach to control the network topology, this paper aims at improving the network connectivity and fault-tolerant capability in response to node failures, while taking into account that the control approach has to be localized and energy efficient. Two fuzzy controllers are proposed in this paper: one is Learning-based Fuzzy-logic Topology Control (LFTC), of which the fuzzy controller is learnt from a training data set; another one is Rules-based Fuzzy-logic Topology Control (RFTC), of which the fuzzy controller is obtained through designing if-then rules and membership functions. Both LFTC and RFTC do not rely on location information, and they are localized. Comparing them with other three representative algorithms (LTRT, List-based, and NONE) through extensive simulations, our two proposed fuzzy controllers have been proved to be very energy efficient to achieve desired node degree and improve the network connectivity when sensor nodes run out of battery or are subject to random attacks.

Conchoid surfaces of spheres

The conchoid of a surface F with respect to given fixed point O is roughly speaking the surface obtained by increasing the radius function with respect to O by a constant. This paper studies conchoid surfaces of spheres and shows that these surfaces admit rational parameterizations. Explicit parameterizations of these surfaces are constructed using the relations to pencils of quadrics in R^3 and R^4. Moreover we point to remarkable geometric properties of these surfaces and their construction.

Resilient Wireless Sensor Networks Using Topology Control: A Review

Wireless sensor networks (WSNs) may be deployed in failure-prone environments, and WSNs nodes easily fail due to unreliable wireless connections, malicious attacks and resource-constrained features. Nevertheless, if WSNs can tolerate at most losing k − 1 nodes while the rest of nodes remain connected, the network is called k − connected. k is one of the most important indicators for WSNs’ self-healing capability. Following a WSN design flow, this paper surveys resilience issues from the topology control and multi-path routing point of view. This paper provides a discussion on transmission and failure models, which have an important impact on research results. Afterwards, this paper reviews theoretical results and representative topology control approaches to guarantee WSNs to be k − connected at three different network deployment stages: pre-deployment, post-deployment and re-deployment. Multi-path routing protocols are discussed, and many NP-complete or NP-hard problems regarding topology control are identified. The challenging open issues are discussed at the end. This paper can serve as a guideline to design resilient WSNs.

On the performance of the approximate parametrization algorithm for curves

In Perez-Diaz et al. (2009) [5], the authors present an algorithm to parametrize approximately @e-rational curves, and they show that the Hausdorff distance, w.r.t. the Euclidean distance, between the input and output curves is finite. In this paper, we analyze this distance for a family of curves randomly generated and we empirically find a reasonable upper bound of the Hausdorff distance between each input and output curve of the family.