Polynomials and degrees of maps in real normed algebras

summary:Let $\mathcal{A}$ be the algebra of quaternions $\mathbb{H}$ or octonions $\mathbb{O}$. In this manuscript an elementary proof is given, based on ideas of Cauchy and D'Alembert, of the fact that an ordinary polynomial $f(t) \in \mathcal{A} [t]$ has a root in $\mathcal{A}$. As a consequence, the Jacobian determinant $\lvert J(f)\rvert$ is always non-negative in $\mathcal{A}$. Moreover, using the idea of the topological degree we show that a regular polynomial $g(t)$ over $\mathcal{A}$ has also a root in $\mathcal{A}$. Finally, utilizing multiplication ($*$) in $\mathcal{A}$, we prove various results on the topological degree of products of maps. In particular, if $S$ is the unit sphere in $\mathcal{A}$ and $h_1, h_2\colon S \to S$ are smooth maps, it is shown that $\deg (h_1 * h_2)=\deg (h_1) + \deg (h_2)$

Construction of rational curves with rational arc lengths by direct integration

A methodology for the construction of rational curves with rational arc length functions, by direct integration of hodographs, is developed. For a hodograph of the form r′(ξ)=(u2(ξ)−v2(ξ),2u(ξ)v(ξ))/w2(ξ), where w(ξ) is a monic polynomial defined by prescribed simple roots, we identify conditions on the polynomials u(ξ) and v(ξ) which ensure that integration of r′(ξ) produces a rational curve with a rational arc length function s(ξ). The method is illustrated by computed examples, and a generalization to spatial rational curves is also briefly discussed. The results are also compared to existing theory, based upon the dual form of rational Pythagorean-hodograph curves, and it is shown that direct integration produces simple low-degree curves which otherwise require a symbolic factorization to identify and cancel common factors among the curve homogeneous coordinates

Towards an Accessible Personal Health Record

Patient empowerment frameworks, including personal health records (PHR), actively engage technology empowered citizens in their healthcare. Particularly today, with the current increase of chronic diseases, the high growth rate of the elderly and disabled populations and at the same time the much higher cross-border patient mobility,such systems may prove to be lifesaving, cost effective and time saving. Currently, there are many different online applications promoted as being functional, user-friendly and detailed enough to provide a complete and accurate summary of an individual’s medical history. However, it seems that most of the Web services available do not fully adhere to well known accessibility standards, such as those promoted by the W3C, thus turning them away from people with disability and elderly people, who most probably need them most. Additionally, support for mobile devices introduces additional obstacles to users with disability when trying to operate such services. This paper presents fundamental (design for all) guidelines for the successful implementation of an accessible ePHR service that can be operated by any patient including people with disabilities irrespective of the device they use to access this service

On the asymptotic and practical complexity of solving bivariate systems over the reals

This paper is concerned with exact real solving of well-constrained, bivariate polynomial systems. The main problem is to isolate all common real roots in rational rectangles, and to determine their intersection multiplicities. We present three algorithms and analyze their asymptotic bit complexity, obtaining a bound of \sOB(N^{14}) for the purely projection-based method, and \sOB(N^{12}) for two subresultant-based methods: this notation ignores polylogarithmic factors, where $N$ bounds the degree and the bitsize of the polynomials. The previous record bound was \sOB(N^{14}). Our main tool is signed subresultant sequences. We exploit recent advances on the complexity of univariate root isolation, and extend them to sign evaluation of bivariate polynomials over two algebraic numbers, and real root counting for polynomials over an extension field. Our algorithms apply to the problem of simultaneous inequalities; they also compute the topology of real plane algebraic curves in \sOB(N^{12}), whereas the previous bound was \sOB(N^{14}). All algorithms have been implemented in MAPLE, in conjunction with numeric filtering. We compare them against FGB/RS, system solvers from SYNAPS, and MAPLE libraries INSULATE and TOP, which compute curve topology. Our software is among the most robust, and its runtimes are comparable, or within a small constant factor, with respect to the C/C++ libraries. Key words: real solving, polynomial systems, complexity, MAPLE softwareComment: 17 pages, 4 algorithms, 1 table, and 1 figure with 2 sub-figure

Comparative analysis of time-frequency methods estimating the time-varying microstructure of sleep EEG spindles

Proceedings of the Information Technology Applications in Biomedicine, Ioannina - Epirus, Greece, October 26-28, 2006Parameter estimation for an assumed sleep EEG spindle model (AM-FM signal) is performed by using four time-frequency analysis methods. Results from simulated as well as from real data are presented. In simulated data, the Hilbert Transform-based method has the lowest average percentage error but produces considerable signal distortion. The Complex Demodulation and the Matching Pursuit-based methods have error rates below 10%, but the Matching Pursuit-based method produces considerable signal distortion as well. The Wavelet Transform-based method has the poorest performance. In real data, all methods produce reasonable parameter values. However, the Hilbert Transform and the Matching Pursuitbased methods may not be applicable for sleep spindles shorter than about 0.8 sec. Matching Pursuit-based curve fitting is utilized as part of the parameter estimation process

On a theorem of H. Hopf

A simple proof of a theorem of H. Hopf [1], via Morse theory, is given

BrainNetVis: An Open-Access Tool to Effectively Quantify and Visualize Brain Networks

This paper presents BrainNetVis, a tool which serves brain network modelling and visualization, by providing both quantitative and qualitative network measures of brain interconnectivity. It emphasizes the needs that led to the creation of this tool by presenting similar works in the field and by describing how our tool contributes to the existing scenery. It also describes the methods used for the calculation of the graph metrics (global network metrics and vertex metrics), which carry the brain network information. To make the methods clear and understandable, we use an exemplar dataset throughout the paper, on which the calculations and the visualizations are performed. This dataset consists of an alcoholic and a control group of subjects

Linear and Nonlinear Synchronization Analysis and Visualization during Altered States of Consciousness

Non

Ontology-driven monitoring of patient's vital signs enabling personalized medical detection and alert

C