For an ideal I ⊆ R [x] given by a set of generators, a new semidefinite characterization of its real radical I(V R (I)) is presented, provided it is zero-dimensional (even if I is not). Moreover we propose an algorithm using numerical linear algebra and semidefinite optimization techniques, to compute all (finitely many) points of the real variety V R (I) as well as a set of generators of the real radical ideal. The latter is obtained in the form of a border or Gröbner basis. The algorithm is based on moment relaxations and, in contrast to other existing methods, it exploits the real algebraic nature of the problem right from the beginning and avoids the computation of complex components.

In this paper, we describe new methods to compute the radical (resp. real radical) of an ideal, assuming it complex (resp. real) variety is finte. The aim is to combine approaches for solving a system of polynomial equations with dual methods which involve moment matrices and semi-definite programming. While the border basis algorithms of [17] are efficient and numerically stable for computing complex roots, algorithms based on moment matrices [12] allow the incorporation of additional polynomials, e.g., to restrict the computation to real roots or to eliminate multiple solutions. The proposed algorithm can be used to compute a border basis of the input ideal and, as opposed to other approaches, it can also compute the quotient structure of the (real) radical ideal directly, i.e., without prior algebraic techniques such as Gröbner bases. It thus combines the strength of existing algorithms and provides a unified treatment for the computation of border bases for the ideal, the radical ideal and the real radical ideal.J.B. Lasserre, LAAS,

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Abstract. In this paper we propose a unified methodology for computing the setWe show how moment matrices, defined in terms of a given set of generators of the ideal I, can be used to (numerically) find not only the real variety V R (I), as shown in the authors' previous work, but also the complex variety V C (I), thus leading to a unified treatment of the algebraic and real algebraic problem. In contrast to the real algebraic version of the algorithm, the complex analogue only uses basic numerical linear algebra because it does not require positive semidefiniteness of the moment matrix and so avoids semidefinite programming techniques. The links between these algorithms and other numerical algebraic methods are outlined and their stopping criteria are related.

Background Inspiratory patient effort under assisted mechanical ventilation is an important quantity for assessing patient–ventilator interaction and recognizing over and under assistance. An established clinical standard is respiratory muscle pressure $$\textit{P}_{\mathrm{mus}}$$ P mus , derived from esophageal pressure ($$\textit{P}_{\mathrm{es}}$$ P es ), which requires the correct placement and calibration of an esophageal balloon catheter. Surface electromyography (sEMG) of the respiratory muscles represents a promising and straightforward alternative technique, enabling non-invasive monitoring of patient activity. Methods A prospective observational study was conducted with patients under assisted mechanical ventilation, who were scheduled for elective bronchoscopy. Airway flow and pressure, esophageal/gastric pressures and sEMG of the diaphragm and intercostal muscles were recorded at four levels of pressure support ventilation. Patient efforts were quantified via the $$\textit{P}_{\mathrm{mus}}$$ P mus -time product ($${\mathrm{PTP}}_{\mathrm{mus}}$$ PTP mus ), the transdiaphragmatic pressure-time product ($${\mathrm{PTP}}_{\mathrm{di}}$$ PTP di ) and the EMG-time products (ETP) of the two sEMG channels. To improve the signal-to-noise ratio, a method for automatically selecting the more informative of the sEMG channels was investigated. Correlation between ETP and $${\mathrm{PTP}}_{\mathrm{mus}}$$ PTP mus was assessed by determining a neuromechanical conversion factor $$\textit{K}_{\mathrm{EMG}}$$ K EMG between the two quantities. Moreover, it was investigated whether this scalar can be reliably determined from airway pressure during occlusion maneuvers, thus allowing to quantify inspiratory effort based solely on sEMG measurements. Results In total, 62 patients with heterogeneous pulmonary diseases were enrolled in the study, 43 of which were included in the data analysis. The ETP of the two sEMG channels was well correlated with $${\mathrm{PTP}}_{\mathrm{mus}}$$ PTP mus ($$\textit{r}={0.79\pm 0.25}$$ r = 0.79 ± 0.25 and $$\textit{r}={0.84\pm 0.16}$$ r = 0.84 ± 0.16 for diaphragm and intercostal recordings, respectively). The proposed automatic channel selection method improved correlation with $${\mathrm{PTP}}_{\mathrm{mus}}$$ PTP mus ($$\textit{r}={0.87\pm 0.09}$$ r = 0.87 ± 0.09 ). The neuromechanical conversion factor obtained by fitting ETP to $${\mathrm{PTP}}_{\mathrm{mus}}$$ PTP mus varied widely between patients ($$\textit{K}_{\mathrm{EMG}}= {4.32\pm 3.73}\,{\hbox {cm}\hbox {H}_{2}\hbox {O}/\upmu \hbox {V}}$$ K EMG = 4.32 ± 3.73 cm 2 O / μ V ) and was highly correlated with the scalar determined during occlusions ($$\textit{r}={0.95}$$ r = 0.95 , $$\textit{p}<{.001}$$ p < . 001 ). The occlusion-based method for deriving $${\mathrm{PTP}}_{\mathrm{mus}}$$ PTP mus from ETP showed a breath-wise deviation to $${\mathrm{PTP}}_{\mathrm{mus}}$$ PTP mus of $${0.43\pm 1.73}\,{\hbox {cm}\hbox {H}_{2}\hbox {O}\,\hbox {s}}$$ 0.43 ± 1.73 cm 2 O s across all datasets. Conclusion These results support the use of surface electromyography as a non-invasive alternative for monitoring breath-by-breath inspiratory effort of patients under assisted mechanical ventilation.