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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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.

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.

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Neuromuscular physiology is a vibrant research field that has recently seen exciting advances. Previous publications have focused on thorough analyses of particular aspects of neuromuscular physiology, yet an integration of the various novel findings into a single, comprehensive model is missing. In this article, we provide a unified description of a comprehensive mathematical model of surface electromyographic (EMG) measurements and the corresponding force signal in skeletal muscles, both consolidating and extending the results of previous studies regarding various components of the neuromuscular system. The model comprises motor unit (MU) pool organization, recruitment and rate coding, intracellular action potential generation and the resulting EMG measurements, as well as the generated muscular force during voluntary isometric contractions. Mathematically, it consists of a large number of linear PDEs, ODEs, and various stochastic nonlinear relationships, some of which are solved analytically, others numerically. A parameterization of the electrical and mechanical components of the model is proposed that ensures a physiologically meaningful EMG-force relation in the simulated signals, in particular taking the continuous, size-dependent distribution of MU parameters into account. Moreover, a novel nonlinear transformation of the common drive model input is proposed, which ensures that the model force output equals the desired target force. On a physiological level, this corresponds to adjusting the rate coding model to the force generating capabilities of the simulated muscle, while from a control theoretic point of view, this step is equivalent to an exact linearizing transformation of the controlled neuromuscular system. Finally, an alternative analytical formulation of the EMG model is proposed, which renders the physiological meaning of the model more clear and facilitates a mathematical proof that muscle fibers in this model at no point in time represent a net current source or sink. A consistent description of a complete physiological model as presented here, including thorough justification of model component choices, will facilitate the use of these advanced models in future research. Results of a numerical simulation highlight the model's capability to reproduce many physiological effects observed in experimental measurements, and to produce realistic synthetic data that are useful for the validation of signal processing algorithms.

We provide a real algebraic symbolic-numeric algorithm for computing the real variety V R (I) of an ideal I ⊆ R[x], assuming V R (I) is finite (while V C (I) could be infinite). Our approach uses sets of linear functionals on R[x], vanishing on a given set of polynomials generating I and their prolongations up to a given degree, as well as on polynomials of the real radical ideal R √ I obtained from the kernel of a suitably defined moment matrix assumed to be positive semidefinite and of maximum rank. We formulate a condition on the dimensions of projections of these sets of linear functionals, which serves as stopping criterion for our algorithm; this new criterion is satisfied earlier than the previously used stopping criterion based on a rank condition for moment matrices. This algorithm is based on standard numerical linear algebra routines and semidefinite optimization and combines techniques from previous work of the authors together with an existing algorithm for the complex variety.