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We derive an explicit k-dependence in Wm,p error estimates for Pk Lagrange finite elements. Two laws of probability are established to measure the relative accuracy between Pk1 and Pk2 finite eleme...

This paper deals with the applications of data mining techniques in the evaluation of numerical solutions of Vlasov–Maxwell models. This is part of the topic of characterizing the model and approximation errors via learning techniques. We give two examples of application. The first one aims at comparing two Vlasov–Maxwell approximate models. In the second one, a scheme based on data mining techniques is proposed to characterize the errors between a P 1 and a P 2 finite element Particle-In-Cell approach. Beyond these examples, this original approach should operate in all cases where intricate numerical simulations like for the Vlasov–Maxwell equations take a central part.

In this Note, we propose a new methodology based on exploratory data mining techniques to evaluate the errors due to the description of a given real system. First, we decompose this description error into four types of sources. Then, we construct databases of the entire information produced by different numerical approximation methods, to assess and compare the significant differences between these methods, using techniques like decision trees, Kohonenʼs cards, or neural networks. As an example, we characterize specific states of the real system for which we can locally appreciate the accuracy between two kinds of finite elements methods. In this case, this allowed us to precise the classical Bramble–Hilbert theorem that gives a global error estimate, whereas our approach gives a local error estimate.

This paper proposes a statistical and probabilistic approach to compare and analyze the errors of two different approximation methods. We introduce the principle of numerical uncertainty in such a process, and we illustrate it by considering the discretization difference between two different approximation orders, e.g., first and second order Lagrangian finite element. Then, we derive a probabilistic approach to define and to qualify equivalent results. We illustrate our approach on a model problem on which we built the two above mentioned finite element approximations. We consider some variables as physical “predictors”, and we characterize how they influence the odds of the approximation methods to be locally “same order accurate”.

We introduce a new paraxial asymptotic model to approximate the Vlasov–Maxwell equations. This formulation is fourth order accurate in a parameter η which denotes the ratio between the transverse characteristic velocity of the beam and the speed of light. The model is interesting because it is simpler than the complete Vlasov–Maxwell equations, and is an accurate approximation of them. This model should give an accurate, fast and easy to implement numerical method of solution.

We report a mathematical model that describes the growth of superficial bladder cancer and the effect thereupon of immunotherapy based on the administration of Bacillus Calmette-Guerin (BCG) combined or not with interleukin-2 (IL-2). Intravesical instillations of BCG performed after surgical removal of tumors represents an established treatment with approximately 50% success rate. So far, attempts to improve this efficiency have not led to essential changes. However, convincing clinical results have been reported on the combination of IL-2 to BCG, even though this is still not applied in current practice. The present model provides insights into the dynamical outcomes arising in the bladder from the interactions of immune cells with tumor cells in the course of BCG therapy associated or not with IL-2. Specifically, from the simulations performed using seven ordinary and non-linear differential equations we obtained indications on the conditions that would result in successful bladder cancer treatment. We show that immune cells -effector lymphocytes and antigen-presenting cells-expand and reach a sustainable plateau under BCG treatment, which may account for its beneficial effect, resulting from inflammatory "side-effects" which eliminate residual or eventual newly arising tumor cells, providing thus protection from further cancer development. We find, however, that IL-2 does not actually potentiate the effect of BCG as regards tumor cell eradication. Hence, associating both under the conditions simulated should not result in more efficient treatment of bladder cancer patients.

The aim of this work is to present a method to find and analyze maximum propulsive efficiency kinematics for a birdlike flapping-wing unmanned aerial vehicle using multiobjective evolutionary optimization and data-mining tools. For the sake of clarity and simplicity, simple geometry (rectangular wings with the same profile along the span) and simple kinematics (symmetrical harmonic dihedral motion) are used. In addition, it is assumed that the birdlike aerial vehicle (for which the span and surface area are, respectively, 1 m and 0.15 m 2 ) is in horizontal motion at low cruise speed (6 m/s). The aerodynamic performances of the flapping-wing vehicle are evaluated with a semiempirical flight physics model and the problem is solved using an efficient multiobjective evolutionary algorithm called ∈-MOEA. Groups of attractive solutions are defined on the Pareto surface, and the most efficient solutions within these groups are characterized. Given the high dimensionality of the Pareto surface in the kinematic parameters space, data-mining techniques are used to conduct the study. First, it is shown that these groups can be qualified versus the whole Pareto surface by accurate mathematical relations on the kinematic parameters. Second, the inner structure of each group is studied and highly accurate mathematical relations are found on the optimized parameters describing the most efficient solutions.

Background Obstructive Sleep Apnoea Syndrome (OSAS) and insomnia are common pathologies sharing a high comorbidity. CPAP is a cumbersome treatment. Yet, CPAP compliance must remain optimal in order to reverse excessive daytime sleepiness and prevent the cardiovascular consequences of OSAS. But chronic insomnia could negatively affect CPAP compliance. Objective To assess the consequences of insomnia symptoms on long-term CPAP use. Methods A prospective study was conducted on 148 OSAS patients (RDI=39.0±21.3/h), age=54.8±11.8years, BMI=29.1±6.3kg/m 2 , Epworth Score=12.2±5.4, on CPAP. Using the Insomnia Severity Index (ISI) as an indicator of insomnia (ISI⩾14=moderate to severe insomnia) and baseline data (anthropometric data, sleeping medication intakes, CPAP compliance, Epworth, Pittsburgh Sleep Quality and ISI scores, polygraphic recording data), Data Mining analysis identified the major rules explaining the features "High" or "Low ISI" and "High" or "Low Use" in the groups defined, according to the median values of the ISI and the 6th month-compliance, respectively. Results Median ISI was 15 and median 6th month-compliance was 4.38h/night. Moderate to severe insomnia complaint was found in 50% of patients. In the "High" and "Low ISI," the 6th month-compliance was not significantly different (3.7±2.3 vs 4.2±2.3h/night). In the classification models of compliance, the ISI was not a predictor of CPAP rejection or of long-term use, the predictor for explaining CPAP abandonment being the RDI, and the predictor of the 6th month-compliance being the one month-compliance. Conclusion Insomnia symptoms were highly prevalent in OSAS patients, but had no impact on CPAP rejection or on long-term compliance.

We propose a novel approach that consists in using data mining techniques to perform a sensitivity analysis of approximate models. We give here the example of asymptotic solutions to Vlasov–Maxwell equations, obtained with a paraxial model for relativistic short beams. This new heuristic approach offers new potential applications to treat numerical solutions to mathematical models.

Measuring the complexity of physical systems has been traditionally a problem in numerous engineering applications. Lin [Entropy 10 (1) (2008) 1–5] showed that the structural complexity is related to other properties of a solid such as symmetry and its stability over time. In Ratsaby [Entropy 10 (1) (2008) 6–14] a model was introduced which defines the complexity of a solid structure not by a qualitative notion of entropy but by an algorithmic notion of description complexity. According to the model, a dynamic structure in a random surrounding acts as an interfering entity that deforms randomness. In the current Note we report on the results of an empirical study that analyzes the output response of a simulated elastic beam subjected to a field of external random forces input. The relationship between the complexity of the system and the stochasticity of the output is shown to support this model and is a first indication that solids act similar to algorithmic selection rules.

Important features which are considered in media research are media efficiency and return on investments. To appreciate the powerful level of several channels of communication, one has to firstly consider the basic measure of the impact for a given advertising campaign, that is to say, the individual distribution of exposure and therefore the estimation of the global exposure of a given target. Having pointed out the known models, we will show how one can release part of hypotheses, under which these models are founded to describe new behaviors of media exposure. This will be particularly the case by introducing the Markov chains to describe an unsteady dependency between several issues or TV spots regarding a given advertising campaign. As a consequence we will show that the new models estimate more precisely the net cover of a given campaign composed by N advertising spots.

One of the objectives of geomarketing is to determine the optimal sitting of points of sale for a brand in a given territory. The aim of the optimisation is to develop sales, the flow of potential clients to the site, and to improve all the marketing indicators, which are used to describe the success of a brand within its market. Whatever the marketing objective adopted, there are many variables within the geolocal success criteria mentioned above. These can be socio-demographic or socio-professional variables, characteristic of structural changes in the population from one catchment area to another. However, when considering this wemust include the impact of other players in a given market, whose competitive influence is crucial to understanding each one’s activity. At the end of the 19th century, several specialists in population migration developed mathematical models in order to quantify the influence of attractive geographic points – urban centres in geography and points of sale in geomarket...

Background and purpose Despite correct treatment with positive airway pressure (PAP), obstructive sleep apnea (OSA) patients sometimes remain subjectively somnolent. The reliability of the Epworth Sleepiness Scale (ESS) has been established for healthy subjects and patients under stable conditions; the ESS may eventually vary among treated OSA patients, biasing the results of a cross-sectional analysis of persisting sleepiness. The objective of this study was to depict the evolution of subjective vigilance under treatment using an index of ESS variability (ΔESS). Methods In 80 OSA patients (apnea–hypopnea index [AHI]=54±26/h), initially somnolent (ESS=15±3) and treated with auto-titrating PAP (APAP) (oxyhaemoglobin desaturation index 3% [ODIapap]=3.4±2.2/h; daily APAP use=5.3±1.5h) during 434±73 days, ESS scores were regularly collected four times every 109±36 days. DESS was calculated and data mining methods (Segmentation and Decision Tree) were used to determine homogeneous groups according to the evolution of ESS scores. Results When assessed cross-sectionally, 14–25% of the subjects were recognized as somnolent, depending on the moment when ESS was administered. Using data mining methods, three groups were clearly identifiable: two without residual somnolence – group 1, n =38 (47%), with high ΔESS=−2.9±0.8, baseline ESS=16.3±3.3, AHI=58.5±26.1/h, mean ESSapap=5.1±2.4 and group 2, n =31 (39%), with low ΔESS=−1.1±0.5, baseline ESS=13.2±1.4, AHI=53±27.3/h, mean ESSapap=8.8±1.9; and one with persisting sleepiness; group 3, n =11 (14%), with low ΔESS=−0.3±0.8, baseline ESS=16.3±3, AHI=38.7±10.8/h, mean ESSapap=14.1±1.9. Compliance to PAP was high and comparable in the three groups. Age and body mass index (BMI) did not differ. Conclusion Data mining methods helped to identify 14% of subjects with persisting sleepiness. Validation needs to be done on a larger population in order to determine predictive rules.

International audience; In awake humans, breathing depends on automatic brainstem pattern generators. It is also heavily influenced by cortical networks. For example, functional magnetic resonance imaging and electroencephalographic data show that the supplementary motor area becomes active when breathing is made difficult by inspiratory mechanical loads like resistances or threshold valves, which is associated with perceived respiratory discomfort. We hypothesized that manipulating the excitability of the supplementary motor area with repetitive transcranial magnetic stimulation would modify the breathing pattern response to an experimental inspiratory load and possibly respiratory discomfort. Seven subjects (three men, age 25 ± 4) were studied. Breathing pattern and respiratory discomfort during inspiratory loading were described before and after conditioning the supplementary motor area with repetitive stimulation, using an excitatory paradigm (5 Hz stimulation), an inhibitory paradigm, or sham stimulation. No significant change in breathing pattern during loading was observed after sham conditioning. Excitatory conditioning shortened inspiratory time (p = 0.001), decreased tidal volume (p = 0.016), and decreased ventilation (p = 0.003), as corroborated by an increased end-tidal expired carbon dioxide (p = 0.013). Inhibitory conditioning did not affect ventilation, but lengthened expiratory time (p = 0.031). Respiratory discomfort was mild under baseline conditions, and unchanged after conditioning of the supplementary motor area. This is the first study to show that repetitive transcranial magnetic stimulation conditioning of the cerebral cortex can alter breathing pattern. A 5 Hz conditioning protocol, known to enhance corticophrenic excitability, can reduce the amount of hyperventilation induced by inspiratory threshold loading. Further studies are needed to determine whether and under what circumstances rTMS can have an effect on dyspnoea.

Exact solutions of Navier–Stokes equations enable to describe nice features of atmospherical flows as tornadoes. For this very particular and singular kind of fluid motions, the flows are very often modelled by conical solutions. Famous authors as Burgers (1948) [ J.M. Burgers, A mathematical model illustrating the theory of turbulence, Advan. Appl. Mech. 1 (1948) 197–199 ], Rott (1958) [ N. Rott, On the viscous core of a line vortex, Z. Angew. Math. Phys. 96 (1958) 543–553 ], Serrin (1972) [J. Serrin, The swirling vortex, Phil. Trans. Roy. Soc. Lond. A 271 (1972) 320–360; J. Serrin, The swirling vortex, Phil. Trans. Roy. Soc. Lond. A 271 (1972) 357–358] and Goldshtik and Shtern (1990) [M. Goldshtik, V. Shtern, J. Fluid Mech. 218 (1990) 483–508], improved greatly research. Most of the time, except Serrin’s model, these models applied to tornadoes, rather produced fields of velocity, with orders of magnitudes which are way too low to be appropriate for describing thunderstorms observed in meteorology. Moreover, these models do not include any explicit mechanisms which take into account the potential actions of dust particles often present in tornadoes. Here, we do suggest a new way of modelling the mature phase of a tornado by considering two swirling cells separated by a intermediate cone, which position results by the equilibrium of normal stresses exerted on the two sides of the cone. This choice of modelling is motivated by considering two kinds of flows, inside and outside the cone, to get significant and different characteristics of the associated resulting vortices. This equilibrium condition, completed by the discontinuity of the tangential stresses on the cone, leads to realistic magnitude of the velocity field, i.e., about 102 m s−1. This discontinuity is motivated by several observations which show the ejection of dust particles along a specific cone direction. At our modelling scale, we do want to model the integral of the dragging forces, resulting from the motion of those particles inside the fluid, by a discontinuity of the tangential stresses on the cone which split the flows into two separate cells.

Several mathematical models applied to tornadoes consist of exact and axisymmetric solutions of the steady and incompressible Navier–Stokes equations. These models studied by Serrin,9 Goldshtik and Shtern8 describe families of fluid motions vanishing at the ground and are restricted not to develop a source nor a sink near the vortex line. Therefore, Serrin showed that the flow patterns of the resulting velocity field may have some realistic characteristics to model the mature phase of the lifetime of a tornado, in comparison with atmospheric observations. On the other hand, no reason has been given to motivate the restriction of the absence of a source/sink vortex line. Therefore, we present here the construction and the analysis of a fluid motion driven by the vertical shear near the ground, the rate of the azimuthal rotation and by the intensity of a central source/sink line. We prove the local existence and uniqueness of a family of fluid motions, leading to the genesis of such source/sink lines inside a non-rotating updraft which does not develop, before perturbation, a source nor a sink.

These last ten years, observations of atmospheric tornadoes knew a very large development. The main goal of this kind of exploration is to determine singular conditions that are responsible of the genesis of this variety of thunderstorms. These investigations made it possible to conclude that one of the essential mechanisms of this phenomenon is due to a thermomechanical process, resulting of a strong gradient of temperature, driving the fluid between the ground and the atmosphere. In this paper, we present exact solutions of Navier-Stokes equations linked with the energy equation, which modeled this type of thermomechanical process. Our model consists of stationary and axisymmetrical flows of a viscous fluid in a half-space, limited by a horizontal plane, in the Boussinesq approximation. These solutions, which satisfy the no-slip condition at the ground, belong to a family of fields of velocity and temperature that depends on five parameters. The mathematical problem we consider is derived by the streamfunction-vorticity method in order to substitute the classical formulation of Navier-Stokes equations

Mathematical models applied to tornadoes describe these kinds of flows as an axisymmetric fluid motion which is restricted for not developing a source or a sink near the vortex line. Here, we propose the genesis of a family of a source/sink line into a singular updraft which can modeled one of the step of the genesis of a tornado. This model consists of a three-parameter family of fluid motions, satisfying the steady and incompressible Navier–Stokes equations, which vanish at the ground. We establish the local existence and uniqueness for these fields, at the neighborhood of a nonrotating singular updraft.

Several mathematical models applied to tornadoes consist of exact and axisymmetric solutions of the steady and incompressible Navier—Stokes equations. These models studied by J. Serrin [5] and M.A. Gol'dshtik and V.N. Sthern (1995) describe families of fluid motions vanishing at the ground and are restricted for not developing neither a source nor a sink near the vortex line. Therefore, J. Serrin showed that the flow patterns of the resulting velocity field may have some realistic characteristics to model the mature phase of the lifetime of a tornado, in comparison with atmospheric observations. On another hand, no reason has been explicated to motivate the absence of a source/sink vortex line. We present here a result of local existence and uniqueness of a family of fluid motions, leading to the genesis of such lines inside a non-rotating updraft which do not develops neither a source nor a sink.

A homogeneous and isotropic elastic solid with given Lame parameters\(\lambda >0\) and \(\mu >0\) occupies a region \(\varOmega \) represented by a bounded open subset of the 3D space with coordinates \((x_1,x_2,x_3)\) (see Fig. 5.1).