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Concept# Linearity

Summary

In mathematics, the term linear is used in two distinct senses for two different properties:

- linearity of a function (or mapping );
- linearity of a polynomial. An example of a linear function is the function defined by f(x)=(ax,bx) that maps the real line to a line in the Euclidean plane R2 that passes through the origin. An example of a linear polynomial in the variables X, Y and Z is aX+bY+cZ+d.

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MATH-302: Functional analysis I

Concepts de base de l'analyse fonctionnelle linéaire: opérateurs bornés, opérateurs compacts, théorie spectrale pour les opérateurs symétriques et compacts, le théorème de Hahn-Banach, les théorèmes de l'application ouverte et du graphe fermé.

MATH-111(c): Linear Algebra

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications.

MATH-111(f): Linear Algebra

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications.

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In the last years, sub-nanosecond time-resolved particle detectors have been object of research by many companies and institutes since they represent an efficient tool to improve the performance of detecting systems for various applications such as High Energy Physics (HEP) and medical imaging. The work summarized in this thesis focuses on the design, implementation and testing of high-performance electronics in SiGe BiCMOS technology for the development of timing systems such as gamma-photons detectors for PET scanners or Long-Lived Particles (LLPs) detectors for HEP applications. Design solutions and architectures are presented and analysed and their impact on the performance of timing detectors are emphasized. The main contributions of the thesis also include the development of a non-linearity model for the analysis of the impact of mismatches in free-running ring-oscillator based Time-to-Digital Converters (TDCs) and the implementation of various Genetic Algorithms (GAs) for circuit tuning. A comparison of different types of GAs is presented and the impact of their properties on the performance of the circuits to optimize is analysed. This work describes the design solutions adopted for the implementation of two pre-production ASICs for the upgrade of detecting system of the FASER experiment at CERN. One of them was produced to perform an extensive study on different level of integration of the front-end system inside the pixel area. The second represents a smaller version of the final full-reticle ASIC for the FASER experiment. This work reports and highlights the main design challenges related to the implementation of a monolithic 23.2x15.3 mm2 detector. Part of the thesis is focused on the TT-PET project. The latter aims to develop a monolithic pixel detector characterized by a ~30 ps time resolution to be integrated in a small-animal PET scanner. The design of a compact TDC for the TT-PET project is presented. Its architecture is based on a multi-path free-running Ring-Oscillator (RO) featuring a PLL-less event-by-event calibration system. This system is characterized by a ~33 ps Least Significant Bit (LSB) and its compact area, simplicity and power consumption makes this solution particularly suitable for systems in which the integration of many converters is required. Moreover, the aforementioned non-linearity model allowed demonstrating the source of the performance superiority (in terms of linearity) of the presented design. The PLL-less synchronization system was also integrated in the

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We introduce a residual-based a posteriori error estimator for contact problems in two- and three-dimensional linear elasticity, discretized with linear and quadratic finite elements and Nitsche’s method. Efficiency and reliability of the estimator are proved under a saturation assumption. Numerical experiments illustrate the theoretical properties and the good performance of the estimator.

2018Today's vehicle must be efficient in terms of gas (CO2, NOx) emissions and fuel consumption. Due to improvements in material and oil, the continuous variable transmission (CVT) is now making a breakthrough in the automotive market. The CVT decouples the engine from the wheel speed. CVT enables significant fuel gains by shifting the engine operating point for specific power demands. This optimization of the operating point enables a reduction of this fuel consumption. A CVT is constituted of two pulley sheaves, one fixed and the other one movable in its axial direction when subjected to an external axial force, in general hydraulic. The transition from the minimum to the maximum speed ratio is continuous and an infinite numbers of ratio is available between these two limits. An intermediate element (a metallic belt or chain) transmits the power from the input (the primary) to the exit (the secondary) of the CVT or variator. Further improvements of the fuel consumption and gas emission are still required for example by improving the variator efficiency. Increasing hydraulic performance or decreasing mechanical losses by reducing the axial forces are some solutions. The latter method is not without risks. The diminution of the clamping forces increases the slip between pulley sheaves and the intermediate element. If the axial forces decrease too much, high slip values can be reached and cause damage to the pulleys and the intermediate element. Control of the slip is an attractive solution to decrease the clamping forces in order to safely improve the variator efficiency. The objective of this thesis is to understand and model the slip of each pulley and establish analytic tools dedicate to the variator control. The slip study and the theoretical approach of the CVT variator is applied to the slip control of the variator with a chain. The contribution of this work is threefold. Firstly, the slip and the traction coefficient are analyzed for each pulley. The slip analysis of each pulley is then used to define a new slip synthesis as the summation of the slip of each pulley. It is demonstrated that the slip and the traction coefficient are different for each pulley and depend on the speed ratio of the variator. In low ratios, both the secondary pulley and the primary pulley slip, but only the primary reaches macro slip. For middle or higher ratios, only the secondary pulley slips and reaches high values of slip. Experiments show that the pulley with the smallest clamping force limits the system. Secondly, based on kinematics, force equilibrium, elastic deformations of the pulleys and the intermediate element, a detail model of the variator is proposed. The principal results are the estimation of the clamping forces, of the traction curve for each pulley and of the chain efficiency. These results are implemented in a simpler model that describes the variator dynamics. This last model considers the two pulleys and the intermediate element as free bodies. The hydraulic circuit and the actuators, which are important to take into account for control, are also modeled. Thirdly, the new slip synthesis and the results of the dynamic models are applied to the slip control of the variator in order to improve the efficiency. A pole placement law is applied to the actuators to control the flow that enters or exits the pulleys. With this law, the actuators are decoupled and the bandwidth is increased sufficiently for actuators dynamics to be neglected. The primary and the secondary pressures are decoupled and linearized by an input-output feedback linearization. The resulting system is linear and linear control theory can be applied to control the two pressures. The speed ratio is controlled by the primary clamping force. The secondary pressure is chosen as a function of the control mode of the variator: standard mode or slip mode. In standard mode, the intermediate element is overclamped by 30%, whereas in slip mode, the secondary clamping force is set as a function of the desired slip. By controlling the slip at 2%, the mechanical efficiency was increased by more than 2% and the clamping forces reduced by more than 30%. For the slip control, a proportional-integrator law and a model reference adaptive control (MRAC) are presented and the performances compared. The MRAC gives slightly better results.