Concept# Continuous function

Summary

In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity.
Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general c

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This thesis proposes a framework for haptic devices interacting with virtual environments. As application of the theoretical results a haptic device for the surgical training of hysteroscopy, a gynecologic intervention, is presented. Surgery simulators, which can be described as 'flight simulators for surgeons', comprise a virtual reality for modeling and visualization of organs, and the haptic device: While the surgeon interactively follows the scene on the computer screen, he perceives the contact forces at the tool handle of the surgical instrument. The haptic device tracks the position of the surgery tool and transforms the virtual contact forces to real forces, which can be actually felt by the surgeon. This thesis treats haptic devices with application to surgery simulators for minimally-invasive surgery. Haptic interfaces are complex mechatronic systems and show inherent trade-offs between haptic realism and system stability. Human factors play an important role, as the human operator can both stabilize or destabilize the system. The haptic perception, which can vary between individuals, as a criterion for haptic rendering quality is still not formalized. The optimization of haptic interfaces, within focus of research since 15 years ago, has remained a complex task. However, with look at the emerging applications for haptics, it has become an even more important topic. For this reason it is necessary to integrate mechanical design, control design, mechatronic elements and human factors in a unifying approach. This thesis highlights the principal trade-offs of haptic interfaces and works out design guidelines for new developments and quantitative information on where and how to optimize haptic systems. As application example a haptic interface for the simulation of hysteroscopy, a gynecologic intervention on the female uterus, has been realized as prototype. The device takes the characteristics of hysteroscopy into account and introduces features like insertion and complete removal of the surgery tool from the haptic interface. A design methodology is proposed, which combines aspects of theoretical kinematics based on Lie-Groups with integrated product development – a systematic approach for engineering tasks. The result, a novel kinematic design, has been realized as haptic interface prototype. The well-known affine parameterization for all stabilizing controllers is shown to be a well-suited tool for haptic control synthesis. It provides insights into the control trade-offs and allows to integrate human factors at controller design stage. The resulting controller, demonstrated by experimental results, can reduce parasitic effects (friction, device dynamics) close to and below the human perception threshhold, thus making the device transparent within the specified bandwidth and suitable for real-time implementation. For numerical controller representation the δ-operator is discussed. It unifies continuous and discrete time domain and has beneficial numerical properties. As there is still no standardized procedure for technical performance evaluation of haptic interfaces methods proposed in literature are summarized. In the near future a large market for haptic technology can be expected with applications from surgery-assist devices and drive-by-wire interfaces in automobiles to entertainment, which highlights the relevance of the treated topic.

Let G a locally compact group, H a closed subgroup and 1 < p < ∞. It's well-known that the restriction of the functions from G to H is a surjective linear contraction from Ap(G) onto Ap(H). We prove, when H is amenable, that every element in Ap(H) with compact support can be extended to an element in Ap(G) of which we can check norm and support. This result is already known in the case of normal subgroups and also for compact subgroups. We obtain the existence of a quasi-coretract in the BAN category, as a substitute of a morphism ΓH such that ResH ◦ ΓH = idAp(H). Indeed, for an amenable subsgroup, the morphism ΓH, a priori, doesn't exist. So, we construct a net of morphismes in BAN from Ap(H) into Ap(G), that converge to idAp(H) for the strong operator's topology on Ap(H) (that's for us the notion of a quasi-coretract in BAN). Furthermore, if H is metrizable and σ-compact we obtain, more precisely, a sequence. Moreover, our approach allows us to extend to the non-abelian case some works of H. Reiter and C. Herz concerning the spectral synthesis of bounded uniformly continuous functions. My results are new even for the Fourier algebra.

The ability to notice erroneous behavior is crucial for effective training. Within the framework of neuroprosthetics, numerous studies in electroencephalography (EEG) confirm the existence of neural correlates when humans perceive the erroneous actions of the device. Subsequently, the decoding of this correlate has been used to correct the erroneous behavior performed by the agent or to tune the behavioral strategy of the agent, among others. However, a main limitation of current approaches is that the actions of the agent were discretized, thus restraining the usability of such systems. The main objective of this PhD study is to study, and decode, the neural correlates of error evaluation under continuous trajectories performed by external agents; and to use this decoding to tune the continuous behavior of the agent for individual users. To accomplish this goal, two essential questions will be investigated: (i) whether it is possible to infer individual preference under continuous state-action scenarios, and (ii) how to create a reliable decoding pipeline in a continuous fashion. Results obtained during the first year of the PhD have confirmed the existence of such correlates under continuous motions of a robotic arm. Furthermore, such correlates encode individual preferences, indicating that the neural prosthesis can be also customized for individual users, which may play an important rule to increase the quality of brain-computer based assistance. This property not only will increase the level of perceived assistance provided by a brain-computer interface, but also may facilitate embodiment of the brain-controlled device.