A fourth derivative test for exponential sums

We give an upper bound for the exponential $\sum_{m=1}^M \exp( 2i\pi f (m))$ in terms of $M$ and $\lambda$, where $\lambda$ is a small positive number which denotes the size of the fourth derivative of the real valued function $f$. The classical van der Corput's exponent 1/14 is improved into 1/13 by reducing the problem to a mean square value theorem for triple exponential sums

‘This restless enemy of all fertility’: exploring paradigms of coastal dune management in Western Europe over the last 700 years

Drifting sand has inundated settlements and damaged agricultural land along the coasts of Western Europe for the last 700 years. The need to control sand migration has been an important driver of the management of coastal sand dunes and here we analyse original archival materials to provide new insights into historically changing coastal dune management practices. Records of coastal sand movement in Denmark, The Netherlands, Britain, Ireland and France were reviewed and three distinct management approaches were identified. The ways in which these approaches have played out in space and time were examined with particular reference to records from landed estates in Britain and Ireland. We demonstrate how historical evidence can be used to inform contemporary debates on dune management strategy and practice. We propose a new place-based approach to the future management of coastal dunes that can incorporate both expert and locally produced ‘knowledges’ and that is underpinned by an understanding of how both natural forces and human interventions have shaped these dune landscapes over time

A fourth derivative test for exponential sums

International audienceWe give an upper bound for the exponential $\sum_{m=1}^M \exp( 2iπf (m)) in terms of M and λ, where λ is a small positive number which denotes the size of the fourth derivative of the real valued function f. The classical van der Corput's exponent 1/14 is improved into 1/13 by reducing the problem to a mean square value theorem for triple exponential sums

Etude analytique du fonctionnement des moteurs à réluctance alimentés à fréquence variable

In switched reluctance motors fed by a constant voltage source (like a battery) at high frequencies, the current becomes unpredictable and often cannot reach a given reference value, because of the variation of the inductances with the rotor position ; the “motional” m.m.f. generates commutation troubles which increase with the frequency. An optimal control as well as an approximate design of the motor require a quick and simple calculation of currents, powers and losses ; now, in principle, the non-linear electrical equation needs a numerical resolution, whose results cannot be extrapolated. By linearizing this equation by intervals, the method proposed here allows to express analytically, in any case, the phase currents, the torque and the copper losses, when the feeding voltage itself is constant by intervals. The model neglects saturation, but a simple adjustment of the inductance (chosen ad libitum) allows to deal with it. The calculation is immediate and perfectly accurate as long as the machine parameters themselves are well defined. Some results are given as examples for two usual feeding modes.Dans les machines à réluctance alimentées à haute fréquence par une source à tension constante, comme une batterie, le courant varie de manière difficilement prévisible, à cause de la variation des inductances avec la position du rotor, et souvent ne parvient pas à s'établir à une valeur de consigne imposée ; la f.é.m. “motionnelle” engendre des difficultés de communication qui s'aggravent avec l'augmentation de fréquence jusqu'à empêcher le fonctionnement. Tant pour optimiser la commande que pour dimensionner approximativement un moteur ; on doit pouvoir calculer simplement et rapidement le courant et la puissance ; or l'équation électrique, non linéaire, doit en principe être résolue numériquement et les résultats ne sont pratiquement pas extrapolables. En linéarisant par intervalles cette équation, la méthode proposée ici permer d'exprimer analytiquement et dans tous les cas les courants de phase, la puissance fournie et les pertes Joule, lorsque la tension aux bornes de l'enroulement est constante par morceaux. Le modèle utilisé néglige la saturation ; mais il est possible de tenir compte de celle-ci par des ajustements, facilement calculables, de la courbe d'inductance, quelle que soit son allure. Les calculs sont immédiats et parfaitement précis pour autant que les paramètres soient bien définis. Quelques résultats sont donnés à titre d'exemple, pour deux modes d'alimentation usuels