Magnonics as an emerging nanotechnology offers functionalities beyond current semiconductor technology. Spin waves used in cellular nonlinear networks are expected to speed up technologically, demanding tasks such as image processing and speech recognition at low power consumption. However, efficient coupling to microelectronics poses a vital challenge. Previously developed techniques for spin-wave excitation (for example, by using parametric pumping in a cavity) may not allow for the relevant downscaling or provide only individual point-like sources. Here we demonstrate that a grating coupler of periodically nanostructured magnets provokes multidirectional emission of short-wavelength spin waves with giantly enhanced amplitude compared with a bare microwave antenna. Exploring the dependence on ferromagnetic materials, lattice constants and the applied magnetic field, we find the magnonic grating coupler to be more versatile compared with gratings in photonics and plasmonics. Our results allow one to convert, in particular, straight microwave antennas into omnidirectional emitters for short-wavelength spin waves, which are key to cellular nonlinear networks and integrated magnonics.

Wave control in the solid state has opened new avenues in modern information technology. Surface-acoustic-wave-based devices are found as mass market products in 100 millions of cellular phones. Spin waves (magnons) would offer a boost in today's data handling and security implementations, i.e., image processing and speech recognition. However, nanomagnonic devices realized so far suffer from the relatively short damping length in the metallic ferromagnets amounting to a few 10 micrometers typically. Here we demonstrate that nm-thick YIG films overcome the damping chasm. Using a conventional coplanar waveguide we excite a large series of short-wavelength spin waves (SWs). From the data we estimate a macroscopic of damping length of about 600 micrometers. The intrinsic damping parameter suggests even a record value about 1 mm allowing for magnonics-based nanotechnology with ultra-low damping. In addition, SWs at large wave vector are found to exhibit the non-reciprocal properties relevant for new concepts in nanoscale SW-based logics. We expect our results to provide the basis for coherent data processing with SWs at GHz rates and in large arrays of cellular magnetic arrays, thereby boosting the envisioned image processing and speech recognition.

Seven decades after the discovery of collective spin excitations in microwave-irradiated ferromagnets, there has been a rebirth of magnonics. However, magnetic nanodevices will enable smart GHz-to-THz devices at low power consumption only, if such spin waves (magnons) are generated and manipulated on the sub-100 nm scale. Here we show how magnons with a wavelength of a few 10 nm are exploited by combining the functionality of insulating yttrium iron garnet and nanodisks from different ferromagnets. We demonstrate magnonic devices at wavelengths of 88 nm written/read by conventional coplanar waveguides. Our microwave-to-magnon transducers are reconfigurable and thereby provide additional functionalities. The results pave the way for a multi-functional GHz technology with unprecedented miniaturization exploiting nanoscale wavelengths that are otherwise relevant for soft X-rays. Nanomagnonics integrated with broadband microwave circuitry offer applications that are wide ranging, from nanoscale microwave components to nonlinear data processing, image reconstruction and wave-based logic.

The work we present in this paper initiated the formal study of fractional hedonic games, coalition formation games in which the utility of a player is the average value he ascribes to the members of his coalition. Among other settings, this covers situations in which players only distinguish between friends and non-friends and desire to be in a coalition in which the fraction of friends is maximal. Fractional hedonic games thus not only constitute a natural class of succinctly representable coalition formation games, but also provide an interesting framework for network clustering. We propose a number of conditions under which the core of fractional hedonic games is non-empty and provide algorithms for computing a core stable outcome. By contrast, we show that the core may be empty in other cases, and that it is computationally hard in general to decide non-emptiness of the core. the partitions in the strict core, and give a polynomial time algorithm to compute a unique finest partition in the strict core.• We discuss how computing desirable outcomes in fractional hedonic games provides an interesting game-theoretic perspective to community detection [see, e.g., Fortunato, 2010, Newman, 2004 and network clustering. 2 RELATED WORKFractional hedonic games are related to additively separable hedonic games [see, e.g., , Olsen, 2009, Sung and Dimitrov, 2010. In both fractional hedonic games and additively separable hedonic games, each player ascribes a cardinal value to every other player. In additively separable hedonic games, utility in a coalition is derived by adding the values for the other players. By contrast, in fractional hedonic games, utility in a coalition is derived by adding the values for the other players and then dividing the sum by the total number of players in the coalition. Although conceptually, additively separable and fractional hedonic games are similar, their formal properties are quite different. As neither of the two models is obviously superior, this shows how slight modeling decisions may affect the formal analysis. For example, in unweighted and undirected graphs, the grand coalition is trivially core stable for additively separable hedonic games.On the other hand, this is not the case for fractional hedonic games. 3 A fractional hedonic game approach to social networks with only non-negative weights may help detect like-minded and densely connected communities. In comparison, when the network only has non-negative weights for the edges, any reasonable solution for the corresponding additively separable hedonic game returns the grand coalition, which is not informative. The difference between additively separable and fractional hedonic games is reminiscent of some issues in population ethics (see, e.g., Arrhenius et al., 2017), which concerns the evaluation of states of the world with different numbers of individuals alive. Two prominent principles in population ethics are total utilitarianism and average utilitarianism. The former claims that a state of the world is better than another...

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