Schur functions and their realizations in the slice hyperholomorphic setting

we start the study of Schur analysis in the quaternionic setting using the theory of slice hyperholomorphic functions. The novelty of our approach is that slice hyperholomorphic functions allows to write realizations in terms of a suitable resolvent, the so called S-resolvent operator and to extend several results that hold in the complex case to the quaternionic case. We discuss reproducing kernels, positive definite functions in this setting and we show how they can be obtained in our setting using the extension operator and the slice regular product. We define Schur multipliers, and find their co-isometric realization in terms of the associated de Branges-Rovnyak space

Positive and generalized positive real lemma for slice hyperholomorphic functions

In this paper we prove a quaternionic positive real lemma as well as its generalized version, in case the associated kernel has negative squares for slice hyperholomorphic functions. We consider the case of functions with positive real part in the half space of quaternions with positive real part, as well as the case of (generalized) Schur functions in the open unit ball

Electrolytic pretreatment unit gaseous effluent conditioning

The electrolytic pretreatment of urine is an advanced process that eliminates the need for handling and storing the highly corrosive chemicals that are normally used in water reclamation systems. The electrolytic pretreatment process also converts the organic materials in urine to gases (N2 and O2) that can be used to replenish those lost to space by leakage, venting, and air lock operations. The electrolytic process is more than a pretreatment, since it decreases the urine solids content by approximately one third, thus reducing the load and eventual solids storage requirements of the urine processing system. The evolved gases from the pretreatment step cannot, however, be returned directly to the atmosphere of a spacecraft without first removing several impurities including hydrogen, chlorine, and certain organic compounds. A treatment concept was developed that would decrease the impurities in the gas stream that emanates from an electrolysis unit to levels sufficiently low to allow the conditioned gas stream to be safely discharged to a spacecraft atmosphere. Two methods were experimentally demonstrated that can accomplish the desired cleanup. The bases of the two methods are, repectively: (1) raw urine scrubbing and (2) silica gel sorption

RoboJam: A Musical Mixture Density Network for Collaborative Touchscreen Interaction

RoboJam is a machine-learning system for generating music that assists users of a touchscreen music app by performing responses to their short improvisations. This system uses a recurrent artificial neural network to generate sequences of touchscreen interactions and absolute timings, rather than high-level musical notes. To accomplish this, RoboJam's network uses a mixture density layer to predict appropriate touch interaction locations in space and time. In this paper, we describe the design and implementation of RoboJam's network and how it has been integrated into a touchscreen music app. A preliminary evaluation analyses the system in terms of training, musical generation and user interaction

Regular Moebius transformations of the space of quaternions

Let H be the real algebra of quaternions. The notion of regular function of a quaternionic variable recently presented by G. Gentili and D. C. Struppa developed into a quite rich theory. Several properties of regular quaternionic functions are analogous to those of holomorphic functions of one complex variable, although the diversity of the quaternionic setting introduces new phenomena. This paper studies regular quaternionic transformations. We first find a quaternionic analog to the Casorati-Weierstrass theorem and prove that all regular injective functions from H to itself are affine. In particular, the group Aut(H) of biregular functions on H coincides with the group of regular affine transformations. Inspired by the classical quaternionic linear fractional transformations, we define the regular fractional transformations. We then show that each regular injective function from the Alexandroff compactification of H to itself is a regular fractional transformation. Finally, we study regular Moebius transformations, which map the unit ball B onto itself. All regular bijections from B to itself prove to be regular Moebius transformations.Comment: 12 page

Second fundamental form of the Prym map in the ramified case

In this paper we study the second fundamental form of the Prym map $P_{g,r}: R_{g,r} \rightarrow {\mathcal A}^{\delta}_{g-1+r}$ in the ramified case $r>0$. We give an expression of it in terms of the second fundamental form of the Torelli map of the covering curves. We use this expression to give an upper bound for the dimension of a germ of a totally geodesic submanifold, and hence of a Shimura subvariety of ${\mathcal A}^{\delta}_{g-1+r}$, contained in the Prym locus.Comment: To appear in Galois Covers, Grothendieck-Teichmueller Theory and Dessins d'Enfants - Interactions between Geometry, Topology, Number Theory and Algebra. Springer Proceedings in Mathematics & Statistics. arXiv admin note: text overlap with arXiv:1711.0342

k-Dirac operator and parabolic geometries

The principal group of a Klein geometry has canonical left action on the homogeneous space of the geometry and this action induces action on the spaces of sections of vector bundles over the homogeneous space. This paper is about construction of differential operators invariant with respect to the induced action of the principal group of a particular type of parabolic geometry. These operators form sequences which are related to the minimal resolutions of the k-Dirac operators studied in Clifford analysis