On Kapteyn-Kummer Series' Integral Form

In this short research note we obtain double definite integral expressions for the Kapteyn type series built by Kummer's $M$ (or confluent hypergeometric ${}_1F_1$) functions. These kind of series unify in natural way the similar fashion results for Neumann-, Schl\"omilch- and Kapteyn-Bessel series recently established by Pog\'any, S\"uli, Baricz and Jankov Ma\v{s}irevi\'c

Necessary condition for the existence of an intertwining operator and classification of transmutations on its basis

The authors study second-order ordinary differential operators with functional coefficients for all derivatives and the Volterra integral operator with a definite kernel. Results of the paper establish a hyperbolic equation and additional conditions that allow one to construct a kernel according to the OD

History of Mathematics: Models and Visualization in the Mathematical and Physical Sciences

This workshop brought together historians of mathematics and science as well as mathematicians to explore important historical developments connected with models and visual elements in the mathematical and physical sciences. It addressed the larger question of what has been meant by a model, a notion that has seldom been subjected to careful historical study. Most of the talks dealt with case studies from the period 1800 to 1950 that covered a number of analytical, geometrical, mechanical, astronomical, and physical phenomena. The workshop also considered the role of visual thinking as a component of mathematical creativity and understanding

Storage and Control of a Single Photon Wave Packet

Quantum optical memory implies the storage of quantum state of light in an atomic ensemble and its retrieval at the later moment of time on demand. It is one of the key elements of both quantum communication and quantum computing. Two types of quantum optical memory techniques have been developed in the last decade. The first one is based on an optimal temporal shaping of the amplitude of a strong coherent control field, forming along with the quantum field a three-level configuration in atomic medium (such as EIT and Raman quantum memories). The second one is based on photon echo mechanism [such as atomic frequency comb (AFC) and gradient echo memory (GEM)]. Each method has its advantages and disadvantages, but in general, an experiment-friendly, reliable, high speed, low loss, broad band quantum storage of a single-photon wave packet with large efficiency and fidelity remains a very challenging task. Here we propose two new quantum optical storage techniques to resolve some of the difficulties and to introduce more controllability over the single-photon processing. The first method is based on phase matching control in Raman configuration (via the modulation in time of the control field's refractive index, propagation direction, and/or carrier frequency chirp). The second method is based on (continues or discrete) spatial frequency chirp of a control field. In order to overcome some general problems inherent to light-atoms interfaces, we propose also a new quantum interface based on γ -ray-nuclear transitions, which looks promising for quantum information processing

Anisotropy Across Fields and Scales

This open access book focuses on processing, modeling, and visualization of anisotropy information, which are often addressed by employing sophisticated mathematical constructs such as tensors and other higher-order descriptors. It also discusses adaptations of such constructs to problems encountered in seemingly dissimilar areas of medical imaging, physical sciences, and engineering. Featuring original research contributions as well as insightful reviews for scientists interested in handling anisotropy information, it covers topics such as pertinent geometric and algebraic properties of tensors and tensor fields, challenges faced in processing and visualizing different types of data, statistical techniques for data processing, and specific applications like mapping white-matter fiber tracts in the brain. The book helps readers grasp the current challenges in the field and provides information on the techniques devised to address them. Further, it facilitates the transfer of knowledge between different disciplines in order to advance the research frontiers in these areas. This multidisciplinary book presents, in part, the outcomes of the seventh in a series of Dagstuhl seminars devoted to visualization and processing of tensor fields and higher-order descriptors, which was held in Dagstuhl, Germany, on October 28–November 2, 2018