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This work is about random measures stationary with respect to a possibly non-transitive group action. It contains chapters on Palm Theory, the Mass-Transport Principle and Ergodic Theory for such random measures. The thesis ends with discussions of several new models in Stochastic Geometry (Cox Delauney mosaics, isometry stationary random partitions on Riemannian manifolds). These make crucial use of the previously developed techniques and objects.

Random measure, Palm theory, mass-transport principle, ergodic theory, Stochastic Geometry

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This Special Issue of the journal Entropy, titled “Information Geometry I”, contains a collection of 17 papers concerning the foundations and applications of information geometry. Based on a geometrical interpretation of probability, information geometry has become a rich mathematical field employing the methods of differential geometry. It has numerous applications to data science, physics, and neuroscience. Presenting original research, yet written in an accessible, tutorial style, this collection of papers will be useful for scientists who are new to the field, while providing an excellent reference for the more experienced researcher. Several papers are written by authorities in the field, and topics cover the foundations of information geometry, as well as applications to statistics, Bayesian inference, machine learning, complex systems, physics, and neuroscience.

Markov random fields --- information theory --- Fisher information --- entropy --- maximum pseudo-likelihood estimation --- Bezout matrix --- Sylvester matrix --- tensor Sylvester matrix --- Stein equation --- Vandermonde matrix --- stationary process --- matrix resultant --- Fisher information matrix --- information geometry --- dually flat structure --- decomposable divergence --- (?,?)

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Cryptography lies at the heart of most technologies deployed today for secure communications. At the same time, mathematics lies at the heart of cryptography, as cryptographic constructions are based on algebraic scenarios ruled by group or number theoretical laws. Understanding the involved algebraic structures is, thus, essential to design robust cryptographic schemes. This Special Issue is concerned with the interplay between group theory, symmetry and cryptography. The book highlights four exciting areas of research in which these fields intertwine: post-quantum cryptography, coding theory, computational group theory and symmetric cryptography. The articles presented demonstrate the relevance of rigorously analyzing the computational hardness of the mathematical problems used as a base for cryptographic constructions. For instance, decoding problems related to algebraic codes and rewriting problems in non-abelian groups are explored with cryptographic applications in mind. New results on the algebraic properties or symmetric cryptographic tools are also presented, moving ahead in the understanding of their security properties. In addition, post-quantum constructions for digital signatures and key exchange are explored in this Special Issue, exemplifying how (and how not) group theory may be used for developing robust cryptographic tools to withstand quantum attacks.

cryptography --- non-commutative cryptography --- one-way functions --- NP-Completeness --- key agreement protocol --- group theory --- symmetry --- Engel words --- alternating group --- WalnutDSA --- digital signatures --- post-quantum cryptography --- cryptanalysis --- braid groups --- algorithms in groups --- group-based cryptography --- Reed–Solomon codes --- key equation --- Berlekamp–Massey algorithm --- Sugiyama et al. algorithm --- euclidean algorithm --- numerical semigroup --- Weierstrass semigroup --- semigroup ideal --- error-correcting code --- algebraic-geometry code --- lightweight cryptography --- permutation group --- block cipher --- generalized self-shrinking generator --- t-modified self-shrinking generator --- pseudo-random number generator --- statistical randomness tests --- cryptography --- pseudorandom permutation --- block cipher --- ideal cipher model --- beyond birthday bound --- provable security --- group key establishment --- group theory --- provable security --- protocol compiler

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The last few years have been characterized by a tremendous development of quantum information and probability and their applications, including quantum computing, quantum cryptography, and quantum random generators. In spite of the successful development of quantum technology, its foundational basis is still not concrete and contains a few sandy and shaky slices. Quantum random generators are one of the most promising outputs of the recent quantum information revolution. Therefore, it is very important to reconsider the foundational basis of this project, starting with the notion of irreducible quantum randomness. Quantum probabilities present a powerful tool to model uncertainty. Interpretations of quantum probability and foundational meaning of its basic tools, starting with the Born rule, are among the topics which will be covered by this issue. Recently, quantum probability has started to play an important role in a few areas of research outside quantum physics—in particular, quantum probabilistic treatment of problems of theory of decision making under uncertainty. Such studies are also among the topics of this issue.

quantum logic --- groups --- partially defined algebras --- quasigroups --- viable cultures --- quantum information theory --- bit commitment --- protocol --- entropy --- entanglement --- orthogonality --- quantum computation --- Gram–Schmidt process --- quantum probability --- potentiality --- complementarity --- uncertainty relations --- Copenhagen interpretation --- indefiniteness --- indeterminism --- causation --- randomness --- quantum information --- quantum dynamics --- entanglement --- algebra --- causality --- geometry --- probability --- quantum information theory --- realism --- reality --- entropy --- correlations --- qubits --- probability representation --- Bayes’ formula --- quantum entanglement --- three-qubit random states --- entanglement classes --- entanglement polytope --- anisotropic invariants --- quantum random number --- vacuum state --- maximization of quantum conditional min-entropy --- quantum logics --- quantum probability --- holistic semantics --- epistemic operations --- Bell inequalities --- algorithmic complexity --- Borel normality --- Bayesian inference --- model selection --- random numbers --- quantum-like models --- operational approach --- information interpretation of quantum theory --- social laser --- social energy --- quantum information field --- social atom --- Bose–Einstein statistics --- bandwagon effect --- social thermodynamics --- resonator of social laser --- master equation for socio-information excitations --- quantum contextuality --- Kochen–Specker sets --- MMP hypergraphs --- Greechie diagrams --- quantum foundations --- probability --- irreducible randomness --- random number generators --- quantum technology --- entanglement --- quantum-like models for social stochasticity --- contextuality

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The Internet of Things (IoT) has attracted much attention from society, industry and academia as a promising technology that can enhance day to day activities, and the creation of new business models, products and services, and serve as a broad source of research topics and ideas. A future digital society is envisioned, composed of numerous wireless connected sensors and devices. Driven by huge demand, the massive IoT (mIoT) or massive machine type communication (mMTC) has been identified as one of the three main communication scenarios for 5G. In addition to connectivity, computing and storage and data management are also long-standing issues for low-cost devices and sensors. The book is a collection of outstanding technical research and industrial papers covering new research results, with a wide range of features within the 5G-and-beyond framework. It provides a range of discussions of the major research challenges and achievements within this topic.

narrowband --- IoT --- PHY --- NB-IoT --- MAC --- deployment --- survey --- mMTC --- 5G --- grant-free --- scheduling --- URLLC --- ultra-reliable and low-latency communications --- 5G --- deterministic --- time-critical --- reliability --- latency --- aperiodic traffic --- Industry 4.0 --- 5G --- mMTC --- IoT --- CSMA --- SINR --- throughput --- polynomial interpolation --- 5G --- liquid detection --- radio propagation --- dielectric constant --- WCI --- congestion --- estimation --- irregular repetition slotted ALOHA --- medium access control --- random access --- successive interference cancellation --- 5G --- internet of things --- mMTC --- eMBB --- stochastic geometry --- resource allocation --- Internet of Things --- Industry 4.0 --- Internet of Things --- Industrial Internet of Things --- Cyber Physical System --- cloud computing --- fog computing --- edge computing --- smart devices --- smart factory --- industrial automation --- M2M --- heterogeneous networks --- non-orthogonal multiple access --- energy efficiency --- MU association --- power control --- D2D communication --- 5G --- sensor network --- sensor --- end-to-end delay --- USRP --- distributed mechanism --- Raspberry Pi