The Statistical Foundations of Entropy

In the last two decades, the understanding of complex dynamical systems underwent important conceptual shifts. The catalyst was the infusion of new ideas from the theory of critical phenomena (scaling laws, renormalization group, etc.), (multi)fractals and trees, random matrix theory, network theory, and non-Shannonian information theory. The usual Boltzmann–Gibbs statistics were proven to be grossly inadequate in this context. While successful in describing stationary systems characterized by ergodicity or metric transitivity, Boltzmann–Gibbs statistics fail to reproduce the complex statistical behavior of many real-world systems in biology, astrophysics, geology, and the economic and social sciences.The aim of this Special Issue was to extend the state of the art by original contributions that could contribute to an ongoing discussion on the statistical foundations of entropy, with a particular emphasis on non-conventional entropies that go significantly beyond Boltzmann, Gibbs, and Shannon paradigms. The accepted contributions addressed various aspects including information theoretic, thermodynamic and quantum aspects of complex systems and found several important applications of generalized entropies in various systems

Large consequences of quantum coherence in small systems

This thesis is concerned with the theoretical behaviour and interactions of quantum systems. It is composed of three main parts. We begin by investigating the correlations attainable when a bipartite quantum system undergoes unitary dynamics. The correlations are quantified by the quantum mutual information. We fully solve the problem for the smallest system of two qubits and present work towards solving the general case. The optimisation can be applied to thermodynamic scenarios, for example, heat exchange between two quantum systems. More specifically, we find bounds any negative heat flow from cold to hot, which can occur if the systems are initially correlated. We also present related applications such as a generalized collision model approach to thermal equilibrium, and a situation where a global Maxwell demon can play tricks on a local observer by reversing their local arrow of time. Experimental evidence suggests that biology may harness quantum effects to improve the efficiency of some of its processes. One such process is hydrogen transfer, catalysed by an enzyme called soybean lipoxygenase. The observed rates for this reaction strongly indicate that the hydrogen could be tunnelling through the energy barrier. We study this reaction by designing a qualitative model and find that our rates exhibit similar trends to those seen in experiments. The final part of the thesis is concerned with the quantum steering ellipsoid: a faithful, three-dimensional (3D) representation for the state of a two-qubit system. The steering ellipsoid is the set of states that Bob can collapse Alice's qubit to when he performs all possible measurements on his qubit. This formalism leads to numerous new features. We uncover a notion of incomplete steering of a separable state; geometric necessary and sufficient conditions for entanglement and discord; and a volume formula for the ellipsoid that identifies when steering is 3D, giving rise to a new type of correlation called "obesity".Open Acces

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

On Improving Generalization of CNN-Based Image Classification with Delineation Maps Using the CORF Push-Pull Inhibition Operator

Deployed image classification pipelines are typically dependent on the images captured in real-world environments. This means that images might be affected by different sources of perturbations (e.g. sensor noise in low-light environments). The main challenge arises by the fact that image quality directly impacts the reliability and consistency of classification tasks. This challenge has, hence, attracted wide interest within the computer vision communities. We propose a transformation step that attempts to enhance the generalization ability of CNN models in the presence of unseen noise in the test set. Concretely, the delineation maps of given images are determined using the CORF push-pull inhibition operator. Such an operation transforms an input image into a space that is more robust to noise before being processed by a CNN. We evaluated our approach on the Fashion MNIST data set with an AlexNet model. It turned out that the proposed CORF-augmented pipeline achieved comparable results on noise-free images to those of a conventional AlexNet classification model without CORF delineation maps, but it consistently achieved significantly superior performance on test images perturbed with different levels of Gaussian and uniform noise