No Arabic abstract
The recent COSY-11 collaboration measurement of the two-proton correlation function in the pp -> ppeta reaction, reported at this meeting [1], arouse some interest in a simple theoretical description of the correlation function. In these notes we present a pedagogical introduction to the practical methods that can be used for calculating the correlation function.
We discuss a general combinatorial framework for operator ordering problems by applying it to the normal ordering of the powers and exponential of the boson number operator. The solution of the problem is given in terms of Bell and Stirling numbers enumerating partitions of a set. This framework reveals several inherent relations between ordering problems and combinatorial objects, and displays the analytical background to Wicks theorem. The methodology can be straightforwardly generalized from the simple example given herein to a wide class of operators.
Much of our understanding of critical phenomena is based on the notion of Renormalization Group (RG), but the actual determination of its fixed points is usually based on approximations and truncations, and predictions of physical quantities are often of limited accuracy. The RG fixed points can be however given a fully rigorous and non-perturbative characterization, and this is what is presented here in a model of symplectic fermions with a nonlocal (long-range) kinetic term depending on a parameter $varepsilon$ and a quartic interaction. We identify the Banach space of interactions, which the fixed point belongs to, and we determine it via a convergent approximation scheme. The Banach space is not limited to relevant interactions, but it contains all possible irrelevant terms with short-ranged kernels, decaying like a stretched exponential at large distances. As the model shares a number of features in common with $phi^4$ or Ising models, the result can be used as a benchmark to test the validity of truncations and approximations in RG studies. The analysis is based on results coming from Constructive RG to which we provide a tutorial and self-contained introduction. In addition, we prove that the fixed point is analytic in $varepsilon$, a somewhat surprising fact relying on the fermionic nature of the problem.
Based on the high statistics data from the pp -> ppX reaction measured by the COSY-11 collaboration [1] we have derived a two-proton correlation function for the production of the ppeta and pp+pions systems. The measured correlation function normalized to the value simulated for a point-like source was compared with a theoretical prediction in order to estimate the size of the reaction volume.
This paper investigates the role of size in biological organisms. More specifically, how the energy demand, expressed by the metabolic rate, changes according to the mass of an organism. Empirical evidence suggests a power-law relation between mass and metabolic rate, namely allometric law. For vascular organisms, the exponent $beta$ of this power-law is smaller than one, which implies scaling economy; that is, the greater the organism is, the lesser energy per cell it demands. However, the numerical value of this exponent is a theme of an extensive debate and a central issue in comparative physiology. It is presented in this work some empirical data and a detailed discussion about the most successful theories to explain these issues. A historical perspective is also shown, beginning with the first empirical insights in the sec. 19 about scaling properties in biology, passing through the two more important theories that explain the scaling properties quantitatively. Firstly, the Rubner model, that consider organism surface area and heat dissipation to derive $beta = 2/3$. Secondly, the West-Brown-Enquist theory, that explains such scaling properties as a consequence of the hierarchical and fractal nutrient distribution network, deriving $beta = 3/4$.
This paper tries to give a gentle introduction to deep learning in medical image processing, proceeding from theoretical foundations to applications. We first discuss general reasons for the popularity of deep learning, including several major breakthroughs in computer science. Next, we start reviewing the fundamental basics of the perceptron and neural networks, along with some fundamental theory that is often omitted. Doing so allows us to understand the reasons for the rise of deep learning in many application domains. Obviously medical image processing is one of these areas which has been largely affected by this rapid progress, in particular in image detection and recognition, image segmentation, image registration, and computer-aided diagnosis. There are also recent trends in physical simulation, modelling, and reconstruction that have led to astonishing results. Yet, some of these approaches neglect prior knowledge and hence bear the risk of producing implausible results. These apparent weaknesses highlight current limitations of deep learning. However, we also briefly discuss promising approaches that might be able to resolve these problems in the future.