Clifford analysis between continuous and discrete

Some decades ago D. Knuth et al. have coined concrete mathematics as the blending of CONtinuous and disCRETE math, taking into account that problems of standard discrete mathematics can often be solved by methods based on continuous mathematics together with a controlled manipulation of mathematical formulas. Of course, it was not a new idea, but due to the ongoing emergence of computer aided algebraic manipulation tools of that time it emphasized their use for elegant solutions of old problems or even the detection of new important relationships. Our aim is to show that the same philosophy can be successfully applied to Clifford Analysis by taking advantages of its inherent non-commutative algebra to obtain results or develop methods that are di erent from other ones. In particular, we determine new binomial sums by using a hypercomplex generating function for a special type of monogenic polynomials and develop an algorithm for the determination of their scalar and vector part which illustrates well the diifferences to the corresponding complex case.The research of the first author was partially supported by the R&D Unit Matemdtica e Aplicagoes (UIMA) of the University of Aveiro, through the Portuguese Foundation for Science and Technology (FCT)

Simulation

Welcome to this graduate course on Discrete-Event Simulation, a hybrid discipline that combines knowledge and techniques from Operations Research (OR) and Computer Science (CS) (Figure 1). Due to the fast and continuous improvements in computer hardware and software, Simulation has become an emergent research area with practical industrial and services applications. Today, most real-world systems are too complex to be modeled and studied by using analytical methods. Instead, numerical methods such as simulation must be employed in order to study the performance of those systems, to gain insight into their internal behavior and to consider alternative (“what-if”) scenarios. Applications of Simulations are widely spread among different knowledge areas, including the performance analysis of computer and telecommunication systems or the optimization of manufacturing and logistics processes. This course introduces concepts and methods for designing, performing and analyzing experiments conducted using a Simulation approach. Among other concepts, this course discusses the proper collection and modeling of input data and system randomness, the generation of random variables to emulate the behavior of the real system, the verification and validation of models, and the analysis of the experimental outputs. FigurePeer ReviewedPostprint (published version

Mathematical Modeling of Metabolic-Genetic Networks

Systems biology deals with the computational and mathematical modeling of complex biological systems. The aim is to understand the big picture of the system’s dynamics rather than the individual parts by integrating different sciences, e.g., mathematics, physics, biology, computer science, and engineering. In biological systems, mathematical models of biochemical networks are necessary for predicting and optimizing the behavior of cells in culture. Different mathematical models have been discussed, such as discrete models, continuous models, and hybrid models. In a discrete model, the biological system assumes discrete values. A continuous model uses a system of differential equations to describe the change of concentrations of substances in a cell over time. A hybrid model combines both discrete and continuous models. The main challenge in continuous models is to find the kinetic parameter values. In this thesis, we build a kinetic model of a metabolic-genetic network introduced in Covert et al., 2001 that mimics a discrete model of regulatory flux balance analysis (rFBA) which is based on steady-state assumptions. The kinetic model we introduce has unknown parameters, so it is necessary to perform parameter estimation techniques. We perform a parameter estimation technique using data sets generated from a simulation of the rFBA model. In nature, many phenomena of interest are high-dimensional and complex. Thus, model reduction is considered a vital topic in systems biology. Model reduction methods are mathematical techniques that aim to represent a high-dimensional, dynamical system by a low-dimensional system that roughly preserves the main features and characteristics of the original system. The idea of model order reduction is to use the reduced-order model instead of the full-order model in the simulation or optimization of the system to reduce the computational effort and the runtime of the simulations. In this thesis, we discuss two different model reduction methods. The first method assumes a time scale separation, i.e., it assumes two time scales, a fast time scale and a slow time scale, where the fast time scale dynamics converge to a quasi-steady state. The second approach, proper orthogonal decomposition, aims at obtaining low-dimensional approximate descriptions of high-dimensional processes while retaining the most important features of the dynamics. We apply these approaches to different biological system models from the BioModels database

The Statistical Physics of Learning Revisited:Typical Learning Curves in Model Scenarios

The exchange of ideas between computer science and statistical physics has advanced the understanding of machine learning and inference significantly. This interdisciplinary approach is currently regaining momentum due to the revived interest in neural networks and deep learning. Methods borrowed from statistical mechanics complement other approaches to the theory of computational and statistical learning. In this brief review, we outline and illustrate some of the basic concepts. We exemplify the role of the statistical physics approach in terms of a particularly important contribution: the computation of typical learning curves in student teacher scenarios of supervised learning. Two, by now classical examples from the literature illustrate the approach: the learning of a linearly separable rule by a perceptron with continuous and with discrete weights, respectively. We address these prototypical problems in terms of the simplifying limit of stochastic training at high formal temperature and obtain the corresponding learning curves.</p