Statistical mechanics of non equilibrium matter: from minimal models to morphogen gradients

Living systems are by definition far from thermodynamic equilibrium, a condition that can be maintained only at the cost of a continuous injection of energy at the microscale, e.g. via cellular metabolic processes, and dissipation into the surrounding environment. The absence of thermodynamic equilibrium, formalised in the breaking of the global detailed balance condition, allows for a wealth of exotic and often counterintuitive phenomena. Our understanding of the capabilities and limitations of living matter has been greatly informed by thermodynamic approaches, which have to be generalised with respect to their traditional counterparts in order to deal with systems subject to strong random fluctuations. The resulting toolkit of stochastic thermodynamics, in particular the concept of entropy production, gives us a quantitative handle on the degree of "non-equilibriumness" of such stochastic processes. Recently, stochastic thermodynamics has benefitted from cross-contamination with the field-theoretic literature and the techniques developed in the latter for the study of collective behaviour have opened the doors to the thermodynamic characterisation of increasingly complex systems. Starting from minimal mathematical models of single active particles and moving up across scales to the level of morphogenetic processes in real organisms (in particular, the formation of morphogen gradients), this thesis contributes to laying the foundations for a bridge between physical understanding and biological insight. While the focus is here on generic mechanisms and on the development of theoretical tools, the applicability to specific experimental scenarios will be pointed out where relevant.Open Acces

Biological Protein Patterning Systems across the Domains of Life: from Experiments to Modelling

Distinct localisation of macromolecular structures relative to cell shape is a common feature across the domains of life. One mechanism for achieving spatiotemporal intracellular organisation is the Turing reaction-diffusion system (e.g. Min system in the bacterium Escherichia coli controlling in cell division). In this thesis, I explore potential Turing systems in archaea and eukaryotes as well as the effects of subdiffusion. Recently, a MinD homologue, MinD4, in the archaeon Haloferax volcanii was found to form a dynamic spatiotemporal pattern that is distinct from E. coli in its localisation and function. I investigate all four archaeal Min paralogue systems in H. volcanii by identifying four putative MinD activator proteins based on their genomic location and show that they alter motility but do not control MinD4 patterning. Additionally, one of these proteins shows remarkably fast dynamic motion with speeds comparable to eukaryotic molecular motors, while its function appears to be to control motility via interaction with the archaellum. In metazoa, neurons are highly specialised cells whose functions rely on the proper segregation of proteins to the axonal and somatodendritic compartments. These compartments are bounded by a structure called the axon initial segment (AIS) which is precisely positioned in the proximal axonal region during early neuronal development. How neurons control these self-organised localisations is poorly understood. Using a top-down analysis of developing neurons in vitro, I show that the AIS lies at the nodal plane of the first non-homogeneous spatial harmonic of the neuron shape while a key axonal protein, Tau, is distributed with a concentration that matches the same harmonic. These results are consistent with an underlying Turing patterning system which remains to be identified. The complex intracellular environment often gives rise to the subdiffusive dynamics of molecules that may affect patterning. To simulate the subdiffusive transport of biopolymers, I develop a stochastic simulation algorithm based on the continuous time random walk framework, which is then applied to a model of a dimeric molecular motor. This provides insight into the effects of subdiffusion on motor dynamics, where subdiffusion reduces motor speed while increasing the stall force. Overall, this thesis makes progress towards understanding intracellular patterning systems in different organisms, across the domains of life

Three Risky Decades: A Time for Econophysics?

Our Special Issue we publish at a turning point, which we have not dealt with since World War II. The interconnected long-term global shocks such as the coronavirus pandemic, the war in Ukraine, and catastrophic climate change have imposed significant humanitary, socio-economic, political, and environmental restrictions on the globalization process and all aspects of economic and social life including the existence of individual people. The planet is trapped—the current situation seems to be the prelude to an apocalypse whose long-term effects we will have for decades. Therefore, it urgently requires a concept of the planet's survival to be built—only on this basis can the conditions for its development be created. The Special Issue gives evidence of the state of econophysics before the current situation. Therefore, it can provide excellent econophysics or an inter-and cross-disciplinary starting point of a rational approach to a new era

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

Mathematical Economics

This book is devoted to the application of fractional calculus in economics to describe processes with memory and non-locality. Fractional calculus is a branch of mathematics that studies the properties of differential and integral operators that are characterized by real or complex orders. Fractional calculus methods are powerful tools for describing the processes and systems with memory and nonlocality. Recently, fractional integro-differential equations have been used to describe a wide class of economical processes with power law memory and spatial nonlocality. Generalizations of basic economic concepts and notions the economic processes with memory were proposed. New mathematical models with continuous time are proposed to describe economic dynamics with long memory. This book is a collection of articles reflecting the latest mathematical and conceptual developments in mathematical economics with memory and non-locality based on applications of fractional calculus

Model selection in a multi-hypothesis test setting: applications in financial econometrics

In this thesis, we investigate model selection in a general setting and perform several exercises in financial econometrics. We present the multi-hypothesis testing (MHT) framework, with which we design different type of model comparisons. We distinguish between test of model performance significance, of relative and absolute model performance and apply our framework to market risk forecasting model, to latent factor jump-diffusion models employed for the estimation of the statistical measure of an equity index, as well as to equity option pricing models. We develop original tests and, with regard to the proper exercise of model selection from an initial battery of models without any reference to a benchmark model, we combine the MHT approach with the model confidence set (MCS) to deliver a novel test of model comparison that is performed along with the established version of the MCS, as well as with an alternative simplified new MCS test that are detailed in the course of this work. We collect empirical evidence concerning model comparison in several subjects. With respect to market risk forecasting models, we have found that models capturing volatility clustering or targeting directly an auto-correlated conditional distribution percentile, perform better than the target model set and in particular, better than the historical simulation, widely employed by practitioners, and better than the so called RiskMetrics model. With respect to the equity index data dynamics, we have found that the popular affine jump-diffusion model requires a CEV augmentation to perform appropriately and that those models are slightly overperformed by an alternative stochastic volatility model, characterised by stochastic hazard with high frequency small jumps. The test performed over a large model set employed in the option pricing exercise points to a wide similarity of the results obtained by the many model specifications of the superior exponential volatility model, therefore suggesting a more careful adjustment of the model complexity. The model selection framework has proven very flexible in dealing with the varied collection of statistical problems. In particular, our main contribution represented by the generalised MHT based MCS test provides a method for model selection that is robust to finite sample distribution and that has the advantage of an adjustable tolerance for false rejections, allowing conservative to aggressive testing profiles