510 Mathematik
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We prove that the projectivized strata of differentials are not contained in pointed Brill-Noether divisors, with only a few exceptions. For a generic element in a stratum of differentials, we show that many of the associated pointed Brill-Noether loci are of expected dimension. We use our results to study the Auel-Haburcak Conjecture: We obtain new non-containments between maximal Brill-Noether loci in Mg. Our results regarding quadratic differentials imply that the quadratic strata in genus 6 are uniruled.
Bounded rationality is one crucial component in human behaviours. It plays a key role in the typical collective behaviour of evacuation, in which heterogeneous information can lead to deviations from optimal choices. In this study, we propose a framework of deep learning to extract a key dynamical parameter that drives crowd evacuation behaviour in a cellular automaton (CA) model. On simulation data sets of a replica dynamic CA model, trained deep convolution neural networks (CNNs) can accurately predict dynamics from multiple frames of images. The dynamical parameter could be regarded as a factor describing the optimality of path-choosing decisions in evacuation behaviour. In addition, it should be noted that the performance of this method is robust to incomplete images, in which the information loss caused by cutting images does not hinder the feasibility of the method. Moreover, this framework provides us with a platform to quantitatively measure the optimal strategy in evacuation, and this approach can be extended to other well-designed crowd behaviour experiments.
The development of epilepsy (epileptogenesis) involves a complex interplay of neuronal and immune processes. Here, we present a first-of-its-kind mathematical model to better understand the relationships among these processes. Our model describes the interaction between neuroinflammation, blood-brain barrier disruption, neuronal loss, circuit remodeling, and seizures. Formulated as a system of nonlinear differential equations, the model reproduces the available data from three animal models. The model successfully describes characteristic features of epileptogenesis such as its paradoxically long timescales (up to decades) despite short and transient injuries or the existence of qualitatively different outcomes for varying injury intensity. In line with the concept of degeneracy, our simulations reveal multiple routes toward epilepsy with neuronal loss as a sufficient but non-necessary component. Finally, we show that our model allows for in silico predictions of therapeutic strategies, revealing injury-specific therapeutic targets and optimal time windows for intervention.
Changes in the efficacies of synapses are thought to be the neurobiological basis of learning and memory. The efficacy of a synapse depends on its current number of neurotransmitter receptors. Recent experiments have shown that these receptors are highly dynamic, moving back and forth between synapses on time scales of seconds and minutes. This suggests spontaneous fluctuations in synaptic efficacies and a competition of nearby synapses for available receptors. Here we propose a mathematical model of this competition of synapses for neurotransmitter receptors from a local dendritic pool. Using minimal assumptions, the model produces a fast multiplicative scaling behavior of synapses. Furthermore, the model explains a transient form of heterosynaptic plasticity and predicts that its amount is inversely related to the size of the local receptor pool. Overall, our model reveals logistical tradeoffs during the induction of synaptic plasticity due to the rapid exchange of neurotransmitter receptors between synapses.
Changes in the efficacies of synapses are thought to be the neurobiological basis of learning and memory. The efficacy of a synapse depends on its current number of neurotransmitter receptors. Recent experiments have shown that these receptors are highly dynamic, moving back and forth between synapses on time scales of seconds and minutes. This suggests spontaneous fluctuations in synaptic efficacies and a competition of nearby synapses for available receptors. Here we propose a mathematical model of this competition of synapses for neurotransmitter receptors from a local dendritic pool. Using minimal assumptions, the model produces a fast multiplicative scaling behavior of synapses. Furthermore, the model explains a transient form of heterosynaptic plasticity and predicts that its amount is inversely related to the size of the local receptor pool. Overall, our model reveals logistical tradeoffs during the induction of synaptic plasticity due to the rapid exchange of neurotransmitter receptors between synapses.
HbA1c is the gold standard test for monitoring medium/long term glycemia conditions in diabetes care, which is a critical factor in reducing the risk of chronic diabetes complications. Current technologies for measuring HbA1c concentration are invasive and adequate assays are still limited to laboratory-based methods that are not widely available worldwide. The development of a non-invasive diagnostic tool for HbA1c concentration can lead to the decrease of the rate of undiagnosed cases and facilitate early detection in diabetes care. We present a preliminary validation diagnostic study of W-band spectroscopy for detection and monitoring of sustained hyperglycemia, using the HbA1c concentration as reference. A group of 20 patients with type 1 diabetes mellitus and 10 healthy subjects were non-invasively assessed at three different visits over a period of 7 months by a millimeter-wave spectrometer (transmission mode) operating across the full W-band. The relationship between the W-band spectral profile and the HbA1c concentration is studied using longitudinal and non-longitudinal functional data analysis methods. A potential blind discrimination between patients with or without diabetes is obtained, and more importantly, an excellent relation (R-squared = 0.97) between the non-invasive assessment and the HbA1c measure is achieved. Such results support that W-band spectroscopy has great potential for developing a non-invasive diagnostic tool for in-vivo HbA1c concentration monitoring in humans.
Topological phases set themselves apart from other phases since they cannot be understood in terms of the usual Landau theory of phase transitions. This fact, which is a consequence of the property that topological phase transitions can occur without breaking symmetries, is reflected in the complicated form of topological order parameters. While the mathematical classification of phases through homotopy theory is known, an intuition for the relation between phase transitions and changes to the physical system is largely inhibited by the general complexity.
In this thesis we aim to get back some of this intuition by studying the properties of the Chern number (a topological order parameter) in two scenarios. First, we investigate the effect of electronic correlations on topological phases in the Green's function formalism. By developing a statistical method that averages over all possible solutions of the manybody problem, we extract general statements about the shape of the phase diagram and investigate the stability of topological phases with respect to interactions. In addition, we find that in many topological models the local approximation, which is part of many standard methods for solving the manybody lattice model, is able to produce qualitatively correct phase transitions at low to intermediate correlations.
We then extend the statistical method to study the effect of the lattice, where we evaluate possible applications of standard machine learning techniques against our information theoretical approach. We define a measure for the information about particular topological phases encoded in individual lattice parameters, which allows us to construct a qualitative phase diagram that gives a more intuitive understanding of the topological phase.
Finally, we discuss possible applications of our method that could facilitate the discovery of new materials with topological properties.
The Fisher information constitutes a natural measure for the sensitivity of a probability distribution with respect to a set of parameters. An implementation of the stationarity principle for synaptic learning in terms of the Fisher information results in a Hebbian self-limiting learning rule for synaptic plasticity. In the present work, we study the dependence of the solutions to this rule in terms of the moments of the input probability distribution and find a preference for non-Gaussian directions, making it a suitable candidate for independent component analysis (ICA). We confirm in a numerical experiment that a neuron trained under these rules is able to find the independent components in the non-linear bars problem. The specific form of the plasticity rule depends on the transfer function used, becoming a simple cubic polynomial of the membrane potential for the case of the rescaled error function. The cubic learning rule is also an excellent approximation for other transfer functions, as the standard sigmoidal, and can be used to show analytically that the proposed plasticity rules are selective for directions in the space of presynaptic neural activities characterized by a negative excess kurtosis.
The paper will focus on the early texts of Galileo Galilei (1613~1623) and Daniel Bernoulli (1738) as examples of pure combinatorical analysis and perspectively considerations within the mathematical discipline of probability theory. It is argued that Bernoulli's approach needed to be developed further in order to achieve a successful and satisfactory theory of risk. In modern economy the need for a proper definition of a notion of risk is seen and currently discussed within the frame of ISO standards. But as already mentioned this interest is mainly owed to the governmental demands of the Basel II and Solvency standards and therefore an external demand. On the other hand an intrinsic understanding of the meaning of risk, as could be provided by a conclusive theory, could lead to a better success in modelling various risks and help to achieve better prognosis.