A reinterpretation of set differential equations as differential equations in a Banach space

Set differential equations are usually formulated in terms of the Hukuhara differential, which implies heavy restrictions for the nature of a solution. We propose to reformulate set differential equations as ordinary differential equations in a Banach space by identifying the convex and compact subsets of $\R^d$ with their support functions. Using this representation, we demonstrate how existence and uniqueness results can be applied to set differential equations. We provide a simple example, which can be treated in support function representation, but not in the Hukuhara setting

Sample-efficient estimation of entanglement entropy through supervised learning

We explore a supervised machine learning approach to estimate the entanglement entropy of multi-qubit systems from few experimental samples. We put a particular focus on estimating both aleatoric and epistemic uncertainty of the network's estimate and benchmark against the best known conventional estimation algorithms. For states that are contained in the training distribution, we observe convergence in a regime of sample sizes in which the baseline method fails to give correct estimates, while extrapolation only seems possible for regions close to the training regime. As a further application of our method, highly relevant for quantum simulation experiments, we estimate the quantum mutual information for non-unitary evolution by training our model on different noise strengths.Comment: 5 + 1 pages, 4 figure

Lorentz Boost Networks: Autonomous Physics-Inspired Feature Engineering

We present a two-stage neural network architecture that enables a fully autonomous and comprehensive characterization of collision events by exclusively exploiting the four momenta of final-state particles. We refer to the first stage of the architecture as Lorentz Boost Network (LBN). The LBN allows the creation of particle combinations representing rest frames. The LBN also enables the formation of further composite particles, which are then transformed into said rest frames by Lorentz transformation. The properties of the composite, transformed particles are compiled in the form of characteristic variables that serve as input for a subsequent network. This second network has to be configured for a specific analysis task such as the separation of signal and background events. Using the example of the classification of ttH and ttbb events, we compare the separation power of the LBN approach with that of domain-unspecific deep neural networks (DNN). We observe leading performance with the LBN, even though we provide the DNNs with extensive additional input information beyond the particle four momenta. Furthermore, we demonstrate that the LBN forms physically meaningful particle combinations and autonomously generates suitable characteristic variables

Procalcitonin as a biomarker in equine chronic pneumopathies

Background Procalcitonin (PCT), a precursor protein of the hormone calcitonin, is a sensitive inflammatory marker in human medicine, which is primarily used for diagnosis of bacterial sepsis, but is also useful in diagnosis of exacerbation of asthma and COPD. In this study, PCT was evaluated as a potential biomarker for different chronic pneumopathies in the horse using an equine specific ELISA in comparison to established clinical markers and different interleukins. Sixty-four horses were classified as free of respiratory disease, recurrent airway obstruction (RAO), inflammatory airway disease (IAD) or chronic interstitial pneumopathy (CIP) using a scoring system. PCT concentrations were measured in plasma (n = 17) and in the cell- free supernatant of bronchoalveolar lavage (n = 64). PCT concentrations were correlated to interleukins IL-1ß and IL-6 in BALF, clinical findings and BALF cytology. Results The median PCT concentrations in plasma were increased in respiratory disease (174.46 ng/ml, n = 7) compared to controls (13.94 ng/ml, n = 10, P = 0.05) and correlated to PCT in BALF supernatant (rs = 0.48). Compared to controls (5.49 ng/ml, n = 15), median PCT concentrations in BALF supernatant correlated to the overall clinical score (rs = 0.32, P = 0.007) and were significantly increased in RAO (13.40 ng/ml, n = 21) and IAD (16.89 ng/ml, n = 16), while no differences were found for CIP (12.02 ng/ml, n = 12). No significant increases were found for IL-1 and IL-6 between controls and respiratory disease in general as well as different disease groups. Conclusions Although some correlations were found between PCT in plasma, BALF supernatant and clinical scores, PCT in BALF does not seem to be a superior marker compared to established clinical markers. PCT in plasma seems to be more promising and a greater number of samples should be evaluated in further studies