Mean exit time in irregularly-shaped annular and composite disc domains

Calculating the mean exit time (MET) for models of diffusion is a classical problem in statistical physics, with various applications in biophysics, economics and heat and mass transfer. While many exact results for MET are known for diffusion in simple geometries involving homogeneous materials, calculating MET for diffusion in realistic geometries involving heterogeneous materials is typically limited to repeated stochastic simulations or numerical solutions of the associated boundary value problem (BVP). In this work we derive exact solutions for the MET in irregular annular domains, including some applications where diffusion occurs in heterogenous media. These solutions are obtained by taking the exact results for MET in an annulus, and then constructing various perturbation solutions to account for the irregular geometries involved. These solutions, with a range of boundary conditions, are implemented symbolically and compare very well with averaged data from repeated stochastic simulations and with numerical solutions of the associated BVP. Software to implement the exact solutions is available on \href{https://github.com/ProfMJSimpson/Exit_time}{GitHub}.

Holomorphicity, vortex attachment, gauge invariance and the fractional quantum hall effect

A gauge invariant reformulation of nonrelativistic fermions in background magnetic fields is used to obtain the Laughlin and Jain wave functions as exact results in Mean Field Theory (MFT). The gauge invariant framework trades the U(1) gauge symmetry for an emergent holomorphic symmetry and fluxes for vortices. The novel holomorphic invariance is used to develop an analytical method for attaching vortices to particles. Vortex attachment methods introduced in this paper are subsequently employed to construct the Read operator within a second quantized framework and obtain the Laughlin and Jain wave functions as exact results entirely within a mean-field approximation. The gauge invariant framework and vortex attachment techniques are generalized to the case of spherical geometry and spherical counterparts of Laughlin and Jain wave functions are also obtained exactly within MFT.

Bayesian Logical Neural Networks for Human-Centered Applications in Medicine

Medicine is characterized by its inherent ambiguity, i.e., the difficulty to identify and obtain exact outcomes from available data. Regarding this problem, electronic Health Records (EHRs) aim to avoid imprecisions in the data recording, for instance by its recording in an automatic way or by the integration of data that is both readable by humans and machines. However, the inherent biology and physiological processes introduce a constant epistemic uncertainty, which has a deep implication in the way the condition of the patients is estimated. For instance, for some patients, it is not possible to speak about an exact diagnosis, but about the suspicion of a disease, which reveals that the medical practice is often ambiguous. In this work, we report a novel modeling methodology combining explainable models, defined on Logic Neural Networks (LONNs), and Bayesian Networks (BN) that deliver ambiguous outcomes, for instance, medical procedures (Therapy Keys (TK)), depending on the uncertainty of observed data. If epistemic uncertainty is generated from the underlying physiology, the model delivers exact or ambiguous results depending on the individual parameters of each patient. Thus, our model does not aim to assist the customer by providing exact results but is a user-centered solution that informs the customer when a given recommendation, in this case, a therapy, is uncertain and must be carefully evaluated by the customer, implying that the final customer must be a professional who will not fully rely on automatic recommendations. This novel methodology has been tested on a database for patients with heart insufficiency.

OpenPlaning: Open-Source Framework for the Hydrodynamic Design of Planing Hulls

The use of Dr. Savitsky's empirical methods for the hydrodynamic design of planing hulls is widespread in industry and academia. In spite of their common use, their implementations are frequently inconsistent among their users. This makes it difficult to share and replicate exact results, and apply those results to a reader's desired case study. This paper presents an open-source Python-based framework of the Savitsky '64 and Savitksy & Brown '76 papers that is suitable for industry and research purposes, named OpenPlaning. The original Savitsky method implementation required the use of charts and results interpolation to find the boat's equilibrium. OpenPlaning instead uses a root-finding algorithm to determine the equilibrium attitude, automating the process and assuring consistent use of the Savitsky method. This formulaic approach allows the users to change and explore variations among any desired hull characteristic (beam, deadrise, weight, flap size/deflection, etc.), and use Python packages to optimize these parameters. OpenPlaning includes Faltinsen's 2010 planing hull porpoising inception work, which expanded upon the Savitsky method to estimate the vessel's mass, damping, and restoring coefficients. Consequently, control engineers can also use OpenPlaning to obtain a linear system in the heave and pitch degree of freedoms, and obtain initial tuning parameters for control systems. To illustrate the use of OpenPlaning, an example planing craft is designed, explored and optimized with real-world constraints.