1,052 research outputs found

    Numerical methods for computing the discrete and continuous Laplace transforms

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    We propose a numerical method to spline-interpolate discrete signals and then apply the integral transforms to the corresponding analytical spline functions. This represents a robust and computationally efficient technique for estimating the Laplace transform for noisy data. We revisited a Meijer-G symbolic approach to compute the Laplace transform and alternative approaches to extend canonical observed time-series. A discrete quantization scheme provides the foundation for rapid and reliable estimation of the inverse Laplace transform. We derive theoretic estimates for the inverse Laplace transform of analytic functions and demonstrate empirical results validating the algorithmic performance using observed and simulated data. We also introduce a generalization of the Laplace transform in higher dimensional space-time. We tested the discrete LT algorithm on data sampled from analytic functions with known exact Laplace transforms. The validation of the discrete ILT involves using complex functions with known analytic ILTs

    Introduction to Riemannian Geometry and Geometric Statistics: from basic theory to implementation with Geomstats

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    International audienceAs data is a predominant resource in applications, Riemannian geometry is a natural framework to model and unify complex nonlinear sources of data.However, the development of computational tools from the basic theory of Riemannian geometry is laborious.The work presented here forms one of the main contributions to the open-source project geomstats, that consists in a Python package providing efficient implementations of the concepts of Riemannian geometry and geometric statistics, both for mathematicians and for applied scientists for whom most of the difficulties are hidden under high-level functions. The goal of this monograph is two-fold. First, we aim at giving a self-contained exposition of the basic concepts of Riemannian geometry, providing illustrations and examples at each step and adopting a computational point of view. The second goal is to demonstrate how these concepts are implemented in Geomstats, explaining the choices that were made and the conventions chosen. The general concepts are exposed and specific examples are detailed along the text.The culmination of this implementation is to be able to perform statistics and machine learning on manifolds, with as few lines of codes as in the wide-spread machine learning tool scikit-learn. We exemplify this with an introduction to geometric statistics

    Exoteric effects at nanoscopic interfaces - Uncommon negative compressibility of nanoporous materials and unexpected cavitation at liquid/liquid interfaces

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    This PhD thesis is devoted to the investigation of some peculiar effects happening at nanoscopic interfaces between immiscible liquids or liquids and solids via molecular dynamics simulations. The study of the properties of interfaces at a nanoscopic scale is driven by the promise of many interesting technological applications, including: a novel technology for developing both eco-friendly energy storage devices in the form of mechanical batteries, as well as energy dissipation systems and, in particular, shock absorbers for the automotive market; biomedical applications related to cavitation, such as High-Intensity Focused Ultrasound (HIFU) ablation of cancer tissues and localised drug delivery, and many more. The kinetics of phenomena taking places at these scales is typically determined by large free-energy barriers separating the initial and final states, and even intermediate metastable states, when they are present. Because of such barriers, the phenomena we are interested in are "rare events", i.e. the system attempts the crossing of the barrier(s) many times before finally succeeding when an energy fluctuation makes it possible. At the same time, the magnitude of the barrier is determined by the energetics and dynamics of atoms, which forces us to model the system by taking into account both the femtosecond atomistic timescale and the timescale of the relevant phenomena, typically exceeding the former by several orders of magnitude. These longer timescales are inaccessible to standard molecular dynamics, so, in order to tackle this issue, advanced MD techniques need to be employed. The thesis is divided into two parts, corresponding to the main lines of research investigated, which are (I) the interfaces between water and complex nanoporous solids, and (II) planar solid-liquid and liquid-liquid interfaces. Anticipating some results, atomistic simulations helped uncovering the microscopic mechanism behind the (incredibly rare!) giant negative compressibility exhibited by the ZIF-8 metal organic framework (MOF) upon water intrusion. Molecular dynamics simulations also supported experimental results showing how it is possible to change the intermediate intrusion-extrusion performance of ZIF-8 by changing its grain morphology and arrangement, from a fine powder to compact monolith. Free-energy MD calculations allowed to explain the exceptional stability of surface nanobubbles in water, at undersaturated conditions, on a surprisingly wide variety of substrates, characterized by disparate hydrophobicities and gas affinities; and yet, how they catastrophically destabilize in organic solvents. Finally, through simulations, some light was shed upon the working mechanism behind the novelly discovered phenomenon of how the interface between two immiscible liquids can act as a nucleation site for cavitation

    A unified framework for Simplicial Kuramoto models

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    Simplicial Kuramoto models have emerged as a diverse and intriguing class of models describing oscillators on simplices rather than nodes. In this paper, we present a unified framework to describe different variants of these models, categorized into three main groups: "simple" models, "Hodge-coupled" models, and "order-coupled" (Dirac) models. Our framework is based on topology, discrete differential geometry as well as gradient flows and frustrations, and permits a systematic analysis of their properties. We establish an equivalence between the simple simplicial Kuramoto model and the standard Kuramoto model on pairwise networks under the condition of manifoldness of the simplicial complex. Then, starting from simple models, we describe the notion of simplicial synchronization and derive bounds on the coupling strength necessary or sufficient for achieving it. For some variants, we generalize these results and provide new ones, such as the controllability of equilibrium solutions. Finally, we explore a potential application in the reconstruction of brain functional connectivity from structural connectomes and find that simple edge-based Kuramoto models perform competitively or even outperform complex extensions of node-based models.Comment: 36 pages, 11 figure

    Electron Thermal Runaway in Atmospheric Electrified Gases: a microscopic approach

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    Thesis elaborated from 2018 to 2023 at the Instituto de Astrofísica de Andalucía under the supervision of Alejandro Luque (Granada, Spain) and Nikolai Lehtinen (Bergen, Norway). This thesis presents a new database of atmospheric electron-molecule collision cross sections which was published separately under the DOI : With this new database and a new super-electron management algorithm which significantly enhances high-energy electron statistics at previously unresolved ratios, the thesis explores general facets of the electron thermal runaway process relevant to atmospheric discharges under various conditions of the temperature and gas composition as can be encountered in the wake and formation of discharge channels

    AI for time-resolved imaging: from fluorescence lifetime to single-pixel time of flight

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    Time-resolved imaging is a field of optics which measures the arrival time of light on the camera. This thesis looks at two time-resolved imaging modalities: fluorescence lifetime imaging and time-of-flight measurement for depth imaging and ranging. Both of these applications require temporal accuracy on the order of pico- or nanosecond (10−12 − 10−9s) scales. This demands special camera technology and optics that can sample light-intensity extremely quickly, much faster than an ordinary video camera. However, such detectors can be very expensive compared to regular cameras while offering lower image quality. Further, information of interest is often hidden (encoded) in the raw temporal data. Therefore, computational imaging algorithms are used to enhance, analyse and extract information from time-resolved images. "A picture is worth a thousand words". This describes a fundamental blessing and curse of image analysis: images contain extreme amounts of data. Consequently, it is very difficult to design algorithms that encompass all the possible pixel permutations and combinations that can encode this information. Fortunately, the rise of AI and machine learning (ML) allow us to instead create algorithms in a data-driven way. This thesis demonstrates the application of ML to time-resolved imaging tasks, ranging from parameter estimation in noisy data and decoding of overlapping information, through super-resolution, to inferring 3D information from 1D (temporal) data

    Decoupling the time dependent Schr\"{o}dinger equation using recursive Fourier transforms

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    A strategy is developed for writing the time-dependent Schr\"{o}dinger equation (TDSE), and more generally the Dyson Series, as a convolution equation using recursive Fourier transforms, thereby decoupling the second-order integral from the first without using the time ordering operator. The energy distribution is calculated for a number of standard perturbation theory example at first- and second-order. Possible applications include characterization of photonic spectra for bosonic sampling and four-wave mixing in quantum computation, and Bardeen tunneling amplitude in quantum mechanics.Comment: 29 pages,10 figures. Revision: minor modification of title. Inserted additional discussion on four-wave mixin

    Cram\'er-Rao Bound Optimized Subspace Reconstruction in Quantitative MRI

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    We extend the traditional framework for estimating subspace bases that maximize the preserved signal energy to additionally preserve the Cram\'er-Rao bound (CRB) of the biophysical parameters and, ultimately, improve accuracy and precision in the quantitative maps. To this end, we introduce an \textit{approximate compressed CRB} based on orthogonalized versions of the signal's derivatives with respect to the model parameters. This approximation permits singular value decomposition (SVD)-based minimization of both the CRB and signal losses during compression. Compared to the traditional SVD approach, the proposed method better preserves the CRB across all biophysical parameters with negligible cost to the preserved signal energy, leading to reduced bias and variance of the parameter estimates in simulation. In vivo, improved accuracy and precision are observed in two quantitative neuroimaging applications, permitting the use of smaller basis sizes in subspace reconstruction and offering significant computational savings

    Molecular Mechanisms and Therapies of Colorectal Cancer

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    Colorectal cancer (CRC) is currently the third leading cause of cancer-related mortality, with 1.9 million incidence cases and 0.9 million deaths worldwide. The global number of new CRC cases is predicted to reach 3.2 million in 2040, based on the projection of aging, population growth, and human development.In clinics, despite advances of diagnosis and surgical procedures, 20% of the patients with CRC present with metastasis at the time of diagnosis, caused by residual tumor cells that have spread to distant organs prior to surgery, affecting the patient survival rate. Standard systemic chemotherapy, alternative therapies that target mechanisms involved in cancer progression and metastasis, immunotherapy, and combination therapies are the major CRC-treatment strategies. In the advanced stage of CRC the transforming growth factor-beta (TGF-β) plays an oncogenic role by promoting cancer cell proliferation, cancer cell self-renewal, epithelial-to-mesenchymal transition, invasion, tumor progression, metastatic spread, and immune escape. Furthermore, high levels of TGF-β1 confers poor prognosis and is associated with early recurrence after surgery, resistance to chemo- or immunotherapy, and shorter survival. Based on the body of experimental evidence indicating that TGF-β signaling has the potential to be a good therapeutic target in CRC, several anti-TGF-β drugs have been investigated in cancer clinical trials. Here, we presented a comprehensive collection of manuscripts regarding studies on targeting the TGF-β signaling in CRC to improve patient’s prognosis and personalized treatments
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