Predicting evaporation rates of droplet arrays

The evaporation of multiple sessile droplets is both scientifically interesting and practically important, occurring in many natural and industrial applications. Although there are simple analytic expressions to predict evaporation rates of single droplets, there are no such frameworks for general configurations of droplets of arbitrary size, contact angle or spacing. However, a recent theoretical contribution by Masoud, Howell & Stone (J. Fluid Mech., vol. 927, 2021, R4) shows how considerable insight can be gained into the evaporation of arbitrary configurations of droplets without having either to obtain the solution for the concentration of vapour in the atmosphere or to perform direct numerical simulations of the full problem. The theoretical predictions show excellent agreement with simulations for all configurations, only deviating by 25 % for the most confined droplets.

Stationary multi-kinks in the discrete sine-Gordon equation

We consider the existence and spectral stability of static multi-kink structures in the discrete sine-Gordon equation, as a representative example of the family of discrete Klein–Gordon models. The multi-kinks are constructed using Lin’s method from an alternating sequence of well-separated kink and antikink solutions. We then locate the point spectrum associated with these multi-kink solutions by reducing the spectral problem to a matrix equation. For an m-structure multi-kink, there will be m eigenvalues in the point spectrum near each eigenvalue of the primary kink, and, as long as the spectrum of the primary kink is imaginary, the spectrum of the multi-kink will be as well. We obtain analytic expressions for the eigenvalues of a multi-kink in terms of the eigenvalues and corresponding eigenfunctions of the primary kink, and these are in very good agreement with numerical results. We also perform numerical time-stepping experiments on perturbations of multi-kinks, and the outcomes of these simulations are interpreted using the spectral results.

An Analytical Prediction Model of Balanced and Unbalanced Faults in Doubly-Fed Induction Machines

This paper presents the exact transient solution to the unbalanced and balanced faults in the doubly-fed induction machine (DFIM). Stator currents, rotor currents, and stator fluxes have been validated using simulation and experiment. The work is meant to strengthen and fasten the predictability of large DFIMs in the design stage to comply with mechanical constraints or grid fault issues. Moreover, the analytical approach reduces the computational costs of large-scale stability studies and is especially suited to the initial phase where a plethora design computations must be carried out for the DFIM before it is checked for its transient interaction with the power system. The possibility to dynamically estimate DFIM performance is simplified by original equations derived from first principles. First, a case study of a large 265.50 MVA DFIM is used to verify the proposed "large machine approximation" using simulation, achieving an exact match. Then, laboratory measurements were conducted on a 10.96 kVA and a 1.94 kVA DFIM to validate the transient current peaks predicted in the proposed analytic expressions for two-phase and three-phase faults, respectively.

An Analytical Prediction Model of Balanced and Unbalanced Faults in Doubly-Fed Induction Machines

This paper presents the exact transient solution to the unbalanced and balanced faults in the doubly-fed induction machine (DFIM). Stator currents, rotor currents, and stator fluxes have been validated using simulation and experiment. The work is meant to strengthen and fasten the predictability of large DFIMs in the design stage to comply with mechanical constraints or grid fault issues. Moreover, the analytical approach reduces the computational costs of large-scale stability studies and is especially suited to the initial phase where a plethora design computations must be carried out for the DFIM before it is checked for its transient interaction with the power system. The possibility to dynamically estimate DFIM performance is simplified by original equations derived from first principles. First, a case study of a large 265.50 MVA DFIM is used to verify the proposed "large machine approximation" using simulation, achieving an exact match. Then, laboratory measurements were conducted on a 10.96 kVA and a 1.94 kVA DFIM to validate the transient current peaks predicted in the proposed analytic expressions for two-phase and three-phase faults, respectively.

Analytical Prediction of Stretch-Bending Springback Based on the Proportional Kinematic Hardening Model

Pre-stretching and post-bending are the simplest loading methods for the profile stretch-bending technical process. The inner layers of the profile are stretched and then compressed during the loading process. Considering the Bauschinger effect of metal materials, a new material model called the proportional kinematic hardening model was proposed. The stretch-bending mechanical model was established under a pre-stretching and post-bending loading path. The stress and strain on the cross section of profiles were analyzed. The analytic expressions of curvature radius of the strain neutral layer and bending moment were derived after loading. The analytic method for determining the curvature radius of the geometric center layer after unloading and springback during stretch-bending was established. The rectangular section ST12 profile with symmetrical characteristics is adopted, the stretch-bending experimental results show that the new proportional kinematic hardening model is more accurate than the classical kinematic hardening model in predicting the stretch-bending springback.

High-resolution multidimensional parametric estimation for nuclear magnetic resonance spectroscopy

<p>Nuclear Magnetic Resonance spectroscopy (NMR) is a powerful technique for rapid and efficient quantitation of compounds in chemical samples. NMR causes the nuclei in the molecules to resonate and various chemical arrangements appear as peaks in the Fourier spectrum of a free induction decay (FID). The spectral parameters elicited from the peaks serve as a fingerprint of the chemical components contained in the molecule. These fingerprints can be employed to understand the chemical structure. Signal acquired from a NMR spectrometer is ideally modelled as a superposition of multiple damped complex exponentials (cisoids) in Additive White Gaussian Noise (AWGN). The number as well as the spectral parameters of the cisoids need to be estimated for characterisation of the underlying chemicals. The estimation, however, suffers from numerous difficulties in practice. These include: unknown number of cisoids, large signal length, large dynamic range, large peak density, and numerous distortions caused by experimental artefacts. This thesis aims at the development of estimators that, in view of the above-mentioned practical features, are capable of rapid, high-resolution and apriori-information-free quantitation of NMR signals. Moreover, for the analytic evaluation of the performance of such estimators, the thesis aims to derive interpretable analytic results for the fundamental estimation theory tool for assessing the performance of an unbiased estimator: the Cramer Rao Lower Bound (CRLB). By such results, we mean those that analytically allow the determination, in terms of the CRLB, of the impact of the free model parameters on the estimator performance. For the CRLB, we report analytic expressions on the variance of unbiased parameter estimates of damping factors, frequencies and complex amplitudes of an arbitrary number of damped cisoids embedded in AWGN. In addition to the CRLB, analytic expressions for the determinant and the condition number of the associated Fisher Information Matrix (FIM) are also reported. Further results, in similar order, are reported for two special cases of the damped cisosid model: the Magnetic Resonance Relaxometry model and the amplitude-only model (employed in quantitative NMR - qNMR). Some auxiliary results for the above-mentioned models are also presented, i.e., on the multiplicity of the eigenvalues and the factorisation of the characteristic polynomial associated with their respective FIMs. These results have not been previously reported. The reported theoretical results successfully account for various physical and chemical phenomena observed in experimental NMR data, and quantify their impact on the accuracy of an unbiased estimator as a function of both model and experimental parameters, e.g., influence of prior knowledge, peak multiplicity, multiplet symmetry, solvent peak, carbon satellites, etc. For rapid, high-resolution and apriori-information-free quantitation of NMR signals, a sub-band Steiglitz-McBride algorithm is reported. The developed algorithm directly converts the time-domain FID data into a table of estimated amplitudes, phases, frequencies and damping factors, without requiring any previous knowledge or pre-processing. A 2D sub-band Steiglitz-McBride algorithm, for the quantitation of 2D NMR data in a similar manner, is also reported. The performance of the developed algorithms is validated by their application to experimental data, which manifests that they outperform the state-of-the-art in terms of speed, resolution and apriori-information-free operation.</p>

Physical Implementability of Linear Maps and Its Application in Error Mitigation

Completely positive and trace-preserving maps characterize physically implementable quantum operations. On the other hand, general linear maps, such as positive but not completely positive maps, which can not be physically implemented, are fundamental ingredients in quantum information, both in theoretical and practical perspectives. This raises the question of how well one can simulate or approximate the action of a general linear map by physically implementable operations. In this work, we introduce a systematic framework to resolve this task using the quasiprobability decomposition technique. We decompose a target linear map into a linear combination of physically implementable operations and introduce the physical implementability measure as the least amount of negative portion that the quasiprobability must pertain, which directly quantifies the cost of simulating a given map using physically implementable quantum operations. We show this measure is efficiently computable by semidefinite programs and prove several properties of this measure, such as faithfulness, additivity, and unitary invariance. We derive lower and upper bounds in terms of the Choi operator's trace norm and obtain analytic expressions for several linear maps of practical interests. Furthermore, we endow this measure with an operational meaning within the quantum error mitigation scenario: it establishes the lower bound of the sampling cost achievable via the quasiprobability decomposition technique. In particular, for parallel quantum noises, we show that global error mitigation has no advantage over local error mitigation.

High-resolution multidimensional parametric estimation for nuclear magnetic resonance spectroscopy

<p>Nuclear Magnetic Resonance spectroscopy (NMR) is a powerful technique for rapid and efficient quantitation of compounds in chemical samples. NMR causes the nuclei in the molecules to resonate and various chemical arrangements appear as peaks in the Fourier spectrum of a free induction decay (FID). The spectral parameters elicited from the peaks serve as a fingerprint of the chemical components contained in the molecule. These fingerprints can be employed to understand the chemical structure. Signal acquired from a NMR spectrometer is ideally modelled as a superposition of multiple damped complex exponentials (cisoids) in Additive White Gaussian Noise (AWGN). The number as well as the spectral parameters of the cisoids need to be estimated for characterisation of the underlying chemicals. The estimation, however, suffers from numerous difficulties in practice. These include: unknown number of cisoids, large signal length, large dynamic range, large peak density, and numerous distortions caused by experimental artefacts. This thesis aims at the development of estimators that, in view of the above-mentioned practical features, are capable of rapid, high-resolution and apriori-information-free quantitation of NMR signals. Moreover, for the analytic evaluation of the performance of such estimators, the thesis aims to derive interpretable analytic results for the fundamental estimation theory tool for assessing the performance of an unbiased estimator: the Cramer Rao Lower Bound (CRLB). By such results, we mean those that analytically allow the determination, in terms of the CRLB, of the impact of the free model parameters on the estimator performance. For the CRLB, we report analytic expressions on the variance of unbiased parameter estimates of damping factors, frequencies and complex amplitudes of an arbitrary number of damped cisoids embedded in AWGN. In addition to the CRLB, analytic expressions for the determinant and the condition number of the associated Fisher Information Matrix (FIM) are also reported. Further results, in similar order, are reported for two special cases of the damped cisosid model: the Magnetic Resonance Relaxometry model and the amplitude-only model (employed in quantitative NMR - qNMR). Some auxiliary results for the above-mentioned models are also presented, i.e., on the multiplicity of the eigenvalues and the factorisation of the characteristic polynomial associated with their respective FIMs. These results have not been previously reported. The reported theoretical results successfully account for various physical and chemical phenomena observed in experimental NMR data, and quantify their impact on the accuracy of an unbiased estimator as a function of both model and experimental parameters, e.g., influence of prior knowledge, peak multiplicity, multiplet symmetry, solvent peak, carbon satellites, etc. For rapid, high-resolution and apriori-information-free quantitation of NMR signals, a sub-band Steiglitz-McBride algorithm is reported. The developed algorithm directly converts the time-domain FID data into a table of estimated amplitudes, phases, frequencies and damping factors, without requiring any previous knowledge or pre-processing. A 2D sub-band Steiglitz-McBride algorithm, for the quantitation of 2D NMR data in a similar manner, is also reported. The performance of the developed algorithms is validated by their application to experimental data, which manifests that they outperform the state-of-the-art in terms of speed, resolution and apriori-information-free operation.</p>

Extracting multi-way chromatin contacts from Hi-C data

There is a growing realization that multi-way chromatin contacts formed in chromosome structures are fundamental units of gene regulation. However, due to the paucity and complexity of such contacts, it is challenging to detect and identify them using experiments. Based on an assumption that chromosome structures can be mapped onto a network of Gaussian polymer, here we derive analytic expressions for n-body contact probabilities (n > 2) among chromatin loci based on pairwise genomic contact frequencies available in Hi-C, and show that multi-way contact probability maps can in principle be extracted from Hi-C. The three-body (triplet) contact probabilities, calculated from our theory, are in good correlation with those from measurements including Tri-C, MC-4C and SPRITE. Maps of multi-way chromatin contacts calculated from our analytic expressions can not only complement experimental measurements, but also can offer better understanding of the related issues, such as cell-line dependent assemblies of multiple genes and enhancers to chromatin hubs, competition between long-range and short-range multi-way contacts, and condensates of multiple CTCF anchors.

A COLLECTIVE RESERVING MODEL WITH CLAIM OPENNESS

The present paper introduces a simple aggregated reserving model based on claim count and payment dynamics, which allows for claim closings and re-openings. The modelling starts off from individual Poisson process claim dynamics in discrete time, keeping track of accident year, reporting year and payment delay. This modelling approach is closely related to the one underpinning the so-called double chain-ladder model, and it allows for producing separate reported but not settled and incurred but not reported reserves. Even though the introduction of claim closings and re-openings will produce new types of dependencies, it is possible to use flexible parametrisations in terms of, for example, generalised linear models (GLM) whose parameters can be estimated based on aggregated data using quasi-likelihood theory. Moreover, it is possible to obtain interpretable and explicit moment calculations, as well as having consistency of normalised reserves when the number of contracts tend to infinity. Further, by having access to simple analytic expressions for moments, it is computationally cheap to bootstrap the mean squared error of prediction for reserves. The performance of the model is illustrated using a flexible GLM parametrisation evaluated on non-trivial simulated claims data. This numerical illustration indicates a clear improvement compared with models not taking claim closings and re-openings into account. The results are also seen to be of comparable quality with machine learning models for aggregated data not taking claim openness into account.