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Understanding fluid movement in multi-pored materials is vital for energy security and physiology. For instance, shale (a geological material) and bone (a biological material) exhibit multiple pore networks. Double porosity/permeability models provide a mechanics-based approach to describe hydrodynamics in aforesaid porous materials. However, current theoretical results primarily address state-state response, and their counterparts in the transient regime are still wanting. The primary aim of this paper is to fill this knowledge gap. We present three principal properties -- with rigorous mathematical arguments -- that the solutions under the double porosity/permeability model satisfy in the transient regime: backward-in-time uniqueness, reciprocity, and a variational principle. We employ the ``energy method" -- by exploiting the physical total kinetic energy of the flowing fluid -- to establish the first property and Cauchy-Riemann convolutions to prove the next two. The results reported in this paper -- that qualitatively describe the dynamics of fluid flow in double-pored media -- have (a) theoretical significance, (b) practical applications, and (c) considerable pedagogical value. In particular, these results will benefit practitioners and computational scientists in checking the accuracy of numerical simulators. The backward-in-time uniqueness lays a firm theoretical foundation for pursuing inverse problems in which one predicts the prescribed initial conditions based on data available about the solution at a later instance.

As the development of formal proofs is a time-consuming task, it is important to devise ways of sharing the already written proofs to prevent wasting time redoing them. One of the challenges in this domain is to translate proofs written in proof assistants based on impredicative logics to proof assistants based on predicative logics, whenever impredicativity is not used in an essential way. In this paper we present a transformation for sharing proofs with a core predicative system supporting prenex universe polymorphism (like in Agda). It consists in trying to elaborate a potentially impredicative term into a predicative universe polymorphic term as general as possible. The use of universe polymorphism is justified by the fact that mapping each universe to a fixed one in the target theory is not sufficient in most cases. During the algorithm, we need to solve unification problems in the equational theory of universe levels. In order to do this, we give a complete characterization of when a single equation admits a most general unifier. This characterization is then employed in an algorithm which uses a constraint-postponement strategy to solve unification problems. The proposed translation is of course partial, but in practice allows one to translate many proofs that do not use impredicativity in an essential way. Indeed, it was implemented in the tool Predicativize and then used to translate semi-automatically many non-trivial developments from Matita's arithmetic library to Agda, including proofs of Bertrand's Postulate and Fermat's Little Theorem, which (as far as we know) were not available in Agda yet.

By a semi-Lagrangian change of coordinates, the hydrostatic Euler equations describing free-surface sheared flows is rewritten as a system of quasilinear equations, where stability conditions can be determined by the analysis of its hyperbolic structure. This new system can be written as a quasi linear system in time and horizontal variables and involves no more vertical derivatives. However, the coefficients in front of the horizontal derivatives include an integral operator acting on the new vertical variable. The spectrum of these operators is studied in detail, in particular it includes a continuous part. Riemann invariants are then determined as conserved quantities along the characteristic curves. Examples of solutions are provided, in particular stationary solutions and solutions blowing-up in finite time. Eventually, we propose an exact multi-layer $\mathbb{P}_0$-discretization, which could be used to solve numerically this semi-Lagrangian system, and analyze the eigenvalues of the corresponding discretized operator to investigate the hyperbolic nature of the approximated system.

The uncertainties in material and other properties of structures are usually spatially correlated. We introduce an efficient technique for representing and processing spatially correlated random fields in robust topology optimisation of lattice structures. Robust optimisation considers the statistics of the structural response to obtain a design whose performance is less sensitive to the specific realisation of the random field. We represent Gaussian random fields on lattices by leveraging the established link between random fields and stochastic partial differential equations (SPDEs). It is known that the precision matrix, i.e. the inverse of the covariance matrix, of a random field with Mat\'ern covariance is equal to the finite element stiffness matrix of a possibly fractional PDE with a second-order elliptic operator. We consider the discretisation of the PDE on the lattice to obtain a random field which, by design, considers its geometry and connectivity. The so-obtained random field can be interpreted as a physics-informed prior by the hypothesis that the elliptic SPDE models the physical processes occurring during manufacturing, like heat and mass diffusion. Although the proposed approach is general, we demonstrate its application to lattices modelled as pin-jointed trusses with uncertainties in member Young's moduli. We consider as a cost function the weighted sum of the expectation and standard deviation of the structural compliance. To compute the expectation and standard deviation and their gradients with respect to member cross-sections we use a first-order Taylor series approximation. The cost function and its gradient are computed using only sparse matrix operations. We demonstrate the efficiency of the proposed approach using several lattice examples with isotropic, anisotropic and non-stationary random fields and up to eighty thousand random and optimisation variables.

The occurrence of extreme events like heavy precipitation or storms at a certain location often shows a clustering behaviour and is thus not described well by a Poisson process. We construct a general model for the inter-exceedance times in between such events which combines different candidate models for such behaviour. This allows us to distinguish data generating mechanisms leading to clusters of dependent events with exponential inter-exceedance times in between clusters from independent events with heavy-tailed inter-exceedance times, and even allows us to combine these two mechanisms for better descriptions of such occurrences. We investigate a modification of the Cram\'er-von Mises distance for the purpose of model fitting. An application to mid-latitude winter cyclones illustrates the usefulness of our work.

Multiple-input multiple-output (MIMO) systems will play a crucial role in future wireless communication, but improving their signal detection performance to increase transmission efficiency remains a challenge. To address this issue, we propose extending the discrete signal detection problem in MIMO systems to a continuous one and applying the Hamiltonian Monte Carlo method, an efficient Markov chain Monte Carlo algorithm. In our previous studies, we have used a mixture of normal distributions for the prior distribution. In this study, we propose using a mixture of t-distributions, which further improves detection performance. Based on our theoretical analysis and computer simulations, the proposed method can achieve near-optimal signal detection with polynomial computational complexity. This high-performance and practical MIMO signal detection could contribute to the development of the 6th-generation mobile network.

Due to their intrinsic capabilities on parallel signal processing, optical neural networks (ONNs) have attracted extensive interests recently as a potential alternative to electronic artificial neural networks (ANNs) with reduced power consumption and low latency. Preliminary confirmation of the parallelism in optical computing has been widely done by applying the technology of wavelength division multiplexing (WDM) in the linear transformation part of neural networks. However, inter-channel crosstalk has obstructed WDM technologies to be deployed in nonlinear activation in ONNs. Here, we propose a universal WDM structure called multiplexed neuron sets (MNS) which apply WDM technologies to optical neurons and enable ONNs to be further compressed. A corresponding back-propagation (BP) training algorithm is proposed to alleviate or even cancel the influence of inter-channel crosstalk on MNS-based WDM-ONNs. For simplicity, semiconductor optical amplifiers (SOAs) are employed as an example of MNS to construct a WDM-ONN trained with the new algorithm. The result shows that the combination of MNS and the corresponding BP training algorithm significantly downsize the system and improve the energy efficiency to tens of times while giving similar performance to traditional ONNs.

Hotspot detection using thermal imaging has recently become essential in several industrial applications, such as security applications, health applications, and equipment monitoring applications. Hotspot detection is of utmost importance in industrial safety where equipment can develop anomalies. Hotspots are early indicators of such anomalies. We address the problem of hotspot detection in thermal images by proposing a self-supervised learning approach. Self-supervised learning has shown potential as a competitive alternative to their supervised learning counterparts but their application to thermography has been limited. This has been due to lack of diverse data availability, domain specific pre-trained models, standardized benchmarks, etc. We propose a self-supervised representation learning approach followed by fine-tuning that improves detection of hotspots by classification. The SimSiam network based ensemble classifier decides whether an image contains hotspots or not. Detection of hotspots is followed by precise hotspot isolation. By doing so, we are able to provide a highly accurate and precise hotspot identification, applicable to a wide range of applications. We created a novel large thermal image dataset to address the issue of paucity of easily accessible thermal images. Our experiments with the dataset created by us and a publicly available segmentation dataset show the potential of our approach for hotspot detection and its ability to isolate hotspots with high accuracy. We achieve a Dice Coefficient of 0.736, the highest when compared with existing hotspot identification techniques. Our experiments also show self-supervised learning as a strong contender of supervised learning, providing competitive metrics for hotspot detection, with the highest accuracy of our approach being 97%.

Graph representation learning for hypergraphs can be used to extract patterns among higher-order interactions that are critically important in many real world problems. Current approaches designed for hypergraphs, however, are unable to handle different types of hypergraphs and are typically not generic for various learning tasks. Indeed, models that can predict variable-sized heterogeneous hyperedges have not been available. Here we develop a new self-attention based graph neural network called Hyper-SAGNN applicable to homogeneous and heterogeneous hypergraphs with variable hyperedge sizes. We perform extensive evaluations on multiple datasets, including four benchmark network datasets and two single-cell Hi-C datasets in genomics. We demonstrate that Hyper-SAGNN significantly outperforms the state-of-the-art methods on traditional tasks while also achieving great performance on a new task called outsider identification. Hyper-SAGNN will be useful for graph representation learning to uncover complex higher-order interactions in different applications.