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In this work, we generalize the concept of bisimulation metric in order to metrize the behaviour of continuous-time processes. Similarly to what is done for discrete-time systems, we follow two approaches and show that they coincide: as a fixpoint of a functional and through a real-valued logic. The whole discrete-time approach relies entirely on the step-based dynamics: the process jumps from state to state. We define a behavioural pseudometric for processes that evolve continuously through time, such as Brownian motion or involve jumps or both., Comment: arXiv admin note: substantial text overlap with arXiv:2312.16729

Personalized learning represents a promising educational strategy within intelligent educational systems, aiming to enhance learners' practice efficiency. However, the discrepancy between offline metrics and online performance significantly impedes their progress. To address this challenge, we introduce Agent4Edu, a novel personalized learning simulator leveraging recent advancements in human intelligence through large language models (LLMs). Agent4Edu features LLM-powered generative agents equipped with learner profile, memory, and action modules tailored to personalized learning algorithms. The learner profiles are initialized using real-world response data, capturing practice styles and cognitive factors. Inspired by human psychology theory, the memory module records practice facts and high-level summaries, integrating reflection mechanisms. The action module supports various behaviors, including exercise understanding, analysis, and response generation. Each agent can interact with personalized learning algorithms, such as computerized adaptive testing, enabling a multifaceted evaluation and enhancement of customized services. Through a comprehensive assessment, we explore the strengths and weaknesses of Agent4Edu, emphasizing the consistency and discrepancies in responses between agents and human learners. The code, data, and appendix are publicly available at https://github.com/bigdata-ustc/Agent4Edu., Comment: Accepted by AAAI2025

6-Degree of Freedom (6DoF) motion estimation with a combination of visual and inertial sensors is a growing area with numerous real-world applications. However, precise calibration of the time offset between these two sensor types is a prerequisite for accurate and robust tracking. To address this, we propose a universal online temporal calibration strategy for optimization-based visual-inertial navigation systems. Technically, we incorporate the time offset td as a state parameter in the optimization residual model to align the IMU state to the corresponding image timestamp using td, angular velocity and translational velocity. This allows the temporal misalignment td to be optimized alongside other tracking states during the process. As our method only modifies the structure of the residual model, it can be applied to various optimization-based frameworks with different tracking frontends. We evaluate our calibration method with both EuRoC and simulation data and extensive experiments demonstrate that our approach provides more accurate time offset estimation and faster convergence, particularly in the presence of noisy sensor data., Comment: 7 pages

This paper aims to present the time-dependent coupling between the coronal model COolfluid COroNal UnsTructured (COCONUT) and the heliospheric forecasting tool EUHFORIA. We perform six COCONUT simulations where a flux rope is implemented at the solar surface using either the Titov-D\'emoulin CME model or the Regularized Biot-Savart Laws (RBSL) CME model. At regular intervals, the magnetic field, velocity, temperature, and density of the 2D surface $R_{b}=21.5~\;R_{\odot}$ are saved in boundary files. This series of coupling files is read in a modified version of EUHFORIA to update progressively its inner boundary. After presenting the early stage of the propagation in COCONUT, we examine how the disturbance of the solar corona created by the propagation of flux ropes is transmitted into EUHFORIA. In particular, we consider the thermodynamic and magnetic profiles at L1 and compare them with those obtained at the interface between the two models. We demonstrate that the properties of the heliospheric solar wind in EUHFORIA are consistent with those in COCONUT, acting as a direct extension of the coronal domain. Moreover, the disturbances initially created from the propagation of flux ropes in COCONUT continue evolving from the corona in the heliosphere to Earth with a smooth transition at the interface between the two simulations. Looking at the profile of magnetic field components at Earth and different distances from the Sun, we also find that the transient magnetic structures have a self-similar expansion in COCONUT and EUHFORIA. However, the amplitude of the profiles depends on the flux rope model used and its properties, thus emphasizing the important role of the initial properties in solar source regions for accurately predicting the impact of CMEs.

The Schr\"odinger bridge problem (SBP) seeks to find the measure $\hat{\mathbf{P}}$ on a certain path space which interpolates between state-space distributions $\rho_0$ at time $0$ and $\rho_T$ at time $T$ while minimizing the KL divergence (relative entropy) to a reference path measure $\mathbf{R}$. In this work, we tackle the SBP in the case when $\mathbf{R}$ is the path measure of a jump diffusion. Under mild assumptions, with both the operator theory approach and the stochastic calculus techniques, we establish an $h$-transform theory for jump diffusions and devise an approximation method to achieve the jump-diffusion SBP solution $\hat{\mathbf{P}}$ as the strong-convergence limit of a sequence of harmonic $h$-transforms. To the best of our knowledge, these results are novel in the study of SBP. Moreover, the $h$-transform framework and the approximation method developed in this work are robust and applicable to a relatively general class of jump diffusions. In addition, we examine the SBP of particular types of jump diffusions under additional regularity conditions and extend the existing results on the SBP from the diffusion case to the jump-diffusion setting.

Learners sharing similar implicit cognitive states often display comparable observable problem-solving performances. Leveraging collaborative connections among such similar learners proves valuable in comprehending human learning. Motivated by the success of collaborative modeling in various domains, such as recommender systems, we aim to investigate how collaborative signals among learners contribute to the diagnosis of human cognitive states (i.e., knowledge proficiency) in the context of intelligent education. The primary challenges lie in identifying implicit collaborative connections and disentangling the entangled cognitive factors of learners for improved explainability and controllability in learner Cognitive Diagnosis (CD). However, there has been no work on CD capable of simultaneously modeling collaborative and disentangled cognitive states. To address this gap, we present Coral, a Collaborative cognitive diagnosis model with disentangled representation learning. Specifically, Coral first introduces a disentangled state encoder to achieve the initial disentanglement of learners' states. Subsequently, a meticulously designed collaborative representation learning procedure captures collaborative signals. It dynamically constructs a collaborative graph of learners by iteratively searching for optimal neighbors in a context-aware manner. Using the constructed graph, collaborative information is extracted through node representation learning. Finally, a decoding process aligns the initial cognitive states and collaborative states, achieving co-disentanglement with practice performance reconstructions. Extensive experiments demonstrate the superior performance of Coral, showcasing significant improvements over state-of-the-art methods across several real-world datasets. Our code is available at https://github.com/bigdata-ustc/Coral., Comment: Accepted by NeurIPS2024

Solar energetic particles (SEPs) associated with solar flares and coronal mass ejections (CMEs) are key agents of space weather phenomena, posing severe threats to spacecraft and astronauts. Recent observations by Parker Solar Probe (PSP) indicate that the magnetic flux ropes of a CME can trap energetic particles and act as barriers, preventing other particles from crossing. In this paper, we introduce the novel COCONUT+PARADISE model to investigate the confinement of energetic particles within a flux rope and the effects of cross-field diffusion (CFD) on particle transport in the solar corona, particularly in the presence of a CME. Using the global magnetohydrodynamic coronal model COCONUT, we generate background configurations containing a CME modeled as a Titov-D\'emoulin flux rope (TDFR). We then utilize the particle transport code PARADISE to inject monoenergetic 100 keV protons inside one of the TDFR legs near its footpoint and evolve the particles through the COCONUT backgrounds. To study CFD, we employ two different approaches regarding the perpendicular proton mean free path (MFP): a constant MFP and a Larmor radius-dependent MFP. We contrast these results with those obtained without CFD. While particles remain fully trapped within the TDFR without CFD, we find that even relatively small perpendicular MFP values allow particles on the outer layers to escape. In contrast, the initially interior trapped particles stay largely confined. Finally, we highlight how our model and this paper's results are relevant for future research on particle acceleration and transport in an extended domain encompassing both the corona and inner heliosphere., Comment: 16 pages, 8 figures, accepted for publication by ApJ Letters

The historical interaction sequences of users plays a crucial role in training recommender systems that can accurately predict user preferences. However, due to the arbitrariness of user behavior, the presence of noise in these sequences poses a challenge to predicting their next actions in recommender systems. To address this issue, our motivation is based on the observation that training noisy sequences and clean sequences (sequences without noise) with equal weights can impact the performance of the model. We propose a novel self-supervised Auxiliary Task Joint Training (ATJT) method aimed at more accurately reweighting noisy sequences in recommender systems. Specifically, we strategically select subsets from users' original sequences and perform random replacements to generate artificially replaced noisy sequences. Subsequently, we perform joint training on these artificially replaced noisy sequences and the original sequences. Through effective reweighting, we incorporate the training results of the noise recognition model into the recommender model. We evaluate our method on three datasets using a consistent base model. Experimental results demonstrate the effectiveness of introducing self-supervised auxiliary task to enhance the base model's performance.

MHD coronal models are critical in the Sun-to-Earth model chain and the most complex and computationally intensive component, particularly the time-evolving coronal models, typically driven by a series of time-evolving photospheric magnetograms. There is an urgent need to develop efficient and reliable time-evolving MHD coronal models to further improve our ability to predict space weather. COCONUT is a rapidly developing MHD coronal model. Adopting the efficient implicit algorithm makes it suitable for performing computationally intensive time-evolving coronal simulations. This paper aims to extend COCONUT to an efficient time-evolving MHD coronal model. In this MHD model, as usual, an implicit temporal integration algorithm is adopted to avoid the CFL stability restriction and increase computational efficiency by large time steps. The Newton iteration method is applied within each time step to enhance the temporal accuracy. The unstructured geodesic mesh is used for flexibility in mesh division and to avoid degeneracy at the poles. Furthermore, an HLL Riemann solver with a self-adjustable dissipation term accommodates both low- and high-speed flows. A series of time-evolving photospheric magnetograms are utilized to drive the evolution of coronal structures from the solar surface to 25Rs during two Carrington rotations (CRs) around the 2019 eclipse in an inertial coordinate system. It shows that COCONUT can mimic the coronal evolution during a full CR within 9 hours (1080 CPU cores, 1.5M cells). We also compare the simulation results of time-evolving versus quasi-steady-state coronal simulations in the thermodynamic MHD model to validate the time-evolving approach. Additionally, we evaluate the effect of time steps on the simulation results to find an optimal time step that simultaneously maintains high efficiency and necessary numerical stability and accuracy., Comment: 24 pages, 14 figures

In this paper, we present an efficient and time-accurate three-dimensional (3D) single-fluid MHD solar coronal model and employ it to simulate CME evolution and propagation. Based on a quasi-steady-state implicit MHD coronal model, we developed an efficient time-accurate coronal model that can be used to speed up the CME simulation by selecting a large time-step size. We have called it the Solar Interplanetary Phenomena-Implicit Finite Volume Method (SIP-IFVM) coronal model. A pseudo-time marching method was implemented to improve temporal accuracy. A regularised Biot-Savart Laws (RBSL) flux rope, whose axis can be designed into an arbitrary shape, was inserted into the background corona to trigger the CME event. We performed a CME simulation on the background corona of Carrington rotation (CR) 2219 and evaluated the impact of time-step sizes on simulation results. Our study demonstrates that this model is able to simulate the CME evolution and propagation process from the solar surface to $20\; R_s$ in less than 0.5 hours (192 CPU cores, $\sim$ 1 M cells). Compared to the explicit counterpart, this implicit coronal model is not only faster, but it also has improved numerical stability. We also conducted an ad hoc simulation with initial magnetic fields artificially increased. It shows that this model can effectively deal with time-dependent low-$\beta$ problems ($\beta<10^{-4}$). Additionally, an Orszag-Tang MHD vortex flow simulation demonstrates that the pseudo-time-marching method used in this coronal model can simulate small-scale unsteady-state flows. The simulation results show that this MHD coronal model is very efficient and numerically stable. It is a promising approach to simulating time-varying events in the solar corona with low plasma $\beta$ in a timely and accurate manner., Comment: 17 pages, 12 figures. (Accepted by A&A.)

Recent advancements in Graph Neural Networks (GNNs) have spurred an upsurge of research dedicated to enhancing the explainability of GNNs, particularly in critical domains such as medicine. A promising approach is the self-explaining method, which outputs explanations along with predictions. However, existing self-explaining models require a large amount of training data, rendering them unavailable in few-shot scenarios. To address this challenge, in this paper, we propose a Meta-learned Self-Explaining GNN (MSE-GNN), a novel framework that generates explanations to support predictions in few-shot settings. MSE-GNN adopts a two-stage self-explaining structure, consisting of an explainer and a predictor. Specifically, the explainer first imitates the attention mechanism of humans to select the explanation subgraph, whereby attention is naturally paid to regions containing important characteristics. Subsequently, the predictor mimics the decision-making process, which makes predictions based on the generated explanation. Moreover, with a novel meta-training process and a designed mechanism that exploits task information, MSE-GNN can achieve remarkable performance on new few-shot tasks. Extensive experimental results on four datasets demonstrate that MSE-GNN can achieve superior performance on prediction tasks while generating high-quality explanations compared with existing methods. The code is publicly available at https://github.com/jypeng28/MSE-GNN.

Rising concerns about the impact of space weather-related disruptions demand modelling and reliable forecasting of coronal mass ejection (CME) impacts. In this study, we demonstrate the application of the modified Miller-Turner (mMT) model implemented in EUropean Heliospheric FORecasting Information Asset (EUHFORIA), to forecast the geo-effectiveness of observed coronal mass ejection (CME) events in the heliosphere. The goal is to develop a model that not only has a global geometry to improve overall forecasting but is also fast enough for operational space weather forecasting. We test the original full torus implementation and introduce a new three-fourth Torus version called the Horseshoe CME model. This new model has a more realistic CME geometry, and it overcomes the inaccuracies of the full torus geometry. We constrain the torus geometrical and magnetic field parameters using observed signatures of the CMEs before, during, and after the eruption. The assessment of the model's capability to predict the most important Bz component is performed using the advanced Dynamic Time Warping technique. The Horseshoe model prediction of CME arrival time and geo-effectiveness for both validation events compare well to the observations and are weighed with the results obtained with the spheromak and FRi3D models that were already available in EUHFORIA. The runtime of the Horseshoe model simulations is close to that of the spheromak model, which is suitable for operational space weather forecasting. Yet, the capability of the magnetic field prediction at 1~AU of the Horseshoe model is close to that of the FRi3D model. In addition, we demonstrate that the Horseshoe CME model can be used for simulating successive CMEs in EUHFORIA, overcoming a limitation of the FRi3D model., Comment: 26 pages, 14 figures, to be published in A&A

Solar wind modelling has become a crucial area of study due to the increased dependence of modern society on technology, navigation, and power systems. Accurate space weather forecasts can predict upcoming threats to Earth's geospace. In this study, we examine a novel full magnetohydrodynamic (MHD) chain from the Sun to Earth. The goal of this study is to demonstrate the capabilities of the full MHD modelling chain from the Sun to Earth by finalising the implementation of the full MHD coronal model into the COolfluid COroNa UnsTructured (COCONUT) model and coupling it to the MHD heliospheric model Icarus. The resulting coronal model has significant advantages compared to the pre-existing polytropic alternative, as it models a more realistic bi-modal wind, which is crucial for heliospheric studies. In this study, only thermal conduction, radiative losses, and approximated coronal heating function were considered in the energy equation. A realistic specific heat ratio was applied. The output of the coronal model was used to onset the 3D MHD heliospheric model Icarus. A minimum solar activity case was chosen as the first test case for the full MHD model. The numerically simulated data in the corona and the heliosphere were compared to observational products. We present a first attempt to obtain the full MHD chain from the Sun to Earth with COCONUT and Icarus. The coronal model has been upgraded to a full MHD model for a realistic bi-modal solar wind configuration. The approximated heating functions have modelled the wind reasonably well, but simple approximations are not enough to obtain a realistic density-speed balance or realistic features in the low corona and farther, near the outer boundary. The full MHD model was computed in 1.06 h on 180 cores of the Genius cluster of the Vlaams Supercomputing Center, which is only 1.8 times longer than the polytropic simulation., Comment: 12 pages, 13 figures

Let $PU_n$ denote the projective unitary group of rank $n$ and $BPU_n$ be its classifying space, for $n>1$. By using the Serre spectral sequence induced by the fibration $BU_n\to BPU_n\to K(\mathbb{Z},3)$, we compute the integral cohomology of $BPU_n$ in dimensions less than $15$ except for $4\mid n$ in dimension $14$., Comment: arXiv admin note: text overlap with arXiv:2402.09103

Coronal mass ejections (CMEs) stand as intense eruptions of magnetized plasma from the Sun, playing a pivotal role in driving significant changes of the heliospheric environment. Deducing the properties of CMEs from their progenitors in solar source regions is crucial for space weather forecasting. Deducing the properties of CMEs from their progenitors in solar source regions is crucial for space weather forecasting. The primary objective of this paper is to establish a connection between CMEs and their progenitors in solar source regions, enabling us to infer the magnetic structures of CMEs before their full development. To this end, we create a dataset comprising a magnetic flux rope series with varying projection shapes, sizes and toroidal fluxes, using the Regularized Biot-Savart Laws (RBSL). Thereafter, we simulate the propagation of these flux ropes from the solar surface to a distance of 25$R_{\odot}$ with our global coronal MHD model which is named COCONUT. Our parametric survey reveals significant impacts of source flux ropes on the consequent CMEs. We find that the projection shape can influence the magnetic structures of CMEs at 20$R_{\odot}$, albeit with minimal impacts on the propagation speed. However, these impacts diminish as source flux ropes become fat. In terms of toroidal flux, our simulation results demonstrate a pronounced correlation with the propagation speed of CMEs, as well as the successfulness in erupting. This work builds the bridge between the CMEs in the outer corona and their progenitors in solar source regions. Our parametric survey suggests that the projection shape, cross-section radius and toroidal flux of source flux ropes are crucial parameters in predicting magnetic structures and propagation speed of CMEs, providing valuable insights for space weather prediction., Comment: 11 pages, 10 figrues, accepted for publication by A&A

Few-Shot Relation Extraction (FSRE), a subtask of Relation Extraction (RE) that utilizes limited training instances, appeals to more researchers in Natural Language Processing (NLP) due to its capability to extract textual information in extremely low-resource scenarios. The primary methodologies employed for FSRE have been fine-tuning or prompt tuning techniques based on Pre-trained Language Models (PLMs). Recently, the emergence of Large Language Models (LLMs) has prompted numerous researchers to explore FSRE through In-Context Learning (ICL). However, there are substantial limitations associated with methods based on either traditional RE models or LLMs. Traditional RE models are hampered by a lack of necessary prior knowledge, while LLMs fall short in their task-specific capabilities for RE. To address these shortcomings, we propose a Dual-System Augmented Relation Extractor (DSARE), which synergistically combines traditional RE models with LLMs. Specifically, DSARE innovatively injects the prior knowledge of LLMs into traditional RE models, and conversely enhances LLMs' task-specific aptitude for RE through relation extraction augmentation. Moreover, an Integrated Prediction module is employed to jointly consider these two respective predictions and derive the final results. Extensive experiments demonstrate the efficacy of our proposed method.

Trusted Execution Environments (TEEs), such as Intel Software Guard Extensions (SGX), ensure the confidentiality and integrity of user applications when using cloud computing resources. However, in the multi-party cloud computing scenario, how to select a Relying Party to verify the TEE of each party and avoid leaking sensitive data to each other remains an open question. In this paper, we propose SRAS, an open self-governed remote attestation scheme with attestation and verification functions for verifying the trustworthiness of TEEs and computing assets, achieving decentralized unified trusted attestation and verification platform for multi-party cloud users. In SRAS, we design a Relying Party enclave, which can form a virtual verifiable network, capable of local verification on behalf of other participants relying parties without leaking sensitive data to others. We provide an open-source prototype implementation of SRAS to facilitate the adoption of this technology by cloud users or developers.

The optimization of chemical processes is challenging due to the nonlinearities arising from process physics and discrete design decisions. In particular, optimal synthesis and design of chemical processes can be posed as a Generalized Disjunctive Programming (GDP) superstructure problem. Various solution methods are available to address these problems, such as reformulating them as Mixed-Integer Nonlinear Programming (MINLP) problems; nevertheless, algorithms explicitly designed to solve the GDP problem and potentially leverage its structure remain scarce. This paper presents the Logic-based Discrete-Steepest Descent Algorithm (LD-SDA) as a solution method for GDP problems involving ordered Boolean variables. The LD-SDA reformulates these ordered Boolean variables into integer decisions called external variables. The LD-SDA solves the reformulated GDP problem using a two-level decomposition approach where the upper-level subproblem determines external variable configurations. Subsequently, the remaining continuous and discrete variables are solved as a subproblem only involving those constraints relevant to the given external variable arrangement, effectively taking advantage of the structure of the GDP problem. The advantages of LD-SDA are illustrated through a batch processing case study, a reactor superstructure, a distillation column, and a catalytic distillation column, and its open-source implementation is available online. The results show convergence efficiency and solution quality improvements compared to conventional GDP and MINLP solvers., Comment: 36 pages, 14 figures

With the development of legal intelligence, Criminal Court View Generation has attracted much attention as a crucial task of legal intelligence, which aims to generate concise and coherent texts that summarize case facts and provide explanations for verdicts. Existing researches explore the key information in case facts to yield the court views. Most of them employ a coarse-grained approach that partitions the facts into broad segments (e.g., verdict-related sentences) to make predictions. However, this approach fails to capture the complex details present in the case facts, such as various criminal elements and legal events. To this end, in this paper, we propose an Event Grounded Generation (EGG) method for criminal court view generation with cooperative (Large) Language Models, which introduces the fine-grained event information into the generation. Specifically, we first design a LLMs-based extraction method that can extract events in case facts without massive annotated events. Then, we incorporate the extracted events into court view generation by merging case facts and events. Besides, considering the computational burden posed by the use of LLMs in the extraction phase of EGG, we propose a LLMs-free EGG method that can eliminate the requirement for event extraction using LLMs in the inference phase. Extensive experimental results on a real-world dataset clearly validate the effectiveness of our proposed method., Comment: Accepted to SIGIR2024

The remarkable success in neural networks provokes the selective rationalization. It explains the prediction results by identifying a small subset of the inputs sufficient to support them. Since existing methods still suffer from adopting the shortcuts in data to compose rationales and limited large-scale annotated rationales by human, in this paper, we propose a Shortcuts-fused Selective Rationalization (SSR) method, which boosts the rationalization by discovering and exploiting potential shortcuts. Specifically, SSR first designs a shortcuts discovery approach to detect several potential shortcuts. Then, by introducing the identified shortcuts, we propose two strategies to mitigate the problem of utilizing shortcuts to compose rationales. Finally, we develop two data augmentations methods to close the gap in the number of annotated rationales. Extensive experimental results on real-world datasets clearly validate the effectiveness of our proposed method., Comment: Accepted to ICLR 2024

Graph Neural Networks (GNNs) have achieved impressive results in graph classification tasks, but they struggle to generalize effectively when faced with out-of-distribution (OOD) data. Several approaches have been proposed to address this problem. Among them, one solution is to diversify training distributions in vanilla classification by modifying the data environment, yet accessing the environment information is complex. Besides, another promising approach involves rationalization, extracting invariant rationales for predictions. However, extracting rationales is difficult due to limited learning signals, resulting in less accurate rationales and diminished predictions. To address these challenges, in this paper, we propose a Cooperative Classification and Rationalization (C2R) method, consisting of the classification and the rationalization module. Specifically, we first assume that multiple environments are available in the classification module. Then, we introduce diverse training distributions using an environment-conditional generative network, enabling robust graph representations. Meanwhile, the rationalization module employs a separator to identify relevant rationale subgraphs while the remaining non-rationale subgraphs are de-correlated with labels. Next, we align graph representations from the classification module with rationale subgraph representations using the knowledge distillation methods, enhancing the learning signal for rationales. Finally, we infer multiple environments by gathering non-rationale representations and incorporate them into the classification module for cooperative learning. Extensive experimental results on both benchmarks and synthetic datasets demonstrate the effectiveness of C2R. Code is available at https://github.com/yuelinan/Codes-of-C2R., Comment: Accepted to WWW 2024

Let $PU_n$ denote the projective unitary group of rank $n$, and let $BPU_n$ be its classifying space. We extend our previous results to a description of $H^s(BPU_n;\mathbb{Z})_{(p)}$ for $s<2p+9$ by showing that $p$-primary subgroups of $H^s(BPU_n;\mathbb{Z})$ is $\mathbb{Z}/p$ for $s=2p+5$ and are trivial for $s = 2p+7$ and $s = 2p+8$, where $p$ is an odd prime.

In this work we study the one-dimensional stochastic Kimura equation $\partial_{t}u\left(z,t\right)=z\partial_{z}^{2}u\left(z,t\right)+u\left(z,t\right)\dot{W}\left(z,t\right)$ for $z,t>0$ equipped with a Dirichlet boundary condition at $0$, with $\dot{W}$ being a Gaussian space-time noise. This equation can be seen as a degenerate analog of the parabolic Anderson model. We combine the Wiener chaos theory from Malliavin calculus, the Duhamel perturbation technique from PDEs, and the kernel analysis of (deterministic) degenerate diffusion equations to develop a solution theory for the stochastic Kimura equation. We establish results on existence, uniqueness, moments, and continuity for the solution $u\left(z,t\right)$. In particular, we investigate how the stochastic potential and the degeneracy in the diffusion operator jointly affect the properties of $u\left(z,t\right)$ near the boundary. We also derive explicit estimates on the comparison under the $L^{2}-$ norm between $u\left(z,t\right)$ and its deterministic counterpart for $\left(z,t\right)$ within a proper range., Comment: 45 pages

Retraction maps have been generalized to discretization maps in (Barbero Li\~n\'an and and Mart\'{\i}n de Diego, 2022). Discretization maps are used to systematically derive numerical integrators that preserve the symplectic structure, as well as the discrete momemtum map under the assumption of symmetric preservation for the discretization map. The procedure described here gives a geometrical construction that can be easily adapted to discretize dynamics on more general structures and open the door to reduction processes., Comment: 4 pages