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In this paper, we study the null controllability for parabolic SPDEs involving both the state and the gradient of the state. To start with, an improved global Carleman estimate for linear forward (resp. backward) parabolic SPDEs with general random coefficients and square-integrable source terms is derived. Based on this, we further develop a new global Carleman estimate for linear forward (resp. backward) parabolic SPDEs with source terms in the Sobolev space of negative order, which enables us to deal with the global null controllability for linear backward (resp. forward) parabolic SPDEs with gradient terms. As a byproduct, a special weighted energy-type estimate for the controlled system that explicitly depends on the parameters $λ,μ$ and the weighted function $θ$ is obtained, which makes it possible to extend the previous linear null controllability to semi-linear backward (resp. forward) parabolic SPDEs by applying the fixed-point argument in an appropriate Banach space.

Learning-based robotics research driven by data demands a new approach to robot hardware design-one that serves as both a platform for policy execution and a tool for embodied data collection to train policies. We introduce ToddlerBot, a low-cost, open-source humanoid robot platform designed for scalable policy learning and research in robotics and AI. ToddlerBot enables seamless acquisition of high-quality simulation and real-world data. The plug-and-play zero-point calibration and transferable motor system identification ensure a high-fidelity digital twin, enabling zero-shot policy transfer from simulation to the real world. A user-friendly teleoperation interface facilitates streamlined real-world data collection for learning motor skills from human demonstrations. Utilizing its data collection ability and anthropomorphic design, ToddlerBot is an ideal platform to perform whole-body loco-manipulation. Additionally, ToddlerBot's compact size (0.56m, 3.4kg) ensures safe operation in real-world environments. Reproducibility is achieved with an entirely 3D-printed, open-source design and commercially available components, keeping the total cost under 6,000 USD. Comprehensive documentation allows assembly and maintenance with basic technical expertise, as validated by a successful independent replication of the system. We demonstrate ToddlerBot's capabilities through arm span, payload, endurance tests, loco-manipulation tasks, and a collaborative long-horizon scenario where two robots tidy a toy session together. By advancing ML-compatibility, capability, and reproducibility, ToddlerBot provides a robust platform for scalable learning and dynamic policy execution in robotics research.

The key to VI is the selection of a tractable density to approximate the Bayesian posterior. For large and complex models a common choice is to assume independence between multivariate blocks in a partition of the parameter space. While this simplifies the problem it can reduce accuracy. This paper proposes using vector copulas to capture dependence between the blocks parsimoniously. Tailored multivariate marginals are constructed using learnable transport maps. We call the resulting joint distribution a ``dependent block posterior'' approximation. Vector copula models are suggested that make tractable and flexible variational approximations. They allow for differing marginals, numbers of blocks, block sizes and forms of between block dependence. They also allow for solution of the variational optimization using efficient stochastic gradient methods. The approach is demonstrated using four different statistical models and 16 datasets which have posteriors that are challenging to approximate. This includes models that use global-local shrinkage priors for regularization, and hierarchical models for smoothing and heteroscedastic time series. In all cases, our method produces more accurate posterior approximations than benchmark VI methods that either assume block independence or factor-based dependence, at limited additional computational cost. A python package implementing the method is available on GitHub at https://github.com/YuFuOliver/VCVI_Rep_PyPackage.

The spin-polarized surface states in topological insulators offer unique transport characteristics that make them distinguishable from trivial conductors. Here, we detect the impact of these surface states in the topological insulator BiSbTeSe$_{2}$ by electrical means using a non-local transport configuration with ferromagnetic Co/Al$_2$O$_3$ electrodes. We show that the non-local measurement allows to probe the surface currents flowing along the whole surface, i.e.~from the top along the side to the bottom surface and back to the top surface along the opposite side. Increasing the temperature increases the interaction between bulk and surface states, which shortens this non-local current path along the surface and hence leads to a complete disappearance of the non-local signal at around 20K. Interestingly, we observe that the ratio between spin signal to background signal is much larger in the non-local geometry compared to the local one. Given that the observed ratio in the non-local geometry aligns well with expectations for spin-polarized surface states, our findings suggest that an as-yet unresolved mechanism diminishes the spin signal in the local geometry.

Ultrahigh field (UHF) Magnetic Resonance Imaging (MRI) offers an elevated signal-to-noise ratio (SNR), enabling exceptionally high spatial resolution that benefits both clinical diagnostics and advanced research. However, the jump to higher fields introduces complications, particularly transmit radiofrequency (RF) field ($B_{1}^{+}$) inhomogeneities, manifesting as uneven flip angles and image intensity irregularities. These artifacts can degrade image quality and impede broader clinical adoption. Traditional RF shimming methods, such as Magnitude Least Squares (MLS) optimization, effectively mitigate $B_{1}^{+}$ inhomogeneity, but remain time-consuming. Recent machine learning approaches, including RF Shim Prediction by Iteratively Projected Ridge Regression and other deep learning architectures, suggest alternative pathways. Although these approaches show promise, challenges such as extensive training periods, limited network complexity, and practical data requirements persist. In this paper, we introduce a holistic learning-based framework called Fast-RF-Shimming, which achieves a 5000x speed-up compared to the traditional MLS method. In the initial phase, we employ random-initialized Adaptive Moment Estimation (Adam) to derive the desired reference shimming weights from multi-channel $B_{1}^{+}$ fields. Next, we train a Residual Network (ResNet) to map $B_{1}^{+}$ fields directly to the ultimate RF shimming outputs, incorporating the confidence parameter into its loss function. Finally, we design Non-uniformity Field Detector (NFD), an optional post-processing step, to ensure the extreme non-uniform outcomes are identified. Comparative evaluations with standard MLS optimization underscore notable gains in both processing speed and predictive accuracy, which indicates that our technique shows a promising solution for addressing persistent inhomogeneity challenges.