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In this paper, we study the Hermitian hulls of (extended) generalized Reed-Solomon (GRS and EGRS) codes over finite fields. For a given class of (extended) GRS codes, by increasing the length, increasing the dimensions and increasing both the length and the dimensions, we obtain three new classes of (extended) GRS codes with Hermitian hulls of arbitrary dimensions. Furthermore, we obtain several new classes of $q^2$-ary maximum distance separable (MDS) codes with Hermitian hulls of arbitrary dimensions. And the dimension of these MDS codes can be taken from $1$ to $\frac{n}{2}$. By propagation rules, the parameters of the obtained code can be more flexible. As an application, a lot of new (MDS) entanglement-assisted quantum error correction codes (EAQECCs) can be constructed from previous known (extended) GRS codes. We derive three new propagation rules on (MDS) EAQECCs constructed from (extended) GRS codes. Finally, we present several new classes of (MDS) EAQECCs with flexible parameters. Notably, the distance parameters of our codes can range from $2$ to $\frac{n+2}{2}$.

In this second part of our two-part paper, we extend to multiple spatial dimensions the one-dimensional, fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme developed in the first part for the chemically reacting Euler equations. Our primary objective is to enable robust and accurate solutions to complex reacting-flow problems using the high-order discontinuous Galerkin method without requiring extremely high resolution. Variable thermodynamics and detailed chemistry are considered. Our multidimensional framework can be regarded as a further generalization of similar positivity-preserving and/or entropy-bounded discontinuous Galerkin schemes in the literature. In particular, the proposed formulation is compatible with curved elements of arbitrary shape, a variety of numerical flux functions, general quadrature rules with positive weights, and mixtures of thermally perfect gases. Preservation of pressure equilibrium between adjacent elements, especially crucial in simulations of multicomponent flows, is discussed. Complex detonation waves in two and three dimensions are accurately computed using high-order polynomials. Enforcement of an entropy bound, as opposed to solely the positivity property, is found to significantly improve stability. Mass, total energy, and atomic elements are shown to be discretely conserved.

In this paper we introduce a multilevel Picard approximation algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities whose coefficient functions do not need to be constant. We also provide a full convergence and complexity analysis of our algorithm. To obtain our main results, we consider a particular stochastic fixed-point equation (SFPE) motivated by the Feynman-Kac representation and the Bismut-Elworthy-Li formula. We show that the PDE under consideration has a unique viscosity solution which coincides with the first component of the unique solution of the stochastic fixed-point equation. Moreover, the gradient of the unique viscosity solution of the PDE exists and coincides with the second component of the unique solution of the stochastic fixed-point equation. Furthermore, we also provide a numerical example in up to $300$ dimensions to demonstrate the practical applicability of our multilevel Picard algorithm.

Thanks to recent advances in generative AI, we are able to prompt large language models (LLMs) to produce texts which are fluent and grammatical. In addition, it has been shown that we can elicit attempts at grammatical error correction (GEC) from LLMs when prompted with ungrammatical input sentences. We evaluate how well LLMs can perform at GEC by measuring their performance on established benchmark datasets. We go beyond previous studies, which only examined GPT* models on a selection of English GEC datasets, by evaluating seven open-source and three commercial LLMs on four established GEC benchmarks. We investigate model performance and report results against individual error types. Our results indicate that LLMs do not always outperform supervised English GEC models except in specific contexts -- namely commercial LLMs on benchmarks annotated with fluency corrections as opposed to minimal edits. We find that several open-source models outperform commercial ones on minimal edit benchmarks, and that in some settings zero-shot prompting is just as competitive as few-shot prompting.

In this paper, we further investigate the problem of selecting a set of design points for universal kriging, which is a widely used technique for spatial data analysis. Our goal is to select the design points in order to make simultaneous predictions of the random variable of interest at a finite number of unsampled locations with maximum precision. Specifically, we consider as response a correlated random field given by a linear model with an unknown parameter vector and a spatial error correlation structure. We propose a new design criterion that aims at simultaneously minimizing the variation of the prediction errors at various points. We also present various efficient techniques for incrementally building designs for that criterion scaling well for high dimensions. Thus the method is particularly suitable for big data applications in areas of spatial data analysis such as mining, hydrogeology, natural resource monitoring, and environmental sciences or equivalently for any computer simulation experiments. We have demonstrated the effectiveness of the proposed designs through two illustrative examples: one by simulation and another based on real data from Upper Austria.

In this paper, we introduce Libriheavy, a large-scale ASR corpus consisting of 50,000 hours of read English speech derived from LibriVox. To the best of our knowledge, Libriheavy is the largest freely-available corpus of speech with supervisions. Different from other open-sourced datasets that only provide normalized transcriptions, Libriheavy contains richer information such as punctuation, casing and text context, which brings more flexibility for system building. Specifically, we propose a general and efficient pipeline to locate, align and segment the audios in previously published Librilight to its corresponding texts. The same as Librilight, Libriheavy also has three training subsets small, medium, large of the sizes 500h, 5000h, 50000h respectively. We also extract the dev and test evaluation sets from the aligned audios and guarantee there is no overlapping speakers and books in training sets. Baseline systems are built on the popular CTC-Attention and transducer models. Additionally, we open-source our dataset creatation pipeline which can also be used to other audio alignment tasks.

In this paper, we address the inference problem in high-dimensional linear expectile regression. We transform the expectile loss into a weighted-least-squares form and apply a de-biased strategy to establish Wald-type tests for multiple constraints within a regularized framework. Simultaneously, we construct an estimator for the pseudo-inverse of the generalized Hessian matrix in high dimension with general amenable regularizers including Lasso and SCAD, and demonstrate its consistency through a new proof technique. We conduct simulation studies and real data applications to demonstrate the efficacy of our proposed test statistic in both homoscedastic and heteroscedastic scenarios.

This paper presents a new accelerated proximal Markov chain Monte Carlo methodology to perform Bayesian inference in imaging inverse problems with an underlying convex geometry. The proposed strategy takes the form of a stochastic relaxed proximal-point iteration that admits two complementary interpretations. For models that are smooth or regularised by Moreau-Yosida smoothing, the algorithm is equivalent to an implicit midpoint discretisation of an overdamped Langevin diffusion targeting the posterior distribution of interest. This discretisation is asymptotically unbiased for Gaussian targets and shown to converge in an accelerated manner for any target that is $\kappa$-strongly log-concave (i.e., requiring in the order of $\sqrt{\kappa}$ iterations to converge, similarly to accelerated optimisation schemes), comparing favorably to [M. Pereyra, L. Vargas Mieles, K.C. Zygalakis, SIAM J. Imaging Sciences, 13,2 (2020), pp. 905-935] which is only provably accelerated for Gaussian targets and has bias. For models that are not smooth, the algorithm is equivalent to a Leimkuhler-Matthews discretisation of a Langevin diffusion targeting a Moreau-Yosida approximation of the posterior distribution of interest, and hence achieves a significantly lower bias than conventional unadjusted Langevin strategies based on the Euler-Maruyama discretisation. For targets that are $\kappa$-strongly log-concave, the provided non-asymptotic convergence analysis also identifies the optimal time step which maximizes the convergence speed. The proposed methodology is demonstrated through a range of experiments related to image deconvolution with Gaussian and Poisson noise, with assumption-driven and data-driven convex priors. Source codes for the numerical experiments of this paper are available from //github.com/MI2G/accelerated-langevin-imla.

In this paper, we introduce Segmentation-Driven Deformation Multi-View Stereo (SD-MVS), a method that can effectively tackle challenges in 3D reconstruction of textureless areas. We are the first to adopt the Segment Anything Model (SAM) to distinguish semantic instances in scenes and further leverage these constraints for pixelwise patch deformation on both matching cost and propagation. Concurrently, we propose a unique refinement strategy that combines spherical coordinates and gradient descent on normals and pixelwise search interval on depths, significantly improving the completeness of reconstructed 3D model. Furthermore, we adopt the Expectation-Maximization (EM) algorithm to alternately optimize the aggregate matching cost and hyperparameters, effectively mitigating the problem of parameters being excessively dependent on empirical tuning. Evaluations on the ETH3D high-resolution multi-view stereo benchmark and the Tanks and Temples dataset demonstrate that our method can achieve state-of-the-art results with less time consumption.

This paper does not describe a working system. Instead, it presents a single idea about representation which allows advances made by several different groups to be combined into an imaginary system called GLOM. The advances include transformers, neural fields, contrastive representation learning, distillation and capsules. GLOM answers the question: How can a neural network with a fixed architecture parse an image into a part-whole hierarchy which has a different structure for each image? The idea is simply to use islands of identical vectors to represent the nodes in the parse tree. If GLOM can be made to work, it should significantly improve the interpretability of the representations produced by transformer-like systems when applied to vision or language