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The main purpose of this paper is to give an upper bound of Hausdorff dimension of random attractors for a stochastic delayed parabolic equation in Banach spaces. The estimation of dimensions of random attractors are obtained by combining the squeezing property and a covering lemma of finite subspace of Banach spaces, which generalizes the method established in Hilbert spaces. Unlike the existing works, where orthogonal projectors with finite ranks applied for proving the squeezing property of stochastic partial differential equations in Hilbert spaces, we adopt the state decomposition of phase space based on the exponential dichotomy of the the linear deterministic part of the studied SDPE to obtain similar squeezing property due to the lack of smooth inner product geometry structure. The obtained dimension of the random attractors depend only on the spectrum of the linear part and the random Lipschitz constant of the nonlinear term, while not relating to the compact embedding of the phase space to another Banach space as the existing works did.

The main objective of this paper is to investigate exponential attractors for a nonlocal delayed reaction-diffusion equation on an unbounded domain. We first obtain the existence of a globally attractive absorbing set for the dynamical system generated by the equation under the assumption that the nonlinear term is bounded. Then, we construct exponential attractors of the equation directly in its natural phase space, i.e., a Banach space with explicit fractal dimension by combining squeezing properties of the system as well as a covering lemma of finite subspace of Banach spaces. Our result generalizes the methods established in Hilbert spaces and weighted spaces, and the fractal dimension of the obtained exponential attractor does not depend on the entropy number but only depends on some inner characteristic of the studied equation.

Many modern applications require the use of data to both select the statistical tasks and make valid inference after selection. In this article, we provide a unifying approach to control for a class of selective risks. Our method is motivated by a reformulation of the celebrated Benjamini-Hochberg (BH) procedure for multiple hypothesis testing as the iterative limit of the Benjamini-Yekutieli (BY) procedure for constructing post-selection confidence intervals. Although several earlier authors have made noteworthy observations related to this, our discussion highlights that (1) the BH procedure is precisely the fixed-point iteration of the BY procedure; (2) the fact that the BH procedure controls the false discovery rate is almost an immediate corollary of the fact that the BY procedure controls the false coverage-statement rate. Building on this observation, we propose a constructive approach to control extra-selection risk (selection made after decision) by iterating decision strategies that control the post-selection risk (decision made after selection), and show that many previous methods and results are special cases of this general framework. We further extend this approach to problems with multiple selective risks and demonstrate how new methods can be developed. Our development leads to two surprising results about the BH procedure: (1) in the context of one-sided location testing, the BH procedure not only controls the false discovery rate at the null but also at other locations for free; (2) in the context of permutation tests, the BH procedure with exact permutation p-values can be well approximated by a procedure which only requires a total number of permutations that is almost linear in the total number of hypotheses., Comment: 8 figures, 2 tables

The rapid development of open-source large language models (LLMs) has been truly remarkable. However, the scaling law described in previous literature presents varying conclusions, which casts a dark cloud over scaling LLMs. We delve into the study of scaling laws and present our distinctive findings that facilitate scaling of large scale models in two commonly used open-source configurations, 7B and 67B. Guided by the scaling laws, we introduce DeepSeek LLM, a project dedicated to advancing open-source language models with a long-term perspective. To support the pre-training phase, we have developed a dataset that currently consists of 2 trillion tokens and is continuously expanding. We further conduct supervised fine-tuning (SFT) and Direct Preference Optimization (DPO) on DeepSeek LLM Base models, resulting in the creation of DeepSeek Chat models. Our evaluation results demonstrate that DeepSeek LLM 67B surpasses LLaMA-2 70B on various benchmarks, particularly in the domains of code, mathematics, and reasoning. Furthermore, open-ended evaluations reveal that DeepSeek LLM 67B Chat exhibits superior performance compared to GPT-3.5.

The purpose of this paper is to investigate the existence and Hausdorff dimension as well as fractal dimension of global attractors for a delayed reaction-diffusion equation on an unbounded domain. The noncompactness of the domain causes the Laplace operator has a continuous spectrum, the semigroup generated by the linear part and the Sobolev embeddings are no longer compact, making the problem more difficult compared with the equations on bounded domains. We first obtain the existence of an absorbing set for the infinite dimensional dynamical system generated by the equation by a priori estimate of the solutions. Then, we show the asymptotic compactness of the solution semiflow by an uniform a priori estimates for far-field values of solutions together with the Arzel\`a-Ascoli theorem, which facilitates us to show the existence of global attractors. By decomposing the solution into three parts and establishing a squeezing property of each part, we obtain the explicit upper estimation of both Hausdorff and fractal dimension of the global attractors, which only depend on the inner characteristic of the equation, while not related to the entropy number compared with the existing literature.

Ducted fan lift systems (DFLSs) powered by two-stroke aviation piston engines present a challenging control problem due to their complex multivariable dynamics. Current controllers for these systems typically rely on proportional-integral algorithms combined with data tables, which rely on accurate models and are not adaptive to handle time-varying dynamics or system uncertainties. This paper proposes a novel adaptive model predictive control (AMPC) strategy with an associated linear parameter varying (LPV) model for controlling the engine-driven DFLS. This LPV model is derived from a global network model, which is trained off-line with data obtained from a general mean value engine model for two-stroke aviation engines. Different network models, including multi-layer perceptron, Elman, and radial basis function (RBF), are evaluated and compared in this study. The results demonstrate that the RBF model exhibits higher prediction accuracy and robustness in the DFLS application. Based on the trained RBF model, the proposed AMPC approach constructs an associated network that directly outputs the LPV model parameters as an adaptive, robust, and efficient prediction model. The efficiency of the proposed approach is demonstrated through numerical simulations of a vertical take-off thrust preparation process for the DFLS. The simulation results indicate that the proposed AMPC method can effectively control the DFLS thrust with a relative error below 3.5%.

A key puzzle in search, ads, and recommendation is that the ranking model can only utilize a small portion of the vastly available user interaction data. As a result, increasing data volume, model size, or computation FLOPs will quickly suffer from diminishing returns. We examined this problem and found that one of the root causes may lie in the so-called ``item-centric'' formulation, which has an unbounded vocabulary and thus uncontrolled model complexity. To mitigate quality saturation, we introduce an alternative formulation named ``user-centric ranking'', which is based on a transposed view of the dyadic user-item interaction data. We show that this formulation has a promising scaling property, enabling us to train better-converged models on substantially larger data sets., Comment: accepted to publish at KDD 2023

This study explores university students' perceptions of generative AI (GenAI) technologies, such as ChatGPT, in higher education, focusing on familiarity, their willingness to engage, potential benefits and challenges, and effective integration. A survey of 399 undergraduate and postgraduate students from various disciplines in Hong Kong revealed a generally positive attitude towards GenAI in teaching and learning. Students recognized the potential for personalized learning support, writing and brainstorming assistance, and research and analysis capabilities. However, concerns about accuracy, privacy, ethical issues, and the impact on personal development, career prospects, and societal values were also expressed. According to John Biggs' 3P model, student perceptions significantly influence learning approaches and outcomes. By understanding students' perceptions, educators and policymakers can tailor GenAI technologies to address needs and concerns while promoting effective learning outcomes. Insights from this study can inform policy development around the integration of GenAI technologies into higher education. By understanding students' perceptions and addressing their concerns, policymakers can create well-informed guidelines and strategies for the responsible and effective implementation of GenAI tools, ultimately enhancing teaching and learning experiences in higher education.

The aim of this paper is to propose a new method to construct exponential attractors for infinite dimensional dynamical systems in Banach spaces with explicit fractal dimension. The approach is established by combing the squeezing properties and the covering of finite subspace of Banach spaces, which generalize the method established in Hilbert spaces. The constructed exponential attractors possess explicit fractal dimensions which do not depend on the entropy number but only depend on the spectrum of the linear part and Lipschitz constant of the nonlinear part. The method is especially effective for functional differential equations in Banach spaces for which sate decomposition of the linear part can be adopted to prove squeezing property. The theoretical results are applied to the retarded functional differential equation and retarded reaction-diffusion equations., Comment: arXiv admin note: substantial text overlap with arXiv:2303.04094

The main purpose of this work is to characterize the almost sure local structure stability of solutions to a class of linear stochastic partial functional differential equations (SPFDEs) by investigating the Lyapunov exponents and invariant manifolds near the stationary point. It is firstly proved that the trajectory field of the stochastic delayed stochastic partial functional differential equation admits an almost sure continuous version which is compact for $t>\tau$ by a delicate construction based on the random semiflow generated by the diffusion term. Then it is proved that the version generates a random dynamical system(RDS) by the Wong-Zakai approximation of the stochastic partial differential equation constructed by the diffusion term. Subsequently, it is shown that the constructed linear cocycle admits fixed (at most) countable set of Lyapunov exponents and the associate Oseledets random filtration of the Banach space is obtained by adopting the infinite-dimensional multiplicative ergodic theorem in Banach spaces established by Lian and Lu [\textit{Mem Amer Math Soc, 2010, 206: 967}]. As a by product, the stable-manifolds theorem for the linear SPFDE in the hyperbolic case is also established., Comment: The paper needs to be revised

The main objective of this paper is to obtain estimations of Hausdorff dimension as well as fractal dimension of global attractors and pullback attractors for both autonomous and nonautonomous functional differential equations (FDEs) in Banach spaces. New criterions for the finite Hausdorff dimension and fractal dimension of attractors in Banach spaces are firslty proposed by combining the squeezing property and the covering of finite subspace of Banach spaces, which generalize the method established in Hilbert spaces. In order to surmount the barrier caused by the lack of orthogonal projectors with finite rank, which is the key tool for proving the squeezing property of partial differential equations in Hilbert spaces, we adopt the state decomposition of phase space based on the exponential dichotomy of the studied FDEs to obtain similar squeezing property. The theoretical results are applied to a retarded nonlinear reaction-diffusion equation and a non-autonomous retarded functional differential equation in the natural phase space, for which explicit bounds of dimensions that do not depend on the entropy number but only depend on the spectrum of the linear parts and Lipschitz constants of the nonlinear parts are obtained.

The purpose of this paper is to investigate the existence and estimation of Hausdorff and fractal dimension of global attractors for a delayed reaction-diffusion equation on an unbounded domain. The noncompactness of the domain cause the Laplace operator has a continuous spectrum, the semigroup generated by the linear part and the Sobolev embeddings no longer compact, making the problem more difficult compared with the equation on bounded domain. In order to adopt Hausdorff and fractal dimension dimension estimation tools established for dynamical systems in Hilbert space, we recast the equation in an auxiliary Hilbert space. We first obtain the existence of global solutions on unbounded domain by a perturbation semigroup approach which generates an infinite dimensional dynamical system. Then, we show the existence of global attractors by firstly showing the existence of an absorbing set and then give a an uniform a priori estimates for far-field values of solutions which facilitate us to prove the asymptotic compactness of the generated dynamical system. After establishing the variational equation in the auxiliary Hilbert space and the differentiable properties of the generated dynamical system, the upper estimate of both Hausdorff and fractal dimensions of the global attractors is obtained., Comment: Needs to be revised

The global asymptotic behavior of a stochastic Hopfield neural network model (HNNM) with delays is explored by studying the existence and structure of random attractors. It is first proved that the trajectory field of the stochastic delayed HNNM admits an almost sure continuous version, which is compact for $t>\tau$ (where $\tau$ is the delay) by a delicate construction based on the random semiflow generated by the diffusion term. Then, this version is shown to generate a random dynamical system (RDS) by piece-wise linear approximation, after which the existence of a random absorbing set is obtained by a careful uniform apriori estimate of the solutions. Subsequently, the pullback asymptotic compactness of the RDS generated by the stochastic delayed HNNM is proved and hence the existence of random attractors is obtained. Moreover, sufficient conditions under which the attractors turn out to be an exponential attracting stationary solution are given. Numerical simulations are also conducted at last to illustrate the effectiveness of the established results.

The aim of this paper is to obtain an estimation of Hausdorff as well as fractal dimensions of random attractors for a class of stochastic partial differential equations with delay. The stochastic equation is first transformed into a delayed random partial differential equation by means of a random conjugation, which is then recast into an auxiliary Hilbert space. For the obtained equation, it is firstly proved that it generates a random dynamical system (RDS) in the auxiliary Hilbert space. Then it is shown that the equation possesses random attractors by a uniform estimate of the solution and the asymptotic compactness of the generated RDS. After establishing the variational equation in the auxiliary Hilbert space and the $\mathbb{P}$ almost surely differentiable properties of the RDS, an upper estimate of both Hausdorff and fractal dimensions of the random attractors are obtained.

The aim of this paper is to prove the existence and qualitative property of random attractors for a stochastic nonlocal delayed reaction-diffusion equation (SNDRDE) on a semi-infinite interval with a Dirichlet boundary condition on the finite end. This equation models the spatial-temporal evolution of the mature individuals for a two-stage species whose juvenile and adults both diffuse that lives on a semi-infinite domain and subject to random perturbations. By transforming the SNDRDE into a random evolution equation with delay, by means of a stationary conjugate transformation, we first establish the global existence and uniqueness of solutions to the equation, after which we show the solutions generate a random dynamical system. Then, we deduce uniform a priori estimates of the solutions and show the existence of bounded random absorbing sets. Subsequently, we prove the pullback asymptotic compactness of the random dynamical system generated by the SNDRDE with respect to the compact open topology, and hence obtain the existence of random attractors. At last, it is proved that the random attractor is an exponentially attracting stationary solution under appropriate conditions.

The aim of this paper is to prove the existence and smoothness of stable and unstable invariant manifolds for a stochastic delayed partial differential equation of parabolic type. The stochastic delayed partial differential equation is firstly transformed into a random delayed partial differential equation by a conjugation, which is then recast into a Hilbert space. For the auxiliary equation, the variation of constants formula holds and we show the existence of Lipschitz continuous stable and unstable manifolds by the Lyapunov-Perron method. Subsequently, we prove the smoothness of these invariant manifolds under appropriate spectral gap condition by carefully investigating the smoothness of auxiliary equation, after which, we obtain the invariant manifolds of the original equation by projection and inverse transformation. Eventually, we illustrate the obtained theoretical results by their application to a stochastic single-species population model.

Suppose we have available individual data from an internal study and various types of summary statistics from relevant external studies. External summary statistics have been used as constraints on the internal data distribution, which promised to improve the statistical inference in the internal data; however, the additional use of external summary data may lead to paradoxical results: efficiency loss may occur if the uncertainty of summary statistics is not negligible and large estimation bias can emerge even if the bias of external summary statistics is small. We investigate these paradoxical results in a semiparametric framework. We establish the semiparametric efficiency bound for estimating a general functional of the internal data distribution, which is shown to be no larger than that using only internal data. We propose a data-fused efficient estimator that achieves this bound so that the efficiency paradox is resolved. Besides, a debiased estimator is further proposed which has selection consistency property by employing adaptive lasso penalty so that the resultant estimator can achieve the same asymptotic distribution as the oracle one that uses only unbiased summary statistics, which resolves the bias paradox. Simulations and application to a Helicobacter pylori infection dataset are used to illustrate the proposed methods., Comment: 17 pages, 3 figures

In biomedical studies, estimating drug effects on chronic diseases requires a long follow-up period, which is difficult to meet in randomized clinical trials (RCTs). The use of a short-term surrogate to replace the long-term outcome for assessing the drug effect relies on stringent assumptions that empirical studies often fail to satisfy. Motivated by a kidney disease study, we investigate the drug effects on long-term outcomes by combining an RCT without observation of long-term outcome and an observational study in which the long-term outcome is observed but unmeasured confounding may exist. Under a mean exchangeability assumption weaker than the previous literature, we identify the average treatment effects in the RCT and derive the associated efficient influence function and semiparametric efficiency bound. Furthermore, we propose a locally efficient doubly robust estimator and an inverse probability weighted (IPW) estimator. The former attains the semiparametric efficiency bound if all the working models are correctly specified. The latter has a simpler form and requires much fewer model specifications. The IPW estimator using estimated propensity scores is more efficient than that using true propensity scores and achieves the semiparametric efficient bound in the case of discrete covariates and surrogates with finite support. Both estimators are shown to be consistent and asymptotically normally distributed. Extensive simulations are conducted to evaluate the finite-sample performance of the proposed estimators. We apply the proposed methods to estimate the efficacy of oral hydroxychloroquine on renal failure in a real-world data analysis.

Accurate recommendation and reliable explanation are two key issues for modern recommender systems. However, most recommendation benchmarks only concern the prediction of user-item ratings while omitting the underlying causes behind the ratings. For example, the widely-used Yahoo!R3 dataset contains little information on the causes of the user-movie ratings. A solution could be to conduct surveys and require the users to provide such information. In practice, the user surveys can hardly avoid compliance issues and sparse user responses, which greatly hinders the exploration of causality-based recommendation. To better support the studies of causal inference and further explanations in recommender systems, we propose a novel semi-synthetic data generation framework for recommender systems where causal graphical models with missingness are employed to describe the causal mechanism of practical recommendation scenarios. To illustrate the use of our framework, we construct a semi-synthetic dataset with Causal Tags And Ratings (CTAR), based on the movies as well as their descriptive tags and rating information collected from a famous movie rating website. Using the collected data and the causal graph, the user-item-ratings and their corresponding user-item-tags are automatically generated, which provides the reasons (selected tags) why the user rates the items. Descriptive statistics and baseline results regarding the CTAR dataset are also reported. The proposed data generation framework is not limited to recommendation, and the released APIs can be used to generate customized datasets for other research tasks., Comment: 10 pages, 6 figures, 4 tables

Recently, recommender system (RS) based on causal inference has gained much attention in the industrial community, as well as the states of the art performance in many prediction and debiasing tasks. Nevertheless, a unified causal analysis framework has not been established yet. Many causal-based prediction and debiasing studies rarely discuss the causal interpretation of various biases and the rationality of the corresponding causal assumptions. In this paper, we first provide a formal causal analysis framework to survey and unify the existing causal-inspired recommendation methods, which can accommodate different scenarios in RS. Then we propose a new taxonomy and give formal causal definitions of various biases in RS from the perspective of violating the assumptions adopted in causal analysis. Finally, we formalize many debiasing and prediction tasks in RS, and summarize the statistical and machine learning-based causal estimation methods, expecting to provide new research opportunities and perspectives to the causal RS community.

Covariate-specific treatment effects (CSTEs) represent heterogeneous treatment effects across subpopulations defined by certain selected covariates. In this article, we consider marginal structural models where CSTEs are linearly represented using a set of basis functions of the selected covariates. We develop a new approach in high-dimensional settings to obtain not only doubly robust point estimators of CSTEs, but also model-assisted confidence intervals, which are valid when a propensity score model is correctly specified but an outcome regression model may be misspecified. With a linear outcome model and subpopulations defined by discrete covariates, both point estimators and confidence intervals are doubly robust for CSTEs. In contrast, confidence intervals from existing high-dimensional methods are valid only when both the propensity score and outcome models are correctly specified. We establish asymptotic properties of the proposed point estimators and the associated confidence intervals. We present simulation studies and empirical applications which demonstrate the advantages of the proposed method compared with competing ones.

Identification of treatment effects in the presence of unmeasured confounding is a persistent problem in the social, biological, and medical sciences. The problem of unmeasured confounding in settings with multiple treatments is most common in statistical genetics and bioinformatics settings, where researchers have developed many successful statistical strategies without engaging deeply with the causal aspects of the problem. Recently there have been a number of attempts to bridge the gap between these statistical approaches and causal inference, but these attempts have either been shown to be flawed or have relied on fully parametric assumptions. In this paper, we propose two strategies for identifying and estimating causal effects of multiple treatments in the presence of unmeasured confounding. The auxiliary variables approach leverages variables that are not causally associated with the outcome; in the case of a univariate confounder, our method only requires one auxiliary variable, unlike existing instrumental variable methods that would require as many instruments as there are treatments. An alternative null treatments approach relies on the assumption that at least half of the confounded treatments have no causal effect on the outcome, but does not require a priori knowledge of which treatments are null. Our identification strategies do not impose parametric assumptions on the outcome model and do not rest on estimation of the confounder. This paper extends and generalizes existing work on unmeasured confounding with a single treatment and models commonly used in bioinformatics.

Electricity theft, the behavior that involves users conducting illegal operations on electrical meters to avoid individual electricity bills, is a common phenomenon in the developing countries. Considering its harmfulness to both power grids and the public, several mechanized methods have been developed to automatically recognize electricity-theft behaviors. However, these methods, which mainly assess users' electricity usage records, can be insufficient due to the diversity of theft tactics and the irregularity of user behaviors. In this paper, we propose to recognize electricity-theft behavior via multi-source data. In addition to users' electricity usage records, we analyze user behaviors by means of regional factors (non-technical loss) and climatic factors (temperature) in the corresponding transformer area. By conducting analytical experiments, we unearth several interesting patterns: for instance, electricity thieves are likely to consume much more electrical power than normal users, especially under extremely high or low temperatures. Motivated by these empirical observations, we further design a novel hierarchical framework for identifying electricity thieves. Experimental results based on a real-world dataset demonstrate that our proposed model can achieve the best performance in electricity-theft detection (e.g., at least +3.0% in terms of F0.5) compared with several baselines. Last but not least, our work has been applied by the State Grid of China and used to successfully catch electricity thieves in Hangzhou with a precision of 15% (an improvement form 0% attained by several other models the company employed) during monthly on-site investigation., Comment: 11 pages, 8 figures, WWW'20 full paper