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We propose a novel approach to numerically approximate McKean-Vlasov stochastic differential equations (MV-SDE) using stochastic gradient descent (SGD) while avoiding the use of interacting particle systems. The technique of SGD is deployed to solve a Euclidean minimization problem, which is obtained by first representing the MV-SDE as a minimization problem over the set of continuous functions of time, and then by approximating the domain with a finite-dimensional subspace. Convergence is established by proving certain intermediate stability and moment estimates of the relevant stochastic processes (including the tangent ones). Numerical experiments illustrate the competitive performance of our SGD based method compared to the IPS benchmarks. This work offers a theoretical foundation for using the SGD method in the context of numerical approximation of MV-SDEs, and provides analytical tools to study its stability and convergence.

This article investigates a local discontinuous Galerkin (LDG) method for one-dimensional and two-dimensional singularly perturbed reaction-diffusion problems on a Shishkin mesh. During this process, due to the inability of the energy norm to fully capture the behavior of the boundary layers appearing in the solutions, a balanced norm is introduced. By designing novel numerical fluxes and constructing special interpolations, optimal convergences under the balanced norm are achieved in both 1D and 2D cases. Numerical experiments support the main theoretical conclusions.

We present a relational representation of odd Sugihara chains. The elements of the algebra are represented as weakening relations over a particular poset which consists of two densely embedded copies of the rationals. Our construction mimics that of Maddux (2010) where a relational representation of the even Sugihara chains is given. An order automorphism between the two copies of the rationals is the key to ensuring that the identity element of the monoid is fixed by the involution.

The recently-emerging field of higher order MDS codes has sought to unify a number of concepts in coding theory. Such areas captured by higher order MDS codes include maximally recoverable (MR) tensor codes, codes with optimal list-decoding guarantees, and codes with constrained generator matrices (as in the GM-MDS theorem). By proving these equivalences, Brakensiek-Gopi-Makam showed the existence of optimally list-decodable Reed-Solomon codes over exponential sized fields. Building on this, recent breakthroughs by Guo-Zhang and Alrabiah-Guruswami-Li have shown that randomly punctured Reed-Solomon codes achieve list-decoding capacity (which is a relaxation of optimal list-decodability) over linear size fields. We extend these works by developing a formal theory of relaxed higher order MDS codes. In particular, we show that there are two inequivalent relaxations which we call lower and upper relaxations. The lower relaxation is equivalent to relaxed optimal list-decodable codes and the upper relaxation is equivalent to relaxed MR tensor codes with a single parity check per column. We then generalize the techniques of GZ and AGL to show that both these relaxations can be constructed over constant size fields by randomly puncturing suitable algebraic-geometric codes. For this, we crucially use the generalized GM-MDS theorem for polynomial codes recently proved by Brakensiek-Dhar-Gopi. We obtain the following corollaries from our main result. First, randomly punctured AG codes of rate $R$ achieve list-decoding capacity with list size $O(1/\epsilon)$ and field size $\exp(O(1/\epsilon^2))$. Prior to this work, AG codes were not even known to achieve list-decoding capacity. Second, by randomly puncturing AG codes, we can construct relaxed MR tensor codes with a single parity check per column over constant-sized fields, whereas (non-relaxed) MR tensor codes require exponential field size.

In this paper the interpolating rational functions introduced by Floater and Hormann are generalized leading to a whole new family of rational functions depending on $\gamma$, an additional positive integer parameter. For $\gamma = 1$, the original Floater--Hormann interpolants are obtained. When $\gamma>1$ we prove that the new rational functions share a lot of the nice properties of the original Floater--Hormann functions. Indeed, for any configuration of nodes in a compact interval, they have no real poles, interpolate the given data, preserve the polynomials up to a certain fixed degree, and have a barycentric-type representation. Moreover, we estimate the associated Lebesgue constants in terms of the minimum ($h^*$) and maximum ($h$) distance between two consecutive nodes. It turns out that, in contrast to the original Floater-Hormann interpolants, for all $\gamma > 1$ we get uniformly bounded Lebesgue constants in the case of equidistant and quasi-equidistant nodes configurations (i.e., when $h\sim h^*$). For such configurations, as the number of nodes tends to infinity, we prove that the new interpolants ($\gamma>1$) uniformly converge to the interpolated function $f$, for any continuous function $f$ and all $\gamma>1$. The same is not ensured by the original FH interpolants ($\gamma=1$). Moreover, we provide uniform and pointwise estimates of the approximation error for functions having different degrees of smoothness. Numerical experiments illustrate the theoretical results and show a better error profile for less smooth functions compared to the original Floater-Hormann interpolants.

The digitalization of energy sectors has expanded the coding responsibilities for power engineers and researchers. This research article explores the potential of leveraging Large Language Models (LLMs) to alleviate this burden. Here, we propose LLM-based frameworks for different programming tasks in power systems. For well-defined and routine tasks like the classic unit commitment (UC) problem, we deploy an end-to-end framework to systematically assesses four leading LLMs-ChatGPT 3.5, ChatGPT 4.0, Claude and Google Bard in terms of success rate, consistency, and robustness. For complex tasks with limited prior knowledge, we propose a human-in-the-loop framework to enable engineers and LLMs to collaboratively solve the problem through interactive-learning of method recommendation, problem de-composition, subtask programming and synthesis. Through a comparative study between two frameworks, we find that human-in-the-loop features like web access, problem decomposition with field knowledge and human-assisted code synthesis are essential as LLMs currently still fall short in acquiring cutting-edge and domain-specific knowledge to complete a holistic problem-solving project.

Determining the weight distribution of a code is an old and fundamental topic in coding theory that has been thoroughly studied. In 1977, Helleseth, Kl{\o}ve, and Mykkeltveit presented a weight enumerator polynomial of the lifted code over $\mathbb{F}_{q^\ell}$ of a $q$-ary linear code with significant combinatorial properties, which can determine the support weight distribution of this linear code. The Solomon-Stiffler codes are a family of famous Griesmer codes, which were proposed by Solomon and Stiffler in 1965. In this paper, we determine the weight enumerator polynomials of the lifted codes of the projective Solomon-Stiffler codes using some combinatorial properties of subspaces. As a result, we determine the support weight distributions of the projective Solomon-Stiffler codes. In particular, we determine the weight hierarchies of the projective Solomon-Stiffler codes.

We provide a new characterization of both belief update and belief revision in terms of a Kripke-Lewis semantics. We consider frames consisting of a set of states, a Kripke belief relation and a Lewis selection function. Adding a valuation to a frame yields a model. Given a model and a state, we identify the initial belief set K with the set of formulas that are believed at that state and we identify either the updated belief set or the revised belief set, prompted by the input represented by formula A, as the set of formulas that are the consequent of conditionals that (1) are believed at that state and (2) have A as antecedent. We show that this class of models characterizes both the Katsuno-Mendelzon (KM) belief update functions and the AGM belief revision functions, in the following sense: (1) each model gives rise to a partial belief function that can be completed into a full KM/AGM update/revision function, and (2) for every KM/AGM update/revision function there is a model whose associated belief function coincides with it. The difference between update and revision can be reduced to two semantic properties that appear in a stronger form in revision relative to update, thus confirming the finding by Peppas et al. (1996) that, "for a fixed theory K, revising K is much the same as updating K"

We consider the problem of discovering $K$ related Gaussian directed acyclic graphs (DAGs), where the involved graph structures share a consistent causal order and sparse unions of supports. Under the multi-task learning setting, we propose a $l_1/l_2$-regularized maximum likelihood estimator (MLE) for learning $K$ linear structural equation models. We theoretically show that the joint estimator, by leveraging data across related tasks, can achieve a better sample complexity for recovering the causal order (or topological order) than separate estimations. Moreover, the joint estimator is able to recover non-identifiable DAGs, by estimating them together with some identifiable DAGs. Lastly, our analysis also shows the consistency of union support recovery of the structures. To allow practical implementation, we design a continuous optimization problem whose optimizer is the same as the joint estimator and can be approximated efficiently by an iterative algorithm. We validate the theoretical analysis and the effectiveness of the joint estimator in experiments.

Recommender systems are widely used in big information-based companies such as Google, Twitter, LinkedIn, and Netflix. A recommender system deals with the problem of information overload by filtering important information fragments according to users' preferences. In light of the increasing success of deep learning, recent studies have proved the benefits of using deep learning in various recommendation tasks. However, most proposed techniques only aim to target individuals, which cannot be efficiently applied in group recommendation. In this paper, we propose a deep learning architecture to solve the group recommendation problem. On the one hand, as different individual preferences in a group necessitate preference trade-offs in making group recommendations, it is essential that the recommendation model can discover substitutes among user behaviors. On the other hand, it has been observed that a user as an individual and as a group member behaves differently. To tackle such problems, we propose using an attention mechanism to capture the impact of each user in a group. Specifically, our model automatically learns the influence weight of each user in a group and recommends items to the group based on its members' weighted preferences. We conduct extensive experiments on four datasets. Our model significantly outperforms baseline methods and shows promising results in applying deep learning to the group recommendation problem.