Miracle-dz/UniFAR-FTUs huggingface dataset:
The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation

Abstract We consider the existence and nonexistence of the positive solution for the following Brézis-Nirenberg problem with logarithmic perturbation: − Δ u = ∣ u ∣ 2 ∗ − 2 u + λ u + μ u log u 2 x ∈ Ω , u = 0 x ∈ ∂ Ω , \left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}-\Delta u={| u| }^{{2}^{\ast }-2}u+\lambda u+\mu u\log {u}^{2}\hspace{1.0em}& x\in \Omega ,\\ u=0\hspace{1.0em}& x\in \partial \Omega ,\end{array}\right. where Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} is a bounded open domain, λ , μ ∈ R \lambda ,\mu \in {\mathbb{R}} , N ≥ 3 N\ge 3 and 2 ∗ ≔ 2 N N − 2 {2}^{\ast }:= \frac{2N}{N-2} is the critical Sobolev exponent for the embedding H 0 1 ( Ω ) ↪ L 2 ∗ ( Ω ) {H}_{0}^{1}\left(\Omega )\hspace{0.33em}\hookrightarrow \hspace{0.33em}{L}^{{2}^{\ast }}\left(\Omega ) . The uncertainty of the sign of s log s 2 s\log {s}^{2} in ( 0 , + ∞ ) \left(0,+\infty ) has some interest in itself. We will show the existence of positive ground state solution, which is of mountain pass type provided λ ∈ R , μ > 0 \lambda \in {\mathbb{R}},\mu \gt 0 and N ≥ 4 N\ge 4 . While the case of μ < 0 \mu \lt 0 is thornier. However, for N = 3 , 4 N=3,4 , λ ∈ ( − ∞ , λ 1 ( Ω ) ) \lambda \in \left(-\infty ,{\lambda }_{1}\left(\Omega )) , we can also establish the existence of positive solution under some further suitable assumptions. A nonexistence result is also obtained for μ < 0 \mu \lt 0 and − ( N − 2 ) μ 2 + ( N − 2 ) μ 2 log − ( N − 2 ) μ 2 + λ − λ 1 ( Ω ) ≥ 0 -\frac{\left(N-2)\mu }{2}+\frac{\left(N-2)\mu }{2}\log \left(-\frac{\left(N-2)\mu }{2}\right)+\lambda -{\lambda }_{1}\left(\Omega )\ge 0 if N ≥ 3 N\ge 3 . Comparing with the results in the study by Brézis and Nirenberg (Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477), some new interesting phenomenon occurs when the parameter μ \mu on logarithmic perturbation is not zero.
Nuclear star cluster formation in star-forming dwarf galaxies

Nuclear star clusters (NSCs) are massive star clusters found in all types of galaxies from dwarfs to massive galaxies. Recent studies show that while low-mass NSCs in dwarf galaxies ($M_\text{gal}<10^{9} M_\odot$) form predominantly out of the merger of globular clusters (GCs), high-mass NSCs in massive galaxies have assembled most of their mass through central enriched star formation. So far, these results of a transition in the dominant NSC formation channel have been based on studies of early-type galaxies and massive late-type galaxies. Here, we present the first spectroscopic analysis of a sample of nine nucleated late-type dwarf galaxies with the aim of identifying the dominant NSC formation pathway. We use integral-field spectroscopy data obtained with the Multi Unit Spectroscopic Explorer (MUSE) instrument to analyse the ages, metallicities, star formation histories, and star formation rates of the NSCs and their surroundings. Our sample includes galaxies with stellar masses $M_\text{gal} = 10^7 - 10^9 M_\odot$ and NSC masses $M_\text{NSC} = 6 \times 10^4 - 6 \times 10^{6} M_\odot$. Although all NSC spectra show emission lines, this emission is not always connected to star formation within the NSC, but rather to other regions along the line-of-sight. The NSC star formation histories reveal that metal-poor and old populations dominate the stellar populations in five NSCs, possibly stemming from the inspiral of GCs. The NSCs of the most massive galaxies in our sample show significant contributions from young and enriched populations that indicate additional mass growth through central star formation. Our results support previous findings of a transition in the dominant NSC formation channel with galaxy mass, showing that the NSCs in low-mass galaxies predominantly grow through the inspiral of GCs, while central star formation can contribute to NSC growth in more massive galaxies.
VICRegL: Self-Supervised Learning of Local Visual Features

Most recent self-supervised methods for learning image representations focus on either producing a global feature with invariance properties, or producing a set of local features. The former works best for classification tasks while the latter is best for detection and segmentation tasks. This paper explores the fundamental trade-off between learning local and global features. A new method called VICRegL is proposed that learns good global and local features simultaneously, yielding excellent performance on detection and segmentation tasks while maintaining good performance on classification tasks. Concretely, two identical branches of a standard convolutional net architecture are fed two differently distorted versions of the same image. The VICReg criterion is applied to pairs of global feature vectors. Simultaneously, the VICReg criterion is applied to pairs of local feature vectors occurring before the last pooling layer. Two local feature vectors are attracted to each other if their l2-distance is below a threshold or if their relative locations are consistent with a known geometric transformation between the two input images. We demonstrate strong performance on linear classification and segmentation transfer tasks. Code and pretrained models are publicly available at: https://github.com/facebookresearch/VICRegL
Exploring Parameter-Efficient Fine-tuning for Improving Communication Efficiency in Federated Learning

Federated learning (FL) has emerged as a promising paradigm for enabling the collaborative training of models without centralized access to the raw data on local devices. In the typical FL paradigm (e.g., FedAvg), model weights are sent to and from the server each round to participating clients. However, this can quickly put a massive communication burden on the system, especially if more capable models beyond very small MLPs are employed. Recently, the use of pre-trained models has been shown effective in federated learning optimization and improving convergence. This opens the door for new research questions. Can we adjust the weight-sharing paradigm in federated learning, leveraging strong and readilyavailable pre-trained models, to significantly reduce the communication burden while simultaneously achieving excellent performance? To this end, we investigate the use of parameter-efficient fine-tuning in federated learning. Specifically, we systemically evaluate the performance of several parameter-efficient fine-tuning methods across a variety of client stability, data distribution, and differential privacy settings. By only locally tuning and globally sharing a small portion of the model weights, significant reductions in the total communication overhead can be achieved while maintaining competitive performance in a wide range of federated learning scenarios, providing insight into a new paradigm for practical and effective federated systems.
Learning of Dynamical Systems under Adversarial Attacks - Null Space Property Perspective

We study the identification of a linear time-invariant dynamical system affected by large-and-sparse disturbances modeling adversarial attacks or faults. Under the assumption that the states are measurable, we develop necessary and sufficient conditions for the recovery of the system matrices by solving a constrained lasso-type optimization problem. In addition, we provide an upper bound on the estimation error whenever the disturbance sequence is a combination of small noise values and large adversarial values. Our results depend on the null space property that has been widely used in the lasso literature, and we investigate under what conditions this property holds for linear time-invariant dynamical systems. Lastly, we further study the conditions for a specific probabilistic model and support the results with numerical experiments.
Primordial Black Holes as Dark Matter: Recent Developments

Although the dark matter is usually assumed to be made up of some form of elementary particle, primordial black holes (PBHs) could also provide some of it. However, various constraints restrict the possible mass windows to 1016–1017 g, 1020–1024 g, and 10–103 M⊙. The last possibility is contentious but of special interest in view of the recent detection of black hole mergers by LIGO/Virgo. PBHs might have important consequences and resolve various cosmological conundra even if they account for only a small fraction of the dark matter density. In particular, those larger than 103 M⊙ could generate cosmological structures through the seed or Poisson effect, thereby alleviating some problems associated with the standard cold dark matter scenario, and sufficiently large PBHs might provide seeds for the supermassive black holes in galactic nuclei. More exotically, the Planck-mass relics of PBH evaporations or stupendously large black holes bigger than 1012 M⊙ could provide an interesting dark component.
GaIA: Graphical Information Gain based Attention Network for Weakly Supervised Point Cloud Semantic Segmentation

While point cloud semantic segmentation is a significant task in 3D scene understanding, this task demands a time-consuming process of fully annotating labels. To address this problem, recent studies adopt a weakly supervised learning approach under the sparse annotation. Different from the existing studies, this study aims to reduce the epistemic uncertainty measured by the entropy for a precise semantic segmentation. We propose the graphical information gain based attention network called GaIA, which alleviates the entropy of each point based on the reliable information. The graphical information gain discriminates the reliable point by employing relative entropy between target point and its neighborhoods. We further introduce anchor-based additive angular margin loss, ArcPoint. The ArcPoint optimizes the unlabeled points containing high entropy towards semantically similar classes of the labeled points on hypersphere space. Experimental results on S3DIS and ScanNet-v2 datasets demonstrate our framework outperforms the existing weakly supervised methods.
Influence Maximization: Divide and Conquer

The problem of influence maximization, i.e., finding the set of nodes having maximal influence on a network, is of great importance for several applications. In the past two decades, many heuristic metrics to spot influencers have been proposed. Here, we introduce a framework to boost the performance of such metrics. The framework consists in dividing the network into sectors of influence, and then selecting the most influential nodes within these sectors. We explore three different methodologies to find sectors in a network: graph partitioning, graph hyperbolic embedding, and community structure. The framework is validated with a systematic analysis of real and synthetic networks. We show that the gain in performance generated by dividing a network into sectors before selecting the influential spreaders increases as the modularity and heterogeneity of the network increase. Also, we show that the division of the network into sectors can be efficiently performed in a time that scales linearly with the network size, thus making the framework applicable to large-scale influence maximization problems.
Fragment Dynamics in Active Asteroid 331P/Gibbs

We present a dynamical analysis of the fragmented active asteroid 331P/Gibbs. Using archival images taken by the Hubble Space Telescope from 2015 to 2018, we measured the astrometry of the primary and the three brightest (presumably the largest) components. Conventional orbit determination revealed a high degree of orbital similarity between the components. We then applied a fragmentation model to fit the astrometry, obtaining key parameters including the fragmentation epochs and separation velocities. Our best-fit models show that Fragment B separated from the primary body at a speed of ∼1 cm s−1 between 2011 April and May, whereas two plausible scenarios were identified for Fragments A and C. The former split either from the primary or from Fragment B, in 2011 mid June at a speed of ∼8 cm s−1, and the latter split from Fragment B either in late 2011 or between late 2013 and early 2014, at a speed of ∼0.7–0.8 cm s−1. The results are consistent with rotational disruption as the mechanism causing the cascading fragmentation of the asteroid, as suggested by the rapid rotation of the primary. The fragments constitute the youngest known asteroid cluster, providing us with a great opportunity to study asteroid fragmentation and formation of asteroid clusters.
SAM as an Optimal Relaxation of Bayes

Sharpness-aware minimization (SAM) and related adversarial deep-learning methods can drastically improve generalization, but their underlying mechanisms are not yet fully understood. Here, we establish SAM as a relaxation of the Bayes objective where the expected negative-loss is replaced by the optimal convex lower bound, obtained by using the so-called Fenchel biconjugate. The connection enables a new Adam-like extension of SAM to automatically obtain reasonable uncertainty estimates, while sometimes also improving its accuracy. By connecting adversarial and Bayesian methods, our work opens a new path to robustness.
Sharp Analysis of Stochastic Optimization under Global Kurdyka-Lojasiewicz Inequality

We study the complexity of finding the global solution to stochastic nonconvex optimization when the objective function satisfies global Kurdyka-Lojasiewicz (KL) inequality and the queries from stochastic gradient oracles satisfy mild expected smoothness assumption. We first introduce a general framework to analyze Stochastic Gradient Descent (SGD) and its associated nonlinear dynamics under the setting. As a byproduct of our analysis, we obtain a sample complexity of $\mathcal{O}(\epsilon^{-(4-\alpha)/\alpha})$ for SGD when the objective satisfies the so called $\alpha$-PL condition, where $\alpha$ is the degree of gradient domination. Furthermore, we show that a modified SGD with variance reduction and restarting (PAGER) achieves an improved sample complexity of $\mathcal{O}(\epsilon^{-2/\alpha})$ when the objective satisfies the average smoothness assumption. This leads to the first optimal algorithm for the important case of $\alpha=1$ which appears in applications such as policy optimization in reinforcement learning.
Magnetic field, differential rotation and activity of the hot-Jupiter-hosting star HD 179949

HD 179949 is an F8V star, orbited by a giant planet at similar to 8 R? every 3.092 514 d. The system was reported to undergo episodes of stellar activity enhancement modulated by the orbital period, interpreted as caused by starplanet interactions (SPIs). One possible cause of SPIs is the large-scale magnetic field of the host star in which the close-in giant planet orbits. In this paper we present spectropolarimetric observations of HD 179949 during two observing campaigns (2009 September and 2007 June). We detect a weak large-scale magnetic field of a few gauss at the surface of the star. The field configuration is mainly poloidal at both observing epochs. The star is found to rotate differentially, with a surface rotation shear of dO= 0.216 +/- 0.061 rad d-1, corresponding to equatorial and polar rotation periods of 7.62 +/- 0.07 and 10.3 +/- 0.8 d, respectively. The coronal field estimated by extrapolating the surface maps resembles a dipole tilted at similar to 70 degrees. We also find that the chromospheric activity of HD 179949 is mainly modulated by the rotation of the star, with two clear maxima per rotation period as expected from a highly tilted magnetosphere. In 2009 September, we find that the activity of HD 179949 shows hints of low-amplitude fluctuations with a period close to the beat period of the system.
Random orthogonal additive filters: a solution to the vanishing/exploding gradient of deep neural networks

Since the recognition in the early nineties of the vanishing/exploding (V/E) gradient issue plaguing the training of neural networks (NNs), significant efforts have been exerted to overcome this obstacle. However, a clear solution to the V/E issue remained elusive so far. In this manuscript a new architecture of NN is proposed, designed to mathematically prevent the V/E issue to occur. The pursuit of approximate dynamical isometry, i.e. parameter configurations where the singular values of the input-output Jacobian are tightly distributed around 1, leads to the derivation of a NN's architecture that shares common traits with the popular Residual Network model. Instead of skipping connections between layers, the idea is to filter the previous activations orthogonally and add them to the nonlinear activations of the next layer, realising a convex combination between them. Remarkably, the impossibility for the gradient updates to either vanish or explode is demonstrated with analytical bounds that hold even in the infinite depth case. The effectiveness of this method is empirically proved by means of training via backpropagation an extremely deep multilayer perceptron of 50k layers, and an Elman NN to learn long-term dependencies in the input of 10k time steps in the past. Compared with other architectures specifically devised to deal with the V/E problem, e.g. LSTMs for recurrent NNs, the proposed model is way simpler yet more effective. Surprisingly, a single layer vanilla RNN can be enhanced to reach state of the art performance, while converging super fast; for instance on the psMNIST task, it is possible to get test accuracy of over 94% in the first epoch, and over 98% after just 10 epochs.
Derived Character Maps of Groups Representations

In this paper, we construct and study derived character maps of ﬁnite-dimensional representa- tions of ∞ -groups. As models for ∞ -groups we take homotopy simplicial groups, i.e. homotopy simplicial G op -algebras over the algebraic theory of groups (in the sense of [4]). We deﬁne cyclic, symmetric and representation homology for ‘group algebras’ k [Γ] over such groups and construct canonical trace maps relating these homology theories. In the case of one-dimensional representations, we show that our trace maps are of topological origin: they are induced by natural maps of (iterated) loop spaces that are well studied in homotopy theory. Using this topological interpretation, we deduce some algebraic results about representa- tion homology: in particular, we prove that the symmetric homology of group algebras and one-dimensional representation homology are naturally isomorphic, provided the base ring k is a ﬁeld of characteristic zero. We also study the behavior of the derived character maps of n -dimensional representations in the stable limit as n → ∞ , in which case we show that they ‘converge’ to become isomorphisms.
A systematic review of the use of Deep Learning in Satellite Imagery for Agriculture

Agricultural research is essential for increasing food production to meet the requirements of an increasing population in the coming decades. Recently, satellite technology has been improving rapidly and deep learning has seen much success in generic computer vision tasks and many application areas which presents an important opportunity to improve analysis of agricultural land. Here we present a systematic review of 150 studies to find the current uses of deep learning on satellite imagery for agricultural research. Although we identify 5 categories of agricultural monitoring tasks, the majority of the research interest is in crop segmentation and yield prediction. We found that, when used, modern deep learning methods consistently outperformed traditional machine learning across most tasks; the only exception was that Long Short-Term Memory (LSTM) Recurrent Neural Networks did not consistently outperform Random Forests (RF) for yield prediction. The reviewed studies have largely adopted methodologies from generic computer vision, except for one major omission: benchmark datasets are not utilised to evaluate models across studies, making it difficult to compare results. Additionally, some studies have specifically utilised the extra spectral resolution available in satellite imagery, but other divergent properties of satellite images - such as the hugely different scales of spatial patterns - are not being taken advantage of in the reviewed studies.
Is Reinforcement Learning (Not) for Natural Language Processing?: Benchmarks, Baselines, and Building Blocks for Natural Language Policy Optimization

We tackle the problem of aligning pre-trained large language models (LMs) with human preferences. If we view text generation as a sequential decision-making problem, reinforcement learning (RL) appears to be a natural conceptual framework. However, using RL for LM-based generation faces empirical challenges, including training instability due to the combinatorial action space, as well as a lack of open-source libraries and benchmarks customized for LM alignment. Thus, a question rises in the research community: is RL a practical paradigm for NLP? To help answer this, we first introduce an open-source modular library, RL4LMs (Reinforcement Learning for Language Models), for optimizing language generators with RL. The library consists of on-policy RL algorithms that can be used to train any encoder or encoder-decoder LM in the HuggingFace library (Wolf et al. 2020) with an arbitrary reward function. Next, we present the GRUE (General Reinforced-language Understanding Evaluation) benchmark, a set of 6 language generation tasks which are supervised not by target strings, but by reward functions which capture automated measures of human preference. GRUE is the first leaderboard-style evaluation of RL algorithms for NLP tasks. Finally, we introduce an easy-to-use, performant RL algorithm, NLPO (Natural Language Policy Optimization) that learns to effectively reduce the combinatorial action space in language generation. We show 1) that RL techniques are generally better than supervised methods at aligning LMs to human preferences; and 2) that NLPO exhibits greater stability and performance than previous policy gradient methods (e.g., PPO (Schulman et al. 2017)), based on both automatic and human evaluations.
Movement Analytics: Current Status, Application to Manufacturing, and Future Prospects from an AI Perspective

Data-driven decision making is becoming an integral part of manufacturing companies. Data is collected and commonly used to improve efficiency and produce high quality items for the customers. IoT-based and other forms of object tracking are an emerging tool for collecting movement data of objects/entities (e.g. human workers, moving vehicles, trolleys etc.) over space and time. Movement data can provide valuable insights like process bottlenecks, resource utilization, effective working time etc. that can be used for decision making and improving efficiency. Turning movement data into valuable information for industrial management and decision making requires analysis methods. We refer to this process as movement analytics. The purpose of this document is to review the current state of work for movement analytics both in manufacturing and more broadly. We survey relevant work from both a theoretical perspective and an application perspective. From the theoretical perspective, we put an emphasis on useful methods from two research areas: machine learning, and logicbased knowledge representation. We also review their combinations in view of movement analytics, and we discuss promising areas for future development and application. Furthermore, we touch on constraint optimization. 1 ar X iv :2 21 0. 01 34 4v 1 [ cs . A I] 4 O ct 2 02 2 Springer Nature 2021 LATEX template 2 Movement Analytics From an application perspective, we review applications of these methods to movement analytics in a general sense and across various industries. We also describe currently available commercial off-the-shelf products for tracking in manufacturing, and we overview main concepts of digital twins and their applications.