Search

We explore the advection-diffusion of a passive vector described by $\partial_t u + U \cdot \nabla u = - \nabla p + \nu \Delta u$, where both $U$ and $u$ are divergence-free velocity fields. We approach this equation from an input/output perspective, with $U$ as the input and $u$ as the output. This input/output viewpoint has been widely applied in recent studies on passive scalar equation, in the context of anomalous dissipation, mixing, optimal scalar transport, and nonuniqueness problems. What makes the passive vector equation considerably more challenging compared to the passive scalar equation is the lack of a Lagrangian perspective due to the presence of pressure. In this paper, rather than requiring $U$ and $u$ to be identical (as in the Navier-Stokes equation), we require $U$ and $u$ to be identical only in certain characteristics. We focus on the case where the characteristics in question are anomalous and enhanced dissipation. We study the advection-diffusion of a passive vector in a Couette flow configuration. The main result of this paper is a construction of the velocity field $U$ for which the energy dissipation scales as $(\log \nu^{-1})^{-2}$ such that the energy dissipation in velocity field $u$ scales at least as $(\log \nu^{-1})^{-2}$. This means that both $U$ and $u$ exhibit near-anomalous dissipation, where the rate of energy dissipation decreases more slowly than any power-law $\nu^{\alpha}$ for any $\alpha > 0$. The result in this paper is not just a mathematical construct; it closely resembles the behavior of turbulent flow in a channel. The $(\log \nu^{-1})^{-2}$ decrease of energy dissipation is predicted by phenomenological theories of wall-bounded turbulence, a prediction that has been extensively validated through experiments and numerical simulations., Comment: 27 pages, 4 Figures

We report here the structural, micro-structural/morphological and detailed magnetic properties of RECrO3 (RE = La, Nd, Sm, Eu and Gd) Orthocromites. A series of RECrO3 (RE = La, Nd, Sm, Eu and Gd) Orthocromites polycrystalline samples with uniform particle size were synthesized via standard solid-state reaction method. LaCrO3 (LCO), NdCrO3 (NCO), SmCrO3(SCO), and EuCrO3 (ECO) crystallizes in single phase of orthorhombic structure with Pnma (No.62) space group while GdCrO3 (GCO) found to crystallize in orthorhombic phase with Pbnm space group. SEM images confirms the systematic growth of all studied samples grain growth takes place during the sintering process and images shows the polyhedral shape with varying average grain size ranging from approximately 1.0{\mu}m to 3{\mu}m confirm the bulk synthesis of studied RECrO3. Lowest average grain size (~ 1.1{\mu}m) was observed for LaCrO3 and maximum (~ 2.8{\mu}m) for EuCrO3. The shapes of the grain size changes because of crystal lattice distortion induced by various rare earths (La, Nd, Sm, Eu and Gd) at A site. Energy dispersive X-ray Analysis (EDAX) reveals the elements confirmation in different studied RECrO3 samples. Within the detection limit of EDAX no impurity elements were observed in different as synthesized RECrO3 samples. Temperature dependent DC magnetization (M-T) studies showed an antiferromagnetic transition (TN) attributed to Cr3+ spins in all samples at different temperature. At low temperature spin-reorientation (TSR) of rare earth ion was also observed in NCO, SCO at around 50K and the same is observed at around 30K in ECO and the same was missing in GCO because of large moment of Gd3+ ions attributed by ground state spectroscopic term 8S7/2 by external applied magnetic field. The magnetization versus applied magnetic field (M-H) loops exhibited ferromagnetic like behavior at 10K and 100K and paramagnetic at 300K for GCO., Comment: 23 pages, 4 figures

In this paper, we investigate the problem of "generation supervision" in large language models, and present a novel bicameral architecture to separate supervision signals from their core capability, helpfulness. Doppelg\"anger, a new module parallel to the underlying language model, supervises the generation of each token, and learns to concurrently predict the supervision score(s) of the sequences up to and including each token. In this work, we present the theoretical findings, and leave the report on experimental results to a forthcoming publication.

This article establishes estimates on the dimension of the global attractor of the two-dimensional rotating Navier-Stokes equation for viscous, incompressible fluids on the $\beta$-plane. Previous results in this setting by M.A.H. Al-Jaboori and D. Wirosoetisno (2011) had proved that the global attractor collapses to a single point that depends only the longitudinal coordinate, i.e., zonal flow, when the rotation is sufficiently fast. However, an explicit quantification of the complexity of the global attractor in terms of $\beta$ had remained open. In this paper, such estimates are established which are valid across a wide regime of rotation rates and are consistent with the dynamically degenerate regime previously identified. Additionally, a decomposition of solutions is established detailing the asymptotic behavior of the solutions in the limit of large rotation., Comment: 23 pages

Mathematical analysis of the analytic hierarchy process (AHP) led to the development of a mathematical function, usually called the inconsistency index, which has the center role in measuring the inconsistency of the judgements in AHP. Inconsistency index is a mathematical function which maps every pairwise comparison matrix (PCM) into a real number. An inconsistency index can be considered more trustworthy when it satisfies a set of suitable properties. Therefore, the research community has been trying to postulate a set of desirable rules (axioms, properties) for inconsistency indices. Subsequently, many axiomatic frameworks for these functions have been suggested independently, however, the literature on the topic is fragmented and missing a broader framework. Therefore, the objective of this article is twofold. Firstly, we provide a comprehensive review of the advancements in the axiomatization of inconsistency indices' properties during the last decade. Secondly, we provide a comparison and discussion of the aforementioned axiomatic structures along with directions of the future research., Comment: 21 pages, 2 figures

We propose a new convex integration scheme in fluid mechanics, and we provide an application to the two-dimensional Euler equations. We prove the flexibility and nonuniqueness of $L^\infty L^2$ weak solutions with vorticity in $L^\infty L^p$ for some $p>1$, surpassing for the first time the critical scaling of the standard convex integration technique. To achieve this, we introduce several new ideas, including: (i) A new family of building blocks built from the Lamb-Chaplygin dipole. (ii) A new method to cancel the error based on time averages and non-periodic, spatially-anisotropic perturbations., Comment: 46 pages, 3 figures

Retrieval-Augmented Generation (RAG) has recently emerged as a promising solution to alleviate Large Language Model (LLM)'s deficiency in lack of knowledge. Existing RAG datasets, however, do not adequately represent the diverse and dynamic nature of real-world Question Answering (QA) tasks. To bridge this gap, we introduce the Comprehensive RAG Benchmark (CRAG), a factual question answering benchmark of 4,409 question-answer pairs and mock APIs to simulate web and Knowledge Graph (KG) search. CRAG is designed to encapsulate a diverse array of questions across five domains and eight question categories, reflecting varied entity popularity from popular to long-tail, and temporal dynamisms ranging from years to seconds. Our evaluation on this benchmark highlights the gap to fully trustworthy QA. Whereas most advanced LLMs achieve <=34% accuracy on CRAG, adding RAG in a straightforward manner improves the accuracy only to 44%. State-of-the-art industry RAG solutions only answer 63% questions without any hallucination. CRAG also reveals much lower accuracy in answering questions regarding facts with higher dynamism, lower popularity, or higher complexity, suggesting future research directions. The CRAG benchmark laid the groundwork for a KDD Cup 2024 challenge, attracting thousands of participants and submissions within the first 50 days of the competition. We commit to maintaining CRAG to serve research communities in advancing RAG solutions and general QA solutions.

First-principles based calculations were executed to investigate the sensing properties of ammonia gas molecules on two-dimensional pristine black phosphorene towards its application as a gas sensor and related applications. We discuss in detail, the interaction of ammonia gas molecules on the phosphorene single sheet through the structural change analysis, electronic band gap, Bader charge transfer, and density-of-states calculations. Our calculations indicate that the phosphorene could be used as a detector of ammonia, where good sensitivity and very short recovery time at room temperature have confirmed the potential use of phosphorene in the detection of ammonia., Comment: 21 pages, Figures 8

Given a divergence-free vector field ${\bf u} \in L^\infty_t W^{1,p}_x(\mathbb R^d)$ and a nonnegative initial datum $\rho_0 \in L^r$, the celebrated DiPerna--Lions theory established the uniqueness of the weak solution in the class of $L^\infty_t L^r_x$ densities for $\frac{1}{p} + \frac{1}{r} \leq 1$. This range was later improved in [BCDL21] to $\frac{1}{p} + \frac{d-1}{dr} \leq 1$. We prove that this range is sharp by providing a counterexample to uniqueness when $\frac{1}{p} + \frac{d-1}{dr} > 1$. To this end, we introduce a novel flow mechanism. It is not based on convex integration, which has provided a non-optimal result in this context, nor on purely self-similar techniques, but shares features of both, such as a local (discrete) self similar nature and an intermittent space-frequency localization., Comment: 39 pages, 14 figures

We introduce Lumos, the first end-to-end multimodal question-answering system with text understanding capabilities. At the core of Lumos is a Scene Text Recognition (STR) component that extracts text from first person point-of-view images, the output of which is used to augment input to a Multimodal Large Language Model (MM-LLM). While building Lumos, we encountered numerous challenges related to STR quality, overall latency, and model inference. In this paper, we delve into those challenges, and discuss the system architecture, design choices, and modeling techniques employed to overcome these obstacles. We also provide a comprehensive evaluation for each component, showcasing high quality and efficiency., Comment: Accepted to KDD 2024 (ADS Track)