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In this report, we share insights from a Research-Practice-Industry Partnership (RPIP) that explored mathematics instructional practices with support of digital mathematics tools. RPIPs bring together researchers, practitioners, and product developers, with each party having an equal voice, in a rapid-cycle model for edtech research and development. We framed this RPIP using the principles of Universal Design for Learning (UDL): provide multiple means of Engagement, Representation, and Action & Expression. These principles support learner variability, improve accessibility, and are closely aligned with mathematics education research, which highlights the multimodal nature of mathematics and the need to use and connect multiple representations to help students develop understanding of mathematics concepts. [This report was produced in partnership with Creativity Labs with funding support from Texthelp.]

Haptic feedback has been integrated into Virtual and Augmented Reality, complementing acoustic and visual information and contributing to an all-round immersive experience in multiple fields, spanning from the medical domain to entertainment and gaming. Haptic technologies involve complex cross-disciplinary research that encompasses sensing, data representation, interactive rendering, perception, and quality of experience. The standard processing pipeline, consists of (I) sensing physical features in the real world using a transducer, (II) modeling and storing the collected information in some digital format, (III) communicating the information, and finally, (IV) rendering the haptic information through appropriate devices, thus producing a user experience (V) perceptually close to the original physical world. Among these areas, sensing, rendering and perception have been deeply investigated and are the subject of different comprehensive surveys available in the literature. Differently, research dealing with haptic surface modeling and data representation still lacks a comprehensive dissection. In this work, we aim at providing an overview on modeling and representation of haptic surfaces from a signal processing perspective, covering the aspects that lie in between haptic information acquisition on one side and rendering and perception on the other side. We analyze, categorize, and compare research papers that address the haptic surface modeling and data representation, pointing out existing gaps and possible research directions., Comment: 19 pages, 6 figures

Given the increasing scale of model sizes, novel training strategies like gradual stacking [Gong et al., 2019, Reddi et al., 2023] have garnered interest. Stacking enables efficient training by gradually growing the depth of a model in stages and using layers from a smaller model in an earlier stage to initialize the next stage. Although efficient for training, the model biases induced by such growing approaches are largely unexplored. In this work, we examine this fundamental aspect of gradual stacking, going beyond its efficiency benefits. We propose a variant of gradual stacking called MIDAS that can speed up language model training by up to 40%. Furthermore we discover an intriguing phenomenon: MIDAS is not only training-efficient but surprisingly also has an inductive bias towards improving downstream tasks, especially tasks that require reasoning abilities like reading comprehension and math problems, despite having similar or slightly worse perplexity compared to baseline training. To further analyze this inductive bias, we construct reasoning primitives -- simple synthetic tasks that are building blocks for reasoning -- and find that a model pretrained with stacking is significantly better than standard pretraining on these primitives, with and without fine-tuning. This provides stronger and more robust evidence for this inductive bias towards reasoning. These findings of training efficiency and inductive bias towards reasoning are verified at 1B, 2B and 8B parameter language models. Finally, we conjecture the underlying reason for this inductive bias by exploring the connection of stacking to looped models and provide strong supporting empirical analysis., Comment: Accepted at NeurIPS 2024

We examine the fluid flow forced by precession of a rotating cylindrical container using numerical simulations and experimental flow measurements with ultrasonic Doppler velocimetry (UDV). The analysis is based on the decomposition of the flow field into contributions with distinct azimuthal symmetry or analytically known inertial modes and the corresponding calculation of their amplitudes. We show that the predominant fraction of the kinetic energy of the precession-driven fluid flow is contained only within a few large-scale modes. The most striking observation shown by simulations and experiments is the transition from a flow dominated by large-scale structures to a more turbulent behaviour with the small-scale fluctuations becoming increasingly important. At a fixed rotation frequency (parametrized by the Reynolds number, ${\rm{Re}}$) this transition occurs when a critical precession ratio is exceeded and consists of a two-stage collapse of the directly driven flow going along with a massive modification of the azimuthal circulation (the zonal flow) and the appearance of an axisymmetric double-roll mode limited to a narrow range of precession ratios. A similar behaviour is found in experiments which make it possible to follow the transition up to Reynolds numbers of ${\rm{Re}}\approx 2\times 10^6$. We find that the critical precession ratio decreases with rotation, initially showing a particular scaling $\propto {\rm{Re}}^{-\frac{1}{5}}$ but developing an asymptotic behaviour for ${\rm{Re}}\gtrsim 10^5$ which might be explained by the onset of turbulence in boundary layers., Comment: 2024, 29 pages, 16 figures, accepted for publication in Journal of Fluid Mechanics

In this article we obtain uniform effective upper bounds for the projective dimension and the Castelnuovo-Mumford regularity of homogeneous ideals inside a standard graded polynomial ring $S$ over a field. Such bounds are independent of the number of variables of $S$, in the spirit of Stillman's conjecture and of the Ananyan-Hochster's theorem, and depend on partial data extracted from the beginning or the end of the resolution. In this direction, we extend a result of McCullough from 2012 regarding a bound on regularity in terms of half the syzygies to a bound on the projective dimension and the regularity of an ideal in terms of a fraction of the syzygies.

In this paper, we study the pure $\kappa-$sparse gapsets and our focus on getting information about the sequence observed in Table 3 at [1], this sequence is listed in OEIS as A374773. We verify that the cardinality of the set of gapsets with genus $3n+1$ such that the maximum distance between two consecutive elements is $2n$ is equal to the cardinality of the set of gapsets with genus $3n+2$ such that the maximum distance between two consecutive elements is $2n+1$, for all $n\in \mathbb{N}$. In particular, we compute the cardinality of the symmetric and pseudo-symmetric gapsets in these cases., Comment: arXiv admin note: text overlap with arXiv:2106.13296

We give a partial negative answer to a question left open in a previous work by Brasco and the first and third-named authors concerning the sharp constant in the fractional Hardy inequality on convex sets. Our approach has a geometrical flavor and equivalently reformulates the sharp constant in the limit case $p=1$ as the Cheeger constant for the fractional perimeter and the Lebesgue measure with a suitable weight. As a by-product, we obtain new lower bounds on the sharp constant in the $1$-dimensional case, even for non-convex sets, some of which optimal in the case $p=1$., Comment: 25 pages

Let $(R,\mathfrak{m})$ be a complete local ring, and $G={\rm gr}_{\mathfrak{m}}(R)$ be its associated graded ring. We introduce a homogenization technique which allows to relate $G$ to the special fiber and $R$ to the generic fiber of a "Gr\"obner-like" deformation. Using this technique we prove sharp results concerning the connectedness of $R$ and $G$. We also construct a family of local domains which fail to satisfy Abhyankar's inequality for the Hilbert-Samuel multiplicity. However, we prove a version of the inequality which holds when $R$ is connected in codimension one.

Distributed systems can be subject to various kinds of partial failures, therefore building fault-tolerance or failure mitigation mechanisms for distributed systems remains an important domain of research. In this paper, we present a calculus to formally model distributed systems subject to crash failures with recovery. The recovery model considered in the paper is weak, in the sense that it makes no assumption on the exact state in which a failed node resumes its execution, only its identity has to be distinguishable from past incarnations of itself. Our calculus is inspired in part by the Erlang programming language and in part by the distributed $\pi$-calculus with nodes and link failures (D$\pi$F) introduced by Francalanza and Hennessy. In order to reason about distributed systems with failures and recovery we develop a behavioral theory for our calculus, in the form of a contextual equivalence, and of a fully abstract coinductive characterization of this equivalence by means of a labelled transition system semantics and its associated weak bisimilarity. This result is valuable for it provides a compositional proof technique for proving or disproving contextual equivalence between systems., Comment: 50 pages

Recently, there has been increasing interest in efficient pretraining paradigms for training Transformer-based models. Several recent approaches use smaller models to initialize larger models in order to save computation (e.g., stacking and fusion). In this work, we study the fundamental question of how to select the best growing strategy from a given pool of growing strategies. Prior works have extensively focused on loss- and/or function-preserving behavior at initialization or simply performance at the end of training. Instead, we identify that behavior at initialization can be misleading as a predictor of final performance and present an alternative perspective based on early training dynamics, which we call "landscape-aware growing (LAG)". We perform extensive analysis of correlation of the final performance with performance in the initial steps of training and find early and more accurate predictions of the optimal growing strategy (i.e., with only a small "lag" after initialization). This perspective also motivates an adaptive strategy for gradual stacking.

The Moons And Jupiter Imaging Spectrometer (MAJIS) is the visible and near-infrared imaging spectrometer onboard ESA s Jupiter Icy Moons Explorer (JUICE) mission. Before its integration into the spacecraft, the instrument undergoes an extensive ground calibration to establish its baseline performances. This process prepares the imaging spectrometer for flight operations by characterizing the behavior of the instrument under various operative conditions and uncovering instrumental distortions that may depend on instrumental commands. Two steps of the on-ground calibration campaigns were held at the instrument level to produce the data. Additional in-flight measurements have recently been obtained after launch during the Near-Earth Commissioning Phase. In this article, we present the analyses of these datasets, focusing on the characterization of the spectral performances. First, we describe and analyze the spectral calibration datasets obtained using both monochromatic sources and polychromatic sources coupled with solid and gas samples. Then, we derive the spectral sampling and the spectral response function over the entire field of view. These spectral characteristics are quantified for various operational parameters of MAJIS, such as temperature and spectral binning. The derived on-ground performances are then compared with in-flight measurements obtained after launch and presented in the framework of the MAJIS performance requirements.

SEERNet is a network of Digital Learning Platforms, researchers, and practitioners with the vision to leverage DLPs as research infrastructure, enabling researchers to ask and answer important questions about learning in ways that are grounded in realistic, widespread use of learning technologies. A first wave of work established the foundation for the research community. The network looks now towards a second wave of work that clarifies research opportunities and builds capacity for the types of research that can be supported. A third wave seeks to expand the network's efforts towards broader aspirations for better science, engineering, practitioner engagement, and community.

In this paper, we present numerical and experimental results on helicity oscillations in a liquid-metal Rayleigh-B\'enard (RB) convection cell, with an aspect ratio of 0.5. We find that helicity oscillations occur during transitions of flow states that are characterised by significant changes in the Reynolds number. Moreover, we also observe helicity oscillations at flow conditions where the temporal gradient of the change in the Reynolds number is significantly smaller than that of the helicity. Notably, the helicity oscillations observed during the transient double-roll state exhibit characteristics remarkably similar to those associated with the Tayler Instability., Comment: 10 pages, 16 figures

The goal of this white paper is to provide a snapshot of the data availability and data needs primarily for the Ariel space mission, but also for related atmospheric studies of exoplanets and brown dwarfs. It covers the following data-related topics: molecular and atomic line lists, line profiles, computed cross-sections and opacities, collision-induced absorption and other continuum data, optical properties of aerosols and surfaces, atmospheric chemistry, UV photodissociation and photoabsorption cross-sections, and standards in the description and format of such data. These data aspects are discussed by addressing the following questions for each topic, based on the experience of the "data-provider" and "data-user" communities: (1) what are the types and sources of currently available data, (2) what work is currently in progress, and (3) what are the current and anticipated data needs. We present a GitHub platform for Ariel-related data, with the goal to provide a go-to place for both data-users and data-providers, for the users to make requests for their data needs and for the data-providers to link to their available data. Our aim throughout the paper is to provide practical information on existing sources of data whether in databases, theoretical, or literature sources., Comment: 58 pages, submitted to RAS Techniques and Instruments (RASTI). The authors welcome feedback: corresponding author emails can be found as footnotes on page 2

Vision tasks are characterized by the properties of locality and translation invariance. The superior performance of convolutional neural networks (CNNs) on these tasks is widely attributed to the inductive bias of locality and weight sharing baked into their architecture. Existing attempts to quantify the statistical benefits of these biases in CNNs over locally connected convolutional neural networks (LCNs) and fully connected neural networks (FCNs) fall into one of the following categories: either they disregard the optimizer and only provide uniform convergence upper bounds with no separating lower bounds, or they consider simplistic tasks that do not truly mirror the locality and translation invariance as found in real-world vision tasks. To address these deficiencies, we introduce the Dynamic Signal Distribution (DSD) classification task that models an image as consisting of $k$ patches, each of dimension $d$, and the label is determined by a $d$-sparse signal vector that can freely appear in any one of the $k$ patches. On this task, for any orthogonally equivariant algorithm like gradient descent, we prove that CNNs require $\tilde{O}(k+d)$ samples, whereas LCNs require $\Omega(kd)$ samples, establishing the statistical advantages of weight sharing in translation invariant tasks. Furthermore, LCNs need $\tilde{O}(k(k+d))$ samples, compared to $\Omega(k^2d)$ samples for FCNs, showcasing the benefits of locality in local tasks. Additionally, we develop information theoretic tools for analyzing randomized algorithms, which may be of interest for statistical research., Comment: 40 pages, 4 figures, Accepted to ICLR 2024, Spotlight

The effects of thermal convection on turbulence in accretion discs, and particularly its interplay with the magnetorotational instability (MRI), are of significant astrophysical interest. Despite extensive theoretical and numerical studies, such an interplay has not been explored experimentally. We conduct linear analysis of the azimuthal version of MRI (AMRI) in the presence of thermal convection and compare the results with our experimental data published before. We show that the critical Hartmann number ($Ha$) for the onset of AMRI is reduced by convection. Importantly, convection breaks symmetry between $m = \pm 1$ instability modes ($m$ is the azimuthal wavenumber). This preference for one mode over the other makes the AMRI-wave appear as a ``one-winged butterfly''., Comment: 11 pages, 5 figures, accepted for publication in the Journal of Fluid Mechanics

Ontology engineering (OE) in large projects poses a number of challenges arising from the heterogeneous backgrounds of the various stakeholders, domain experts, and their complex interactions with ontology designers. This multi-party interaction often creates systematic ambiguities and biases from the elicitation of ontology requirements, which directly affect the design, evaluation and may jeopardise the target reuse. Meanwhile, current OE methodologies strongly rely on manual activities (e.g., interviews, discussion pages). After collecting evidence on the most crucial OE activities, we introduce \textbf{OntoChat}, a framework for conversational ontology engineering that supports requirement elicitation, analysis, and testing. By interacting with a conversational agent, users can steer the creation of user stories and the extraction of competency questions, while receiving computational support to analyse the overall requirements and test early versions of the resulting ontologies. We evaluate OntoChat by replicating the engineering of the Music Meta Ontology, and collecting preliminary metrics on the effectiveness of each component from users. We release all code at https://github.com/King-s-Knowledge-Graph-Lab/OntoChat., Comment: ESWC 2024 Special Track on Large Language Models for Knowledge Engineering

We prove several functional and geometric inequalities only assuming the linearity and a quantitative $\mathrm{L}^\infty$-to-Lipschitz smoothing of the heat semigroup in metric-measure spaces. Our results comprise a Buser inequality, a lower bound on the size of the nodal set of a Laplacian eigenfunction, and different estimates involving the Wasserstein distance. The approach works in large variety settings, including Riemannian manifolds with a variable Kato-type lower bound on the Ricci curvature tensor, $\mathsf{RCD}(K,\infty)$ spaces, and some sub-Riemannian structures, such as Carnot groups, the Grushin plane and the $\mathbb{SU}(2)$ group., Comment: 23 pages

This work presents an original concept directed to implement an unconventional methodology where a device is produced by integrating a solid-state circuitry concept and an electrochemistry cell. In our experimental system an organic semiconductor, (PCPDT-BT), serves both as the gate and working electrode. Gating is obtained via electrochemical polarization exploiting a conventional three electrodes electrochemical cell placed on top of a traditional source/drain/gate solid-state device configuration. Source/drain conduction is probed via impedance measurement (electrochemical impedance spectroscopy, EIS) performed as a function of time at constant frequency, under constant potential control. The conductivity of the PCPDT-BT is due to the polaronic state induced via application of a suitable electrochemical potential (in the oxidation regime), as it is proved by infrared (IR) spectra recorded in-situ/in-operando (in attenuated under total reflection, ATR, mode) upon both electrochemical and chemical ionization/doping of the PCPDT-BT., Comment: Main ms., 15 pages, 4 figures Supp Info, 9 pages, 10 figures, 3 equations

Controlling the flow of excitons between organic molecules holds immense promise for various applications, including energy conversion, spectroscopy, photocatalysis, sensing, and microscopy. DNA nanotechnology has shown promise in achieving this control by using synthetic DNA as a platform for positioning and, very recently, for also orienting organic dyes. In this study, the orientation of doubly-linked dyes in DNA origami structures was manipulated to control energy transfer. By controlling independently the orientation of single donor and acceptor molecules, the average energy transfer efficiency was doubled. This work demonstrates the potential of DNA nanotechnology for precise control of the excitonic energy transfer with implications for artificial light-harvesting antennas., Comment: 19 pages, 4 figures

We study Cheeger and $p$-eigenvalue partition problems depending on a given evaluation function $\Phi$ for $p\in[1,\infty)$. We prove existence and regularity of minima, relations among the problems, convergence, and stability with respect to $p$ and to $\Phi$., Comment: 33 pages

This paper is devoted to bifurcations of periodic solutions of nonlinear measure differential equations with a parameter. Main tools are nonlinear generalized differential equations (in the sense of Kurzweil) and the Kurzweil gauge type generalized integral. We continue the research started by the first author under the supervision of the second one., Comment: Second version of the paper. Mainly some misprints have been corrected

We prove that the boundary of an almost minimizer of the intrinsic perimeter in a plentiful group can be approximated by intrinsic Lipschitz graphs. Plentiful groups are Carnot groups of step~$2$ whose center of the Lie algebra is generated by any co-dimension one horizontal subspace. For example, $H$-type groups not isomorphic to the first Heisenberg group are plentiful. Our results provide the first extension of the regularity theory of intrinsic minimal surfaces beyond the family of Heisenberg groups.

We reveal and analyze an efficient magnetic dynamo action due to precession-driven hydrodynamic turbulence in the local model of a precessional flow, focusing on the kinematic stage of this dynamo. The growth rate of magnetic field monotonically increases with Poincar\'{e} number, $\rm Po$, characterizing precession strength, and magnetic Prandtl number, $\rm Pm$, equal to the ratio of viscosity to resistivity, for the considered ranges of these parameters. The critical ${\rm Po}_c$ for the dynamo onset decreases with increasing $\rm Pm$. To understand the scale-by-scale evolution (growth) of the precession dynamo and its driving processes, we perform spectral analysis by calculating the spectra of magnetic energy and of different terms in the induction equation in Fourier space. To this end, we decompose the velocity field of precession-driven turbulence into 2D vortical and 3D inertial wave modes. It is shown that the dynamo operates across a broad range of scales and exhibits a remarkable transition from a primarily vortex-driven regime at lower $\rm Po$ to a more complex regime at higher $\rm Po$ where it is driven jointly by vortices, inertial waves and the shear of the background precessional flow. The vortices and shear drive the dynamo mostly at large scales, comparable to the flow system size, and at intermediate scales, while at smaller scales it is mainly driven by inertial waves. This study can be important not only for understanding the magnetic dynamo action in precession-driven flows, but also in a general context of flows where vortices emerge and govern the flow dynamics and evolution., Comment: 9 pages, 5 figures, accepted for publication in Physical Review E

Simulation is a powerful tool to easily generate annotated data, and a highly desirable feature, especially in those domains where learning models need large training datasets. Machine learning and deep learning solutions, have proven to be extremely data-hungry and sometimes, the available real-world data are not sufficient to effectively model the given task. Despite the initial skepticism of a portion of the scientific community, the potential of simulation has been largely confirmed in many application areas, and the recent developments in terms of rendering and virtualization engines, have shown a good ability also in representing complex scenes. This includes environmental factors, such as weather conditions and surface reflectance, as well as human-related events, like human actions and behaviors. We present a human crowd simulator, called UniCrowd, and its associated validation pipeline. We show how the simulator can generate annotated data, suitable for computer vision tasks, in particular for detection and segmentation, as well as the related applications, as crowd counting, human pose estimation, trajectory analysis and prediction, and anomaly detection.

Although statistical learning theory provides a robust framework to understand supervised learning, many theoretical aspects of deep learning remain unclear, in particular how different architectures may lead to inductive bias when trained using gradient based methods. The goal of these lectures is to provide an overview of some of the main questions that arise when attempting to understand deep learning from a learning theory perspective. After a brief reminder on statistical learning theory and stochastic optimization, we discuss implicit bias in the context of benign overfitting. We then move to a general description of the mirror descent algorithm, showing how we may go back and forth between a parameter space and the corresponding function space for a given learning problem, as well as how the geometry of the learning problem may be represented by a metric tensor. Building on this framework, we provide a detailed study of the implicit bias of gradient descent on linear diagonal networks for various regression tasks, showing how the loss function, scale of parameters at initialization and depth of the network may lead to various forms of implicit bias, in particular transitioning between kernel or feature learning., Comment: 66 pages, 20 figures