In this paper, we obtain a local limit theorem for the Kemperman's model of oscillating random walk on $\mathbb{Z}$; it extends the existing results for classical random walks on $\mathbb Z$ or reflected random walks on $\mathbb N_0$. The key technical point is to control the long-term behavior of the embedding subprocess that characterizes the oscillations of the original random walk between $\mathbb Z^-$ and $\mathbb Z^+$ in both recurrent and transient cases. Then by combining an extension of \cite[Theorem 1.4]{gouezel} for the convergence of aperiodic sequence of renewal operators acting on a suitable functional Banach space and the decomposition of the trajectories of the random walk, we obtain the exact asymptotic for the return probability under some mild assumptions on the increment moments.

SYK models provide an interesting playground for exploring the $AdS_2/CFT_1$ correspondence. We focus on a class of SYK models that exhibit higher-spin symmetry, whose gravity sector is described by a BF theory generalizing Jackiw--Teitelboim gravity to higher spins. We further develop this framework by constructing consistent interactions between higher-spin gauge fields and scalar matter fields. Two concrete realizations are proposed: Model A, arising from a deformation of the universal enveloping algebra of $\mathfrak{sl}(2,\mathbb{R})$, and Model B, a perturbatively local Poisson sigma-model with an infinite-dimensional target space. While both capture higher-spin dynamics on $(A)dS_2$, they differ in their algebraic structures and locality properties, thus offering complementary perspectives on higher-spin holography.

Applying style transfer to a full 3D environment is a challenging task that has seen many developments since the advent of neural rendering. 3D Gaussian splatting (3DGS) has recently pushed further many limits of neural rendering in terms of training speed and reconstruction quality. This work introduces SGSST: Scaling Gaussian Splatting Style Transfer, an optimization-based method to apply style transfer to pretrained 3DGS scenes. We demonstrate that a new multiscale loss based on global neural statistics, that we name SOS for Simultaneously Optimized Scales, enables style transfer to ultra-high resolution 3D scenes. Not only SGSST pioneers 3D scene style transfer at such high image resolutions, it also produces superior visual quality as assessed by thorough qualitative, quantitative and perceptual comparisons.

Fast radio bursts (FRBs) are enigmatic millisecond-duration signals which encode otherwise unattainable information on the plasma which permeates our Universe, providing insights into magnetic fields and gas distributions. Here we report the discovery of FRB 20240304B originating at redshift 2.148 +/- 0.001 corresponding to just 3 billion years after the Big Bang. FRB 2024030 was detected with the MeerKAT radio telescope and localized to a low-mass, clumpy, star forming galaxy using the James Webb Space Telescope. This discovery doubles the redshift reach of localized FRBs and probes ionized baryons across ~80% of cosmic history. Its sightline, intersecting the Virgo Cluster and a foreground group, reveals magnetic field complexity over many gigaparsec scales. Our observations establish FRB activity during the peak of cosmic star formation and demonstrate that FRBs can probe galaxy formation during the most active era in cosmic time.

Diffusion or score-based models recently showed high performance in image generation. They rely on a forward and a backward stochastic differential equations (SDE). The sampling of a data distribution is achieved by numerically solving the backward SDE or its associated flow ODE. Studying the convergence of these models necessitates to control four different types of error: the initialization error, the truncation error, the discretization error and the score approximation. In this paper, we theoretically study the behavior of diffusion models and their numerical implementation when the data distribution is Gaussian. Our first contribution is to derive the analytical solutions of the backward SDE and the probability flow ODE and to prove that these solutions and their discretizations are all Gaussian processes. Our second contribution is to compute the exact Wasserstein errors between the target and the numerically sampled distributions for any numerical scheme. This allows us to monitor convergence directly in the data space, while experimental works limit their empirical analysis to Inception features. An implementation of our code is available online.

The rapid advancements in satellite remote sensing have enhanced the capability to monitor and analyze the Earth's surface. Among the many variables captured through satellite sensors, Land Surface Temperature (LST) plays a critical role in understanding key environmental processes. However, obtaining high-resolution LST data remains a challenge, as satellite sensors often face a trade-off between spatial and temporal resolutions. In response, Spatio-Temporal Fusion (STF) has emerged as a powerful method to integrate two satellite data sources, one providing high spatial but low temporal resolution, and the other offering high temporal but low spatial resolution. Although a range of STF techniques have been proposed, from traditional methods to cutting-edge deep learning (DL) models, most have focused on surface reflectance, with limited application to LST estimation. DL approaches, in particular, show promise in improving the spatial and temporal resolutions of LST by capturing complex, non-linear relationships between input and output LST data. This paper offers a comprehensive review of the latest advancements in DL-based STF techniques for LST estimation. We analyze key research developments, mathematically formulate the STF problem, and introduce a novel taxonomy for DL-based STF methods. Furthermore, we discuss the challenges faced by current methods and highlight future research directions. In addition, we present the first open-source benchmark STF dataset for LST estimation, consisting of 51 pairs of MODIS-Landsat images spanning from 2013 to 2024. To support our findings, we conduct extensive experiments on state-of-the-art methods and present both quantitative and qualitative assessments. This is the first survey paper focused on DL-based STF for LST estimation. We hope it serves as a valuable reference for researchers and paves the way for future research in this field.

These notes have been prepared for a series of lectures given at the Sarajevo Stochastic Analysis Winter School, from January 28 to February 1, 2019. There already exist several excellent lecture notes and reviews on the subject, such as (Hairer 2009) on (non-singular) stochastic PDEs, and (Hairer 2015, Chandra and Weber 2017) on singular stochastic PDEs and regularity structures. The present notes have two main specificities. The first one is that they focus on a particular example, the Allen-Cahn equation, which allows to introduce several of the difficulties of the theory in a gradual way, by increasing the space dimension step by step. The hope is that while this limits the generality of the theory presented, this limitation is more that made up by a gain in clarity. The second specific aspect of these notes is that they go beyond existence and uniqueness of solutions, by covering a few recent results on convergence to equilibrium and metastability in these system.

This work offers a historical reading of the genesis of special relativity by placing the contributions of Lorentz, Poincare, and Einstein within their scientific and editorial context. It highlights the importance of the German periodical Beiblatter zu den Annalen der Physik as a key channel for the dissemination of international scientific research. The perspective advanced here is that the true revolution did not lie in special relativity itself, but in Maxwell's electrodynamics. Special relativity thus appears as the necessary expression of a framework already transformed by the universality of the speed of light.

A conjecture of Fan and Raspaud [3] asserts that every bridgeless cubic graph con-tains three perfect matchings with empty intersection. Kaiser and Raspaud [6] sug-gested a possible approach to this problem based on the concept of a balanced join in an embedded graph. We give here some new results concerning this conjecture and prove that a minimum counterexample must have at least 32 vertices.

For fixed $c,$ Prolate Spheroidal Wave Functions (PSWFs), denoted by $\psi_{n, c},$ form an orthogonal basis with remarkable properties for the space of band-limited functions with bandwith $c$. They have been largely studied and used after the seminal work of D. Slepian and his co-authors. In several applications, uniform estimates of the $\psi_{n,c}$ in $n$ and $c,$ are needed. To progress in this direction, we push forward the uniform approximation error bounds and give an explicit approximation of their values at $1$ in terms of the Legendre complete elliptic integral of the first kind. Also, we give an explicit formula for the accurate approximation the eigenvalues of the Sturm-Liouville operator associated with the PSWFs.

Numerical simulations of flows are required for numerous applications, and are usually carried out using shallow water equations. We describe the FullSWOF software which is based on up-to-date finite volume methods and well-balanced schemes to solve this kind of equations. It consists of a set of open source C++ codes, freely available to the community, easy to use, and open for further development. Several features make FullSWOF particularly suitable for applications in hydrology: small water heights and wet-dry transitions are robustly handled, rainfall and infiltration are incorporated, and data from grid-based digital topographies can be used directly. A detailed mathematical description is given here, and the capabilities of FullSWOF are illustrated based on analytic solutions and datasets of real cases. The codes, available in 1D and 2D versions, have been validated on a large set of benchmark cases, which are available together with the download information and documentation at this http URL

The rise of Large Language Models (LLMs) has triggered legal and ethical concerns, especially regarding the unauthorized use of copyrighted materials in their training datasets. This has led to lawsuits against tech companies accused of using protected content without permission. Membership Inference Attacks (MIAs) aim to detect whether specific documents were used in a given LLM pretraining, but their effectiveness is undermined by biases such as time-shifts and n-gram overlaps. This paper addresses the evaluation of MIAs on LLMs with partially inferable training sets, under the ex-post hypothesis, which acknowledges inherent distributional biases between members and non-members datasets. We propose and validate algorithms to create ``non-biased'' and ``non-classifiable'' datasets for fairer MIA assessment. Experiments using the Gutenberg dataset on OpenLamma and Pythia show that neutralizing known biases alone is insufficient. Our methods produce non-biased ex-post datasets with AUC-ROC scores comparable to those previously obtained on genuinely random datasets, validating our approach. Globally, MIAs yield results close to random, with only one being effective on both random and our datasets, but its performance decreases when bias is removed.

Contrastive learning has significantly improved representation quality, enhancing knowledge transfer across tasks in continual learning (CL). However, catastrophic forgetting remains a key challenge, as contrastive based methods primarily focus on "soft relationships" or "softness" between samples, which shift with changing data distributions and lead to representation overlap across tasks. Recently, the newly identified Neural Collapse phenomenon has shown promise in CL by focusing on "hard relationships" or "hardness" between samples and fixed prototypes. However, this approach overlooks "softness", crucial for capturing intra-class variability, and this rigid focus can also pull old class representations toward current ones, increasing forgetting. Building on these insights, we propose Focal Neural Collapse Contrastive (FNC^2), a novel representation learning loss that effectively balances both soft and hard relationships. Additionally, we introduce the Hardness-Softness Distillation (HSD) loss to progressively preserve the knowledge gained from these relationships across tasks. Our method outperforms state-of-the-art approaches, particularly in minimizing memory reliance. Remarkably, even without the use of memory, our approach rivals rehearsal-based methods, offering a compelling solution for data privacy concerns.

The $H$-Induced Minor Containment problem ($H$-IMC) consists in deciding if a fixed graph $H$ is an induced minor of a graph $G$ given as input, that is, whether $H$ can be obtained from $G$ by deleting vertices and contracting edges. Equivalently, the problem asks if there exists an induced minor model of $H$ in $G$, that is, a collection of disjoint subsets of vertices of $G$, each inducing a connected subgraph, such that contracting each subgraph into a single vertex results in $H$. It is known that $H$-IMC is NP-complete for several graphs $H$, even when $H$ is a tree. In this work, we investigate which properties of $H$ guarantee the existence of an induced minor model whose structure can be leveraged to solve the problem in polynomial time. This allows us to identify four infinite families of graphs $H$ that enjoy such properties. Moreover, we show that if the input graph $G$ excludes long induced paths, then $H$-IMC is polynomial-time solvable for any fixed graph $H$. As a byproduct of our results, this implies that $H$-IMC is polynomial-time solvable for all graphs $H$ with at most $5$ vertices, except for three open cases.

We present the results of the search for an isotropic stochastic gravitational wave background (GWB) at nanohertz frequencies using the second data release of the European Pulsar Timing Array (EPTA) for 25 millisecond pulsars and a combination with the first data release of the Indian Pulsar Timing Array (InPTA). We analysed (i) the full 24.7-year EPTA data set, (ii) its 10.3-year subset based on modern observing systems, (iii) the combination of the full data set with the first data release of the InPTA for ten commonly timed millisecond pulsars, and (iv) the combination of the 10.3-year subset with the InPTA data. These combinations allowed us to probe the contributions of instrumental noise and interstellar propagation effects. With the full data set, we find marginal evidence for a GWB, with a Bayes factor of four and a false alarm probability of $4\%$. With the 10.3-year subset, we report evidence for a GWB, with a Bayes factor of $60$ and a false alarm probability of about $0.1\%$ ($\gtrsim 3\sigma$ significance). The addition of the InPTA data yields results that are broadly consistent with the EPTA-only data sets, with the benefit of better noise modelling. Analyses were performed with different data processing pipelines to test the consistency of the results from independent software packages. The inferred spectrum from the latest EPTA data from new generation observing systems is rather uncertain and in mild tension with the common signal measured in the full data set. However, if the spectral index is fixed at 13/3, the two data sets give a similar amplitude of ($2.5\pm0.7)\times10^{-15}$ at a reference frequency of $1\,{\rm yr}^{-1}$. By continuing our detection efforts as part of the International Pulsar Timing Array (IPTA), we expect to be able to improve the measurement of spatial correlations and better characterise this signal in the coming years.

In response to ESA Voyage 2050 announcement of opportunity, we propose an ambitious L-class mission to explore one of the most exciting bodies in the Solar System, Saturn largest moon Titan. Titan, a "world with two oceans", is an organic-rich body with interior-surface-atmosphere interactions that are comparable in complexity to the Earth. Titan is also one of the few places in the Solar System with habitability potential. Titan remarkable nature was only partly revealed by the Cassini-Huygens mission and still holds mysteries requiring a complete exploration using a variety of vehicles and instruments. The proposed mission concept POSEIDON (Titan POlar Scout/orbitEr and In situ lake lander DrONe explorer) would perform joint orbital and in situ investigations of Titan. It is designed to build on and exceed the scope and scientific/technological accomplishments of Cassini-Huygens, exploring Titan in ways that were not previously possible, in particular through full close-up and in situ coverage over long periods of time. In the proposed mission architecture, POSEIDON consists of two major elements: a spacecraft with a large set of instruments that would orbit Titan, preferably in a low-eccentricity polar orbit, and a suite of in situ investigation components, i.e. a lake lander, a "heavy" drone (possibly amphibious) and/or a fleet of mini-drones, dedicated to the exploration of the polar regions. The ideal arrival time at Titan would be slightly before the next northern Spring equinox (2039), as equinoxes are the most active periods to monitor still largely unknown atmospheric and surface seasonal changes. The exploration of Titan northern latitudes with an orbiter and in situ element(s) would be highly complementary with the upcoming NASA New Frontiers Dragonfly mission that will provide in situ exploration of Titan equatorial regions in the mid-2030s.

We present a general methodology for structuring textual data, represented as syntax trees enriched with semantic information, guided by a meta-model G defined as an attribute grammar. The method involves an evolution process where both the instance and its grammar evolve, with instance transformations guided by rewriting rules and a similarity measure. Each new instance generates a corresponding grammar, culminating in a target grammar GT that satisfies G. This methodology is applied to build a database populated from textual data. The process generates both a database schema and its instance, independent of specific database models. We demonstrate the approach using clinical medical cases, where trees represent database instances and grammars act as database schemas. Key contributions include the proposal of a general attribute grammar G, a formalization of grammar evolution, and a proof-of-concept implementation for database structuring.

A conjecture of Máucajová and uSkoviera asserts that every bridgeless cubic graph has two perfect matchings whose intersection does not contain any odd edge cut. We prove this conjecture for graphs with few vertices and we give a stronger result for traceable graphs.