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Assume that $R_1,R_2,\dots,R_t$ are disjoint parallel lines in the plane. A $t$-interval (or $t$-track interval) is a set that can be written as the union of $t$ closed intervals, each on a different line. It is known that pairwise intersecting $2$-intervals can be pierced by two points, one from each line. However, it is not true that every set of pairwise intersecting $3$-intervals can be pierced by three points, one from each line. For $k\ge 3$, Kaiser and Rabinovich asked whether $k$-wise intersecting $t$-intervals can be pierced by $t$ points, one from each line. Our main result provides an asymptotic answer: in any set $S_1,\dots,S_n$ of $k$-wise intersecting $t$-intervals, at least $\frac{k-1}{k+1}n$ can be pierced by $t$ points, one from each line. We prove this in a more general form, replacing intervals by subtrees of a tree. This leads to questions and results on covering vertices of edge-colored complete graphs by vertices of monochromatic cliques having distinct colors, where the colorings are chordal, or more generally induced $C_4$-free graphs. For instance, we show that if the edges of a complete graph $K_n$ are colored with red or blue so that both color classes are induced $C_4$-free, then at least ${4n\over 5}$ vertices can be covered by a red and a blue clique, and this is best possible. We conclude by pointing to new Ramsey-type problems emerging from these restricted colorings., Comment: 15 pages, 3 figures

The V1298 Tau system (20-30Myr), is a benchmark young multi-planet system that provides the opportunity to perform comparative exoplanetology between planets orbiting the same star right after their formation. We present the first atmospheric comparison between two planets in the same transiting system: V1298 Tau b and V1298 Tau c. We derive constraints on the mass of planet b and c (<20M$_\oplus$ at 3$\sigma$ confidence level and $17_{-6}^{+13} M_{\oplus}$ respectively) and atmospheric metallicity (logZ/Z$_\odot$=-2.04$_{-0.59}^{0.69}$, -0.16$_{-0.94}^{1.15}$ respectively) from atmospheric retrievals. The V1298 Tau planets, are likely to be similar in terms of mass at the current age, implying that both planets are potential sub-Neptune/super-Earth progenitors. However, planet c is expected to lose a higher fraction of its mass compared to planet b given its close proximity to the host star. Alternatively, the observed spectrum of planet c can be explained by atmospheric hazes, which is in contrast to planet b where efficient haze formation can be ruled out. Higher haze formation efficiency in planet c could be due to differences in atmospheric composition, temperature and higher UV flux incident compared to planet b., Comment: Submitted to Astronomy ad Astrophysics

High-eccentricity gas giant planets serve as unique laboratories for studying the thermal and chemical properties of H/He-dominated atmospheres. One of the most extreme cases is HD 80606b -- a hot Jupiter orbiting a sun-like star with an eccentricity of $0.93$ -- which experiences an increase in incident flux of nearly three orders of magnitude as the star-planet separation decreases from $0.88\,{\rm au}$ at apoastron to $0.03\,{\rm au}$ at periastron. We observed the planet's periastron passage using \emph{JWST}'s NIRSpec/G395H instrument ($2.8-5.2\,{\rm \mu m}$) during a $21\,{\rm hr}$ window centered on the eclipse. We find that, as the planet passes through periastron, its emission spectrum transitions from a featureless blackbody to one in which CO and CH$_4$ absorption features are visible. We obtain significant detections of CH$_4$ during post-periapse phases at $3.7-4.8\sigma$ depending on the phase. Following periapse, CO and H$_2$O are also detected at $3.4\sigma$ and $3.1\sigma$, respectively. Furthermore, we rule out the presence of a strong temperature inversion near the IR photosphere -- predicted by GCMs to form temporarily during periapse passage -- based on the lack of obvious emission features throughout the observing window. Our study demonstrates the feasibility of studying hot Jupiter atmospheres using partial phase curves obtained with NIRSpec/G395H., Comment: 25 pages, 19 figures, 2 tables. Submitted to ApJ

We investigate the three-dimensional compressible Euler-Maxwell system, a model for simulating the transport of electrons interacting with propagating electromagnetic waves in semiconductor devices. First, we show the global well-posedness of classical solutions being a sharp small perturbation of constant equilibrium in a critical regularity setting, uniformly with respect to the relaxation parameter $\varepsilon>0$. Then, for all times $t>0$, we derive quantitative error estimates at the rate $O(\varepsilon)$ between the rescaled Euler-Maxwell system and the limit drift-diffusion model. To the best of our knowledge, this work provides the first global-in-time strong convergence for the relaxation procedure in the case of ill-prepared data. In order to prove our results, we develop a new characterization of the dissipation structure for the linearized Euler-Maxwell system with respect to the relaxation parameter $\varepsilon$. This is done by partitioning the frequency space into three distinct regimes: low, medium and high frequencies, each associated with a different behaviour of the solution. Then, in each regime, the use of efficient unknowns and Lyapunov functionals based on the hypocoercivity theory leads to uniform a priori estimates., Comment: 32pages

Learning a parametric model from a given dataset indeed enables to capture intrinsic dependencies between random variables via a parametric conditional probability distribution and in turn predict the value of a label variable given observed variables. In this paper, we undertake a comparative analysis of generative and discriminative approaches which differ in their construction and the structure of the underlying inference problem. Our objective is to compare the ability of both approaches to leverage information from various sources in an epistemic uncertainty aware inference via the posterior predictive distribution. We assess the role of a prior distribution, explicit in the generative case and implicit in the discriminative case, leading to a discussion about discriminative models suffering from imbalanced dataset. We next examine the double role played by the observed variables in the generative case, and discuss the compatibility of both approaches with semi-supervised learning. We also provide with practical insights and we examine how the modeling choice impacts the sampling from the posterior predictive distribution. With regard to this, we propose a general sampling scheme enabling supervised learning for both approaches, as well as semi-supervised learning when compatible with the considered modeling approach. Throughout this paper, we illustrate our arguments and conclusions using the example of affine regression, and validate our comparative analysis through classification simulations using neural network based models.

Blockchain's decentralization, transparency, and tamper-resistance properties have facilitated the system's use in various application fields. However, the low throughput and high confirmation latency hinder the widespread adoption of Blockchain. Many solutions have been proposed to address these issues, including first-layer solutions (or on-chain solutions) and second-layer solutions (or off-chain solutions). Among the proposed solutions, the blockchain sharding system is the most scalable one, where the nodes in the network are divided into several groups. The nodes in different shards work in parallel to validate the transactions and add them to the blocks, and in such a way, the throughput increases significantly. However, previous works have not adequately summarized the latest achievements in blockchain sharding, nor have they fully showcased its state-of-the-art. Our study provides a systemization of knowledge of public blockchain sharding, including the core components of sharding systems, challenges, limitations, and mechanisms of the latest sharding protocols. We also compare their performance and discuss current constraints and future research directions., Comment: 18 pages

We present a weak-strong uniqueness result for the inhomogeneous Navier-Stokes (INS) equations in $\mathbb{R}^d$ ($d=2,3$) for bounded initial densities that are far from vacuum. Given a strong solution within the class employed in Paicu, Zhang and Zhang (2013) and Chen, Zhang and Zhao (2016), and a Leray-Hopf weak solution, we establish that they coincide if the initial data agree. The strategy of our proof is based on the relative energy method and new $W^{-1,p}$-type stability estimates for the density. A key point lies in proving that every Leray-Hopf weak solution originating from initial densities far from vacuum remains distant from vacuum at all times.

We investigate the Navier-Stokes-Cattaneo-Christov (NSC) system in $\mathbb{R}^d$ ($d\geq3$), a model of heat-conductive compressible flows serving as a finite speed of propagation approximation of the Navier-Stokes-Fourier (NSF) system. Due to the presence of Oldroyd's upper-convected derivatives, the system (NSC) exhibits a \textit{lack of hyperbolicity} which makes it challenging to establish its well-posedness, especially in multi-dimensional contexts. In this paper, within a critical regularity functional framework, we prove the global-in-time well-posedness of (NSC) for initial data that are small perturbations of constant equilibria, uniformly with respect to the approximation parameter $\varepsilon>0$. Then, building upon this result, we obtain the sharp large-time asymptotic behaviour of (NSC) and, for all time $t>0$, we derive quantitative error estimates between the solutions of (NSC) and (NSF). To the best of our knowledge, our work provides the first strong convergence result for this relaxation procedure in the three-dimensional setting and for ill-prepared data. The (NSC) system is partially dissipative and incorporates both partial diffusion and partial damping mechanisms. To address these aspects and ensure the large-time stability of the solutions, we construct localized-in-frequency perturbed energy functionals based on the hypocoercivity theory. More precisely, our analysis relies on partitioning the frequency space into \textit{three} distinct regimes: low, medium and high frequencies. Within each frequency regime, we introduce effective unknowns and Lyapunov functionals, revealing the spectrally expected dissipative structures.

In this paper, we analyze the preservation of asymptotic properties of partially dissipative hyperbolic systems when switching to a discrete setting. We prove that one of the simplest consistent and unconditionally stable numerical methods - the central finite difference scheme - preserves both the asymptotic behaviour and the parabolic relaxation limit of one-dimensional partially dissipative hyperbolic systems which satisfy the Kalman rank condition. The large time asymptotic-preserving property is achieved by conceiving time-weighted perturbed energy functionals in the spirit of the hypocoercivity theory. For the relaxation-preserving property, drawing inspiration from the observation that solutions in the continuous case exhibit distinct behaviours in low and high frequencies, we introduce a novel discrete Littlewood-Paley theory tailored to the central finite difference scheme. This allows us to prove Bernstein-type estimates for discrete differential operators and leads to a new relaxation result: the strong convergence of the discrete linearized compressible Euler system with damping towards the discrete heat equation, uniformly with respect to the mesh parameter.

Young transiting exoplanets offer a unique opportunity to characterize the atmospheres of fresh and evolving products of planet formation. We present the transmission spectrum of V1298 Tau b; a 23 Myr old warm Jovian sized planet orbiting a pre-main sequence star. We detect a primordial atmosphere with an exceptionally large atmospheric scale height and a water vapour absorption at 5$\sigma$ level of significance. We estimate a mass and density upper limit (24$\pm$5$M_{\oplus}$, 0.12gm/$cm^{3}$ respectively). V1298 Tau b is one of the lowest density planets discovered till date. We retrieve a low atmospheric metallicity (logZ=$-0.1^{+0.66}_{-0.72}$ solar), consistent with solar/sub-solar values. Our findings challenge the expected mass-metallicity from core-accretion theory. Our observations can be explained by in-situ formation via pebble accretion together with ongoing evolutionary mechanisms. We do not detect methane, which hints towards a hotter than expected interior from just the formation entropy of this planet. Our observations suggest that V1298 Tau b is likely to evolve into a Neptune/sub-Neptune type of planet., Comment: 37 pages, Submitted Nature Astronomy

We investigate the diffusive relaxation limit and the time-asymptotic stability of the Jin-Xin model toward viscous conservation laws in $\mathbb{R}^d$ with $d\geq1$. First, we establish uniform regularity estimates with respect to both the time and the relaxation parameter $\varepsilon>0$, for initial data in hybrid Besov spaces based on general $L^{p}$-norms. This uniformity enables us to derive $\mathcal{O}(\varepsilon)$ bounds on the difference between solutions of the viscous conservation law and its associated Jin-Xin approximation, thus justifying the strong convergence of the Jin-Xin hyperbolic relaxation. Furthermore, under an additional condition on the initial data, for instance, that the low frequencies belong to $L^{p/2}(\mathbb{R}^{d})$, we show that the $L^{p}(\mathbb{R}^d)$-norm of the solution to the Jin-Xin model decays at the optimal rate $(1+t)^{-d/{2p}}$ while the $L^{p}(\mathbb{R}^d)$-norm of its difference with the solution of the associated viscous conservation law decays at the enhanced rate $\varepsilon(1+t)^{-d/{2p}-1/2}$., Comment: 39 pages

In this paper, we consider the problem of finding an optimal energy management policy for a network of sensor nodes capable of harvesting their own energy and sharing it with other nodes in the network. We formulate this problem in the discounted cost Markov decision process framework and obtain good energy-sharing policies using the Deep Deterministic Policy Gradient (DDPG) algorithm. Earlier works have attempted to obtain the optimal energy allocation policy for a single sensor and for multiple sensors arranged on a mote with a single centralized energy buffer. Our algorithms, on the other hand, provide optimal policies for a distributed network of sensors individually harvesting energy and capable of sharing energy amongst themselves. Through simulations, we illustrate that the policies obtained by our DDPG algorithm using this enhanced network model outperform algorithms that do not share energy or use a centralized energy buffer in the distributed multi-nodal case., Comment: 11 pages, 4 figures

In a graph $G$, the $2$-neighborhood of a vertex set $X$ consists of all vertices of $G$ having at least $2$ neighbors in $X$. We say that a bipartite graph $G(A,B)$ satisfies the double Hall property if $|A|\geq2$, and every subset $X \subseteq A$ of size at least $2$ has a $2$-neighborhood of size at least $|X|$. Salia conjectured that any bipartite graph $G(A,B)$ satisfying the double Hall property contains a cycle covering $A$. Here, we prove the existence of a $2$-factor covering $A$ in any bipartite graph $G(A,B)$ satisfying the double Hall property. We also show Salia's conjecture for graphs with restricted degrees of vertices in $B$. Additionally, we prove a lower bound on the number of edges in a graph satisfying the double Hall property, and the bound is sharp up to a constant factor., Comment: minor corrections; to be published in Discrete Mathematics

Solid-state spin qubits within silicon crystals at mK temperatures show great promise in the realisation of a fully scalable quantum computation platform. Qubit coherence times are limited in natural silicon owing to coupling to the isotope ${ }^{29} \mathrm{Si}$ which has a non-zero nuclear spin. This work presents a method for the depletion of ${ }^{29} \mathrm{Si}$ in localised volumes of natural silicon wafers by irradiation using a 45 keV ${ }^{28} \mathrm{Si}$ focused ion beam with fluences above $1 \times 10^{19} \, \mathrm{ions} \, \mathrm{cm}^{-2}$. Nanoscale secondary ion mass spectrometry analysis of the irradiated volumes shows unprecedented quality enriched silicon that reaches a minimal residual ${ }^{29} \mathrm{Si}$ value of 2.3 $\pm$ 0.7 ppm and with residual C and O comparable to the background concentration in the unimplanted wafer. Transmission electron microscopy lattice images confirm the solid phase epitaxial re-crystallization of the as-implanted amorphous enriched volume extending over 200 nm in depth upon annealing. The ease of fabrication, requiring only commercially available natural silicon wafers and ion sources, opens the possibility for co-integration of qubits in localised highly enriched volumes with control circuitry in the surrounding natural silicon for large-scale devices., Comment: 8 pages, 4 figures, 2 tables

A new framework to obtain time-decay estimates for partially dissipative hyperbolic systems set on the real line is developed. Under the classical Shizuta-Kawashima (SK) stability condition, equivalent to the Kalman rank condition in control theory, the solutions of these systems decay exponentially in time for high frequencies and polynomially for low ones. This allows to derive a sharp description of the space-time decay of solutions for large time. However, such analysis relies heavily on the use of the Fourier transform that we avoid here, developing the "physical space version" of the hyperbolic hypocoercivity approach introduced by Beauchard and Zuazua, to prove new asymptotic results in the linear and nonlinear settings. The new physical space version of the hyperbolic hypocoercivity approach allows to recover the natural heat-like time-decay of solutions under sharp rank conditions, without employing Fourier analysis or $L^1$ assumptions on the initial data. Taking advantage of this Fourier-free framework, we establish new enhanced time-decay estimates for initial data belonging to weighted Sobolev spaces. These results are then applied to the nonlinear compressible Euler equations with linear damping. We also prove the logarithmic stability of the nonlinearly damped $p$-system.

The presence of aerosols is intimately linked to the global energy budget and the composition of a planet's atmospheres. Their ability to reflect incoming light prevents energy from being deposited into the atmosphere, and they shape spectra of exoplanets. We observed five near-infrared secondary eclipses of WASP-80b with the Wide Field Camera 3 (WFC3) aboard the \textit{Hubble Space Telescope} to provide constraints on the presence and properties of atmospheric aerosols. We detect a broadband eclipse depth of $34\pm10$\,ppm for WASP-80b. We detect a higher planetary flux than expected from thermal emission alone at $1.6\sigma$, which hints toward the presence of reflecting aerosols on this planet's dayside, indicating a geometric albedo of $A_g<0.33$ at 3$\sigma$. We paired the WFC3 data with Spitzer data and explored multiple atmospheric models with and without aerosols to interpret this spectrum. Albeit consistent with a clear dayside atmosphere, we found a slight preference for near-solar metallicities and for dayside clouds over hazes. We exclude soot haze formation rates higher than $10^{-10.7}$ g cm$^{-2}$s$^{-1}$ and tholin formation rates higher than $10^{-12.0}$ g cm$^{-2}$s$^{-1}$ at $3\sigma$. We applied the same atmospheric models to a previously published WFC3/Spitzer transmission spectrum for this planet and found weak haze formation. A single soot haze formation rate best fits both the dayside and the transmission spectra simultaneously. However, we emphasize that no models provide satisfactory fits in terms of the chi-square of both spectra simultaneously, indicating longitudinal dissimilarity in the atmosphere's aerosol composition., Comment: Published in ApJ Letters (20 Oct 2023)

In an orientation $O$ of the graph $G$, an arc $e$ is deletable if and only if $O-e$ is strongly connected. For a $3$-edge-connected graph $G$, the Frank number is the minimum $k$ for which $G$ admits $k$ strongly connected orientations such that for every edge $e$ of $G$ the corresponding arc is deletable in at least one of the $k$ orientations. H\"orsch and Szigeti conjectured the Frank number is at most $3$ for every $3$-edge-connected graph $G$. We prove an upper bound of $5$, which improves the previous bound of $7$., Comment: 7 pages

The early K-type T-Tauri star, V1298 Tau ($V=10\,{\rm mag}$, ${\rm age}\approx20-30\,{\rm Myr}$) hosts four transiting planets with radii ranging from $4.9-9.6\,R_\oplus$. The three inner planets have orbital periods of $\approx8-24\,{\rm d}$ while the outer planet's period is poorly constrained by single transits observed with \emph{K2} and \emph{TESS}. Planets b, c, and d are proto-sub-Neptunes that may be undergoing significant mass loss. Depending on the stellar activity and planet masses, they are expected to evolve into super-Earths/sub-Neptunes that bound the radius valley. Here we present results of a joint transit and radial velocity (RV) modelling analysis, which includes recently obtained \emph{TESS} photometry and MAROON-X RV measurements. Assuming circular orbits, we obtain a low-significance ($\approx2\sigma$) RV detection of planet c implying a mass of $19.8_{-8.9}^{+9.3}\,M_\oplus$ and a conservative $2\sigma$ upper limit of $<39\,M_\oplus$. For planets b and d, we derive $2\sigma$ upper limits of $M_{\rm b}<159\,M_\oplus$ and $M_{\rm d}<41\,M_\oplus$. For planet e, plausible discrete periods of $P_{\rm e}>55.4\,{\rm d}$ are ruled out at a $3\sigma$ level while seven solutions with $43.3

Mn$_3$Si$_2$Te$_6$ is a rare example of a layered ferrimagnet. It has recently been shown to host a colossal angular magnetoresistance as the spin orientation is rotated from the in- to out-of-plane direction, proposed to be underpinned by a topological nodal-line degeneracy in its electronic structure. Nonetheless, the origins of its ferrimagnetic structure remain controversial, while its experimental electronic structure, and the role of correlations in shaping this, are little explored to date. Here, we combine x-ray and photoemission-based spectroscopies with first-principles calculations, to probe the elemental-selective electronic structure and magnetic order in Mn$_3$Si$_2$Te$_6$. Through these, we identify a marked Mn-Te hybridisation, which weakens the electronic correlations and enhances the magnetic anisotropy. We demonstrate how this strengthens the magnetic frustration in Mn$_3$Si$_2$Te$_6$, which is key to stabilising its ferrimagnetic order, and find a crucial role of both exchange interactions extending beyond nearest-neighbours and anti-symmetric exchange in dictating its ordering temperature. Together, our results demonstrate a powerful methodology of using experimental electronic structure probes to constrain the parameter space for first-principles calculations of magnetic materials, and through this approach, reveal a pivotal role played by covalency in stabilising the ferrimagnetic order in Mn$_3$Si$_2$Te$_6$.

Let $G(V,E)$ be a finite, simple, isolate-free graph. Two disjoint sets $A,B\subset V$ form a total coalition in $G$, if none of them is a total dominating set, but their union $A\cup B$ is a total dominating set. A vertex partition $\Psi=\{C_1,C_2,\dots,C_k\}$ is a total coalition partition, if none of the partition classes is a total dominating set, meanwhile for every $i\in\{1,2,\dots,k\}$ there exists a distinct $j\in\{1,2,\dots,k\}$ such that $C_i$ and $C_j$ form a total coalition. The maximum cardinality of a total coalition partition of $G$ is the total coalition number of $G$ and denoted by $TC(G)$. We give a general sharp upper bound on the total coalition number as a function of the maximum degree. We further investigate this optimal case and study the total coalition graph. We show that every graph can be realised as a total coalition graph., Comment: This updated version contains a complete answer for Problem 4.7 of the first version