In reverse mathematics, real numbers are traditionally represented by Cauchy sequences with a given rate of convergence. We work without rates and speak of slow Cauchy sequences. It turns out that almost all one-dimensional real analysis from the reverse mathematics book by Simpson can then be developed in theories that are conservative over $\mathsf{RCA}_0$. Specifically, we obtain clusters of equivalences with the infinite pigeonhole principle and the strong cohesive principle. The second cluster includes results like the Bolzano-Weierstrass and Arzelà-Ascoli theorems, which are traditionally associated with the stronger axiom of arithmetical comprehension, but also the Heine-Borel theorem, which is normally separated from these principles. This suggests two things: In elementary analysis, one can avoid logical strength to an extent that the traditional picture seems to forbid. And the division of the so-called reverse mathematics zoo into analytical and combinatorial principles may be less rigid than previously assumed.

We show that the low-energy states of non-Abelian topological orders possess extensive magic which is long-ranged, and cannot be eliminated by a constant-depth local unitary circuit. This refines conventional notions of complexity beyond the linear circuit depth which is required to prepare any topological phase, and provides a new resource-theoretic characterization of topological orders. A central technical result is a no-go theorem establishing that stabilizer states--even up to constant-depth local unitarie--cannot approximate low-energy states of non-Abelian string-net models which satisfy the entanglement bootstrap axioms. Moreover, we show that stabilizer-realizable Abelian string-net phases have mutual braiding phases quantized by the on-site qudit dimension, and that any violation of this condition necessarily implies extensive long-range magic. Extending to higher spatial dimensions, we argue that any state obeying an entanglement area law and hosting excitations with nontrivial fusion spaces must exhibit extensive long-range magic. This applies, in particular, to ground-states and low-energy states of higher-dimensional quantum double models.

We consider a one-parameter family of 1D models for the 3D axisymmetric incompressible Euler equation with $C^α$ vorticity and without swirl near the symmetry axis. For $α= \frac13$, we impose a crucial normalization and construct a $C^{\infty}$ self-similar blowup profile with unbounded 1D stream function and infinite spatial blowup rate, using a fixed-point argument around a numerically constructed approximate profile. For $α< \frac13$ sufficiently close to $\frac13$, we perturb the $\frac13$-profile and analytically construct exact smooth 1D profiles with bounded stream function and finite spatial blowup rate. In the companion work~\cite{chen2026eulerII}, for any $α\in (0,\frac13)$, we lift these 1D blowup profiles to construct exact $C^{1,α}$ self-similar blowup profiles for 3D Euler, and build on them to prove sharp asymptotically self-similar blowup for 3D axisymmetric Euler without swirl from $C_c^α$ initial vorticity and $C^{1,α} \cap L^2$ initial velocity.

Conservation laws of a class of time-dependent damped nonlinear multidimensional wave equations are derived by Noether's theorem. For arbitrary nonzero damping coefficient and nonlinear interaction term, its infinitesimal variational symmetries span a Euclidean algebra $\euclid(n)$ of space translations and rotations. They produce conservation of linear and angular momentums. For some specific forms of these two terms symmetry algebra is enlarged to a subalgebra of the conformal algebra $\conf(1,n)$ and in this case more interesting conservation laws are found.

We show that for $m, r \in \mathbb{N}$ and $N > (2m+1)^r (r!)^{1/m}$, every $r$-coloring of the integers in the interval $[N]$ contains a monochromatic solution to the equation \[ x_1 + \dots + \dots x_{m+1} = y_1 + \dots + y_m. \] This generalizes and improves recent results of Koścuiszko. We also show that if $N \geq 2^{r}$, then every $r$-coloring of the integers in $[N]$ must always determine a monochromatic solution to the above equation for some $m \geq 1$. The latter estimate is optimal.

Neutrino oscillation experiments present anomalous results across a vast range of baselines and energies. Here we show that a 3+1 scenario in which sterile neutrinos feel a novel matter potential $V_s$ proportional to background density of ordinary or (asymmetric) dark matter is able to explain several anomalies. At low-energies ($E\lesssim$ 1 TeV) the model behaves as an effective 3-flavor NSI-like scheme among active flavors and eliminates the tension between the two LBL experiments NOvA and T2K provided that the potential is negative and the two sterile mixing angles $θ_{14}$ and $θ_{24}$ are non-zero. A further indication in favor of a negative non-zero potential comes from the anomalous excess of $ν_e$-like events observed in Super-Kamiokande atmospheric neutrinos, which, in the new scenario is explained by a modification of the 3-flavor resonance at few GeV. A high energies ($E\gtrsim $ 1 TeV) the new framework reveals its 4-flavor nature and produces a resonant behavior at $E \simeq$ 10 TeV as hinted at by IceCube. We identify an irreducible 3-level dynamics generating a new resonance in the $(ν_e, ν_μ)$ sector intertwined with two conventional resonances in the $(ν_e, ν_s$) and $(ν_μ, ν_s)$ systems. The novel amplification mechanism manifests with the emergence of effective mixing angles in matter ($θ_{12}^m$ or $θ_{13}^m$) involving active neutrinos. The scenario requires values of $f = V_s/|V_{NC}| \sim -20 $, $Δm^2_{41} \sim 60 $ eV$^2$, $|U_{e4}|^2\simeq \sin^2θ_{14} \simeq 0.01-0.03$ and $|U_{\mu4}|^2 \simeq \sin^2θ_{24}\simeq 10^{-4}-10^{-3}$. Such a very small size of $|U_{\mu4}|^2$ eliminates the tension between IceCube and the other $ν_μ$ disappearance searches. The model can be directly probed by KATRIN, which is very sensitive to the electron-sterile neutrino admixture in the region of high $Δm^2_{41}$.

It is of a general interest to look for signatures of stellar bodies orbiting supermassive black holes (SMBHs) in galactic nuclei other than the Galactic center. Previously stellar transits were analyzed in UV, optical, and X-ray domains as well as potential microlensing signatures due to more compact bodies orbiting SMBH accretion disks. Here we complement previous studies by considering nearby ($z=0.001$) radio-loud active galactic nuclei targeted by different facilities in the millimeter domain. At these wavelengths the radio core is sufficiently small so that it can be occulted by large evolved stars in dense nuclear star clusters. We find that in the millimeter domain evolved stars with stellar radii of $\gtrsim 500\,R_{\odot}$ can cause eclipses with the relative depth of $\sim 10\%$. Typical recurrence timescales are at least 10 years and the eclipse durations are $\sim 10$ days. Towards lower frequencies the eclipse temporal profiles become shallower and broader while towards higher frequencies they are deeper and narrower. Although expected to be rare due to selection effects and evolved stars being prone to tidal disruption, recurrent eclipses of mm radio cores can be applied to infer SMBH masses and constrain the composition of the Nuclear Star Cluster of the host nucleus.

This paper develops a formal logic for guises based on the work of Héctor-Neri Castañeda, who understood relations from an internalist viewpoint, following Leibniz. We introduce a syntax, model theory, and proof theory for an intensional logic in which guises (taken as bundles of properties equipped with intention) serve as primary semantic objects. The system integrates (i) a Leibnizian containment semantics for singular truths, (ii) an intentional operator that captures internal relations among guises, and (iii) a modal layer for possibility and necessity modeled as maximally consistent closures. We establish core metatheoretic results (e.i. soundness and canonical-model completeness sketches) and analyze hyperintensional phenomena such as substitution failure in intentional contexts, quasi-indexicality, and de se reference. We compare the framework to classical intensional semantics (Montague), property theory (Bealer), hyperintensional logics (Fine), situation semantics (Barwise and Perry), and to the Leibniz program for a calculus of concepts. The result is a selfcontained formal framework that demonstrates that relations are not external causal links but intentional internal structures encoded in the guises through which agents and objects are conceived: i.e., they are perspectives.

This paper investigates the algorithmic safety verification problem of infinite-state parameterized concurrent programs over a rich set of communication topologies. The goal is to automatically produce a proof of correctness in the form of a universally quantified inductive invariant, where the quantification is over the nodes in the topology. We illustrate that under reasonable assumptions on the underlying topology, the problem can be reduced to and solved as a compositional scheme, that is, the verification of the parameterized family is reduced to a set of local proofs, in a complete manner. We propose a verification algorithm, which is implemented as a tool, and demonstrate through a set of benchmarks over several different topologies that our approach is effective in proving parameterized programs safe.

Component network meta-analysis (CNMA) is a statistical methodology that enables estimation of relative effects for multi-component treatments, such as combinations of antidepressants, and individual components, such as single antidepressants, by synthesizing data from multiple studies. A commonly desired output of a systematic review and meta-analysis is a hierarchy of the treatments in terms of a certain performance metric. Methods have been established for standard network meta-analysis (NMA), but have not yet been extended to CNMA. In particular, CNMA presents unique challenges because the set of relative effects that can be uniquely estimated is more complex to determine compared to standard NMA, and a hierarchy involving relative effects that are not uniquely estimable is misleading. We present a step-by-step workflow for answering treatment hierarchy questions in both frequentist and Bayesian CNMA, including explicitly identifying the uniquely estimable relative effects. We illustrate the workflow by posing multiple treatment hierarchy questions in two distinct networks, one concerning primary care of depression and one disconnected network investigating treatment for chronic lymphocytic leukemia.

Ultrafast diffraction is the cutting-edge technique to extract the atomic temperature at femtosecond timescales, and further related quantities - in particular, electron-phonon coupling strength at elevated electronic temperatures. The present work demonstrates limitations of such an analysis, emphasizing the importance of careful evaluation of the evolution of the Debye temperature. It is shown that, due to the sensitivity of the Debye-Waller analysis to this parameter, neglecting its changes under electronic excitation may lead to significant deviations of the atomic temperature extracted from its true values.

A silicon microring resonator with an anisotropic beryllium sulfide (BeS) cladding is proposed for polarization-selective detection of exhaled-breath volatile organic compound biomarkers. The anisotropic dielectric response of BeS enables the transverse-electric (TE) and transverse-magnetic (TM) modes to probe orthogonal components of the cladding permittivity tensor, generating two independent optical observables from a single device. Five clinically relevant biomarkers are investigated: acetone, isoprene, 4-hydroxyhexenal, 2-propenal, and benzene. First-principles optical constants are incorporated into three-dimensional finite-difference time-domain simulations to evaluate the sensing response. The TE mode exhibits a uniform resonance shift of 0.263 nm across all analytes and serves as a concentration reference channel, while the TM mode produces analyte-specific shifts ranging from 0.200 to 0.426 nm. A unique TM amplitude inversion is observed for benzene, enabling additional discrimination. The resulting dual-polarization response forms a two-dimensional optical fingerprint that distinguishes all five biomarkers without requiring a sensor array or multiple functionalized resonators. The device achieves quality factors of 4520 and 3151 for the TE and TM modes, respectively, with sensitivities up to 6.5 nm/RIU, figures of merit up to 14.9 RIU^-1, and detection limits as low as 1.5 mRIU. Cross-sensitivity analysis further shows that CO2 and H2O produce negative TM resonance shifts, separating interferents from target biomarkers in the fingerprint plane. The proposed platform demonstrates a compact route toward array-free photonic breath analysis using intrinsic cladding anisotropy.

We report the discovery of a new candidate pulsating ultraluminous X-ray source (PULX) in NGC 5128 (Centaurus A). The candidate, 4XMM J132542.2-425943, is a transient source, identifiable as a clear X-ray point source for $\sim 8$ months in 2014, during its only major recorded outburst. The source flux exceeded $10^{-12}$ erg cm$^{-2}$ s$^{-1}$ at the peak of the outburst. The long-term light curve of 4XMM J132542.2-425943 shows two further, less luminous detections in 2017 and 2024, but was otherwise in quiescence. This behaviour is similar to the class of pulsating transients with outbursts that reach the ultraluminous regime, which includes the well-studied Galactic PULX, Swift J0243.6+6124. However, 4XMM J132542.2-425943 displays a soft X-ray spectrum, making this source distinct from the existing population of PULXs, which typically show hard spectra below $10$ keV. We searched the 2014 XMM-Newton observations for X-ray pulsations, revealing coherent, sinusoidal X-ray pulsations at a frequency of $1.27$ Hz in one XMM-Newton observation (ObsID 0724060801), at a marginal significance. For this signal we measure a pulsed fraction, PF$\approx~15 - 17~\%$ and $\dot{f}~\sim~4~\times~10^{-9}$ Hz s$^{-1}$. However, we find that the intrinsic randomisation employed by XMM-Newton's Science Analysis Software, XMM-SAS, during the data reduction procedure introduces considerable uncertainty in the strength of our marginal pulsations, which varies significantly between consecutive data reduction iterations. We explore the impact of this randomisation and demonstrate that it can generate widespread false positives and false negatives, which, in the context of PULX searches, may cause viable candidates to be unnecessarily discarded or vice versa.

The theory of log concave polynomials has recently been developed to study objects and problems in combinatorics and other subfields in mathematics. Particular classes of log concave polynomials called Lorentzian polynomials and denormalized and dually Lorentzian polynomials have been used to prove log concavity statements for various combinatorial sequences. This includes the strongest form of Mason's log concavity conjecture on the independent sets of matroids and the log concavity of sequences of Kostka numbers. In this paper, we develop an analogous class of power series called denormalized Lorentzian (DL) Laurent series. This class is the natural generalization of DL polynomials to homogeneous power series with the benefit of capturing a number of combinatorial generating series including the Kostant partition function for integer flows of directed graphs. We then analyze specific DL Laurent series to obtain new bounds for integral flows on general directed acyclic graphs and new bounds for the dimensions of weight spaces of parabolic $\mathfrak{sl}_{n+1}(\mathbb{C})$ Verma modules.

As a critical component of sixth-generation (6G) wireless networks, ultra-reliable and low-latency communication (URLLC) is expected to support real-time and reliable information exchange in low-altitude environments. However, achieving URLLC often incurs significant resource overhead, including increased bandwidth consumption, higher transmit power, and denser access point (AP) deployment, which pose significant challenges to both spectral efficiency (SE) and energy efficiency (EE). Besides, existing iterative optimization algorithms are computationally intensive and struggle to meet the latency requirements of URLLC. To address these challenges, we propose a hybrid aerial-terrestrial cell-free massive MIMO (CF-mMIMO) network to support diverse services, along with a channel prediction network and a deep mixture of experts (MoE) network for uplink optimization. First, we design a channel prediction network (CP-Net) to mitigate channel aging caused by high-mobility user equipment (UE). CP-Net employs three Transformer-based sub-networks for aged channel state information (CSI) prediction, while a channel quality-aware loss function is introduced to improve the prediction accuracy of weak links. Based on the predicted CSI, we develop a deep MoE network (MoE-Net) for power allocation comprising three expert models targeting different objectives. Then, we introduce a weighted gating network (WT-Net) to learn an efficient adaptive combination of expert outputs. The proposed framework better captures heterogeneous UE requirements and improves communication performance under URLLC constraints. Numerical results demonstrate the effectiveness of the proposed method.

For each integer $n \geq 3$, the wheel graph $W_n$ is defined as the graph obtained by connecting a single vertex to all vertices of a cycle of length $n$. In particular, $W_6$ can be uniquely obtained from the Petersen graph by contracting three edges incident to a common vertex. Gubser provided a characterization of all 3-connected planar $W_6$-minor-free graphs. In this paper, we complete the characterization of $W_6$-minor-free graphs by determining the 3-connected nonplanar cases.

This paper investigates User Experience (UX) with prototypes generated by Generative Artificial Intelligence (GenAI) tools. An empirical survey with 92 participants evaluated AI-generated and human-created prototypes without prior identification of authorship. We measured UX using the UEQ-S, covering pragmatic and hedonic dimensions. Results indicate positive evaluations in pragmatic aspects, such as usability and efficiency, and neutral or negative evaluations in hedonic aspects, including originality and innovation. We concluded that GenAI can produce functional interfaces but tends to reinforce visual and structural patterns that affect perceptions of originality.

Efficient and selective excitation of lattice vibrations is central to controlling energy flow at the nanoscale, yet remains challenging under conventional optical excitation. Here, we introduce a mid-infrared-assisted phonon amplification approach, termed MIRAPA, that enables efficient energy injection directly into vibrational bonds. Using surface-enhanced resonant Raman scattering in few-layer $\mathrm{MoS_2}$, we exploit strong exciton--phonon coupling to monitor phonon populations. When mid-infrared (MIR) light is introduced, it couples directly to out-of-plane lattice vibrations, leading to room-temperature phonon amplification exceeding $80\%$. Crucially, MIRAPA bypasses electronic excitation pathways, allowing the MIR power density to be nearly $300\times$ lower than that required for visible excitation to achieve comparable enhancement. The resulting phonon modulation is robust, persisting over more than $2800$ on/off cycles and exceeding $15$ hours of continuous-wave laser illumination without degradation. Quantitative analysis yields an effective noise-equivalent power of approximately $0.3\,\mathrm{nW}/\sqrt{\mathrm{Hz}}$ for MIR detection, highlighting the sensitivity of the approach. By combining vibrational selectivity, low-power operation, and long-term stability, MIRAPA provides a robust platform for probing and amplifying phonons in two-dimensional semiconductors. These results open new opportunities for nanoscale vibrational sensing, mid-infrared detection, and phonon-based coherent devices, including routes toward phonon lasing.

The possibility of laser cooling and the presence of closely spaced rovibrational doublets make polyatomic molecules an attractive platform for the $\mathcal{P}$, $\mathcal{T}$-violation searches. We study the spectrum of the lowest rovibrational state of the AcOCH$_3+$ symmetric top molecule. The electronic structure full-electron computation was performed within a relativistic coupled cluster method with double and perturbative triple excitations. The rovibrational wavefunctions are obtained using a coupled channel technique, taking into account all rovibrational effects and anharmonicities of the potential. As a result, the vibrational frequencies, as well as the values of the electric dipole moments for the rovibrational states, were computed.

This paper develops a difference-in-differences framework for staggered policy adoption when units can be affected by other units' adoption. For each treated cohort and event time, the framework separates the effect of own adoption, the spillover effect generated by other adopters, and the total effect under the realized rollout. Identification uses a prespecified summary of spillover exposure and parallel trends comparisons among units with the same exposure at the baseline and target dates. Spillover effects are learned from never-treated units and evaluated for treated cohorts under the exposure distribution they face. We construct estimators for these effects and an inference procedure that allows for spatial dependence. Monte Carlo simulations illustrate that standard DID estimators that ignore spillovers can miss the total effect, whereas the proposed estimators have small bias for these effects and the associated confidence intervals have coverage close to the nominal level. In an empirical study of the Community Health Centers rollout, estimated spillovers account for a substantial share of the effect on older-adult mortality.

We show that the $x$-axis is the only real algebraic curve in $\mathbb R^2$ whose image via the Gamma function is contained in an algebraic curve. Our proof employs an elegant base-change argument due to Tamiozzo (2023) to deduce the result from the corresponding complex geometric transcendence result of Eterović, Padgett and Zhao (2025). As an application, we use the complex and real geometric transcendence results to study analogues of the Manin--Mumford conjecture for the Gamma function.

Instrumental variables (IV) methods are central to applied microeconomics. While classical approaches assume linear models with constant effects, recent literature has shifted toward the local average treatment effect (LATE) framework to accommodate heterogeneous treatment effects. This paper provides a practical guide to aligning empirical practice with recent theory. We first examine how different specifications with covariates lead to distinct weighted averages of covariate-specific LATEs. We then discuss how parametric misspecification can undermine the causal interpretation of these estimands and suggest flexible specifications as essential robustness checks. Finally, we review formal tests for LATE assumptions and methods robust to monotonicity violations. We provide a guide to software implementations to help researchers apply the methods in practice.

Independent and identically distributed (i.i.d.) states are ubiquitous in quantum information theory. However, in a practical setting, the i.i.d. assumption is too stringent, and possibly not realistic. A physically more compelling class of 'almost i.i.d.' sources was recently proposed by [Mazzola/Sutter/Renner, arXiv:2603.15792]. In this paper, we introduce two alternative definitions of almost i.i.d. states, based on the normalised quantum Wasserstein distance and on the idea of looking at the average $k$-body marginal. We explore some basic properties of these notions and prove a strict hierarchical relation among them, with Mazzola et al.'s notion being the strictest, the one based on $k$-body marginals the loosest, and the one based on the quantum Wasserstein distance in between. Strict separation is established by means of explicit examples.

Solar pores possess irregular and evolving boundaries that are often far from the ideal circular flux tubes assumed in many magnetohydrodynamic (MHD) oscillation models. To tackle this irregularity in a consistent way for wave analysis, we introduce a framework that employs the convex hull of the pore boundary - derived from intensity minimum error thresholding - as the domain to perform further analysis. Using the modal assurance criterion, we find the range of pore shapes for which this approximation is valid. We demonstrate the usefulness of this framework by applying it to multi-height, high-cadence observations (4170 angstrom continuum, G-band, Na~\textsc{i}, and Ca~\textsc{ii}~K) of a solar pore, and apply Proper Orthogonal Decomposition of the convex hull to determine wave modes. The fundamental sausage ($m=1$) and kink ($m=2$) mode is found to remain reliable, while higher-order fluting modes ($m\ge3$) are strongly degraded by small-scale boundary irregularity. As expected, sausage-like modes dominate the variance at all heights and exhibit a systematic upward shift in frequency, consistent with freely propagating compressive waves. In contrast, the kink-like motions appear weak, confined to a persistent low-frequency peak, and most plausibly interpreted as a forced response to granular buffeting rather than a propagating mode. Together, these results establish a practical methodology for boundary-mode analysis in real, highly structured pores and provide new constraints on the nature and height evolution of MHD waves in the lower solar atmosphere.

We consider the displacement contribution to photoresistance in overlapping Landau levels at radiation frequencies much smaller than the cyclotron frequency. We show that in the limit of short-range disorder and high radiation power, this contribution leads to a new class of magneto-resistance oscillations. These oscillations, which we call radiowave-induced resistance oscillations (RIROs), are distinct from the well known microwave-induced resistance oscillations in the following aspects: (i) their amplitude is independent of power, (ii) their period is controlled by the radiation electric field, rather than by the radiation frequency, and (iii) they can be either $1/B$ or $1/B^2$-periodic, depending on $B$, with the crossover point linked to the width of the cyclotron resonance absorption curve. We also show that RIROs should be readily observed in experiments.

The present authors introduced a two-color partition series $S(q)$ and conjectured a Hecke-type formula for the even part of $(q^4;q^4)_\infty S(q)$. Banerjee and Bringmann proved the conjecture by using indefinite theta functions, modular completions, and Sturm's theorem. They also asked whether a direct proof, for instance one based on Bailey-type ideas, could be found, and they suggested that the odd residue classes may be worth studying. We prove a two-variable refinement with an additional parameter $a$. Our proof relies entirely on $q$-series combined with the Bailey pairs The original even identity and the odd identity then follow as corollaries by letting $a=1$. We also record parameter symmetries and cyclotomic companions, including a vanishing result at $a=i$.

Characterizing large quantum systems with minimal assumptions is a central challenge in quantum information science. Self-testing provides the strongest form of certification by identifying the underlying quantum state solely from observed measurement statistics. However, existing self-testing methods for generic $n$-partite states face a scalability barrier, requiring exponentially many samples in the system size. In this work, we overcome this barrier by introducing a protocol that robustly self-tests almost all $n$-qubit states with only polynomial sample complexity. The key ingredient is an efficient scheme for device-independently evaluating multipartite Pauli measurements, which can be implemented using only a linear number of ancillary Bell pairs together with standard projective and Bell measurements, well within the reach of current quantum technology. Beyond self-testing states, our scheme provides a general framework for implementing a wide range of learning and certification protocols in the device-independent setting, thereby opening a scalable route to device-independent quantum information processing in large-scale quantum networks.

For $k\ge 3$, the $(k-2)$-uniform Turán density $π_{k-2}(F)$ of a $k$-graph $F$ is the supremum of $d$ for which there are arbitrarily large $F$-free $k$-graphs that are uniformly $d$-dense with respect to the $k$-vertex cliques of every $(k-2)$-graph on the same vertex set. We develop a \emph{palette framework} for this density. For every family $\mathcal F$ of $k$-graphs, we prove that $π_{k-2}(\mathcal F)$ equals the corresponding palette Turán density. We further establish palette classification tools for the existence of $k$-graphs satisfying prescribed palette colorability constraints. Those together allow us to reduce exact density computations to a palette-homomorphism framework without relying on the hypergraph regularity method. As applications, for all $k\ge 3$ and $r\ge 2$, we establish the following values \[ \frac{r-1}{r},\quad \frac{(r-1)^2}{r^2},\quad \frac{r-1}{2r},\quad \frac{(k-1)^k}{k^k},\quad \frac{4(k-2)^{k-2}}{k^k},\quad \frac{4(k-2)^{k-2}}{3k^k} \] as $(k-2)$-uniform Turán densities of single $k$-graphs. Finally, for every $k\ge3$, we show that there exist $k$-graphs $F_1,F_2$ such that \[ π_{k-2}(\{F_1,F_2\})< \min\{π_{k-2}(F_1),π_{k-2}(F_2)\}, \] which provides the first examples of \emph{non-principal} families for this density.

On the 28th of September, 2018, a tsunami hit the city of Palu in Indonesia, killing 4,340 people. The earthquake preceding the tsunami crippled communication lines and may have rendered the transmission of tsunami warning messages using traditional end-to-end approaches impossible. This paper proposes an alternative approach using Delay Tolerant Networks (DTNs) for tsunami warning message routing given their resilience to disruptions and sparse connections. Both Epidemic and Spray and Wait routing protocols were simulated in a pseudo-realistic environment to evaluate their effectiveness for transmitting tsunami warning messages in Palu. Results indicated that these protocols are not suitable for the tight time constraints of post-earthquake tsunami warnings with the currently available technology. However, they may have promising applications for the earthquakes that precede tsunamis.

Classifying interactions is key in the physical sciences, and bonding mechanisms in matter-antimatter systems remain particularly enigmatic. Here we focus on a paradigmatic example of positronium hydride (PsH) dimer composed of two protons, two positrons, and four electrons, whose bonding nature has been previously described as either ionic, covalent, or van der Waals-like. Accurate quantum Monte Carlo calculations show that the two positrons occupy a delocalized molecular orbital that envelopes the two hydrogen anions and responds as a collective dipole to an applied electric field. This positronic bonding stems from quantum correlations that resemble a single covalent bond formed between negatively charged pseudo-nuclei, but with a bond strength commensurate with the traditional van der Waals interaction. Our findings suggest that the ability to form delocalized proto-bonds is a more general property of quantum systems, and could be present in a broader class of particles, antiparticles, and quasi-particles interacting with matter.