Log-homotopy particle flow filters realize nonlinear Bayesian estimation by continuously migrating samples from the prior to the posterior distribution. This transport is governed by a pseudo-time ordinary differential equation (ODE). A major practical challenge of these filters is the need for numerical integration, which suffers from high computational cost and susceptibility to stiffness. This paper develops an exact, integration-free closed-form solution for the exact Daum--Huang deterministic particle flow under vector linear Gaussian measurements. By transforming the ODE into a specific eigenspace, we derive closed-form algebraic expressions for both the homogeneous state transition matrix and the inhomogeneous forcing term. We prove that this analytic solution is equivalent to the exact Kalman measurement update. We embed this closed-form evaluation within an $N$-step piecewise method for nonlinear measurement models. We further propose a constant contraction rate substep schedule that equalizes the per-step contraction along the eigendirection of $D$ associated with the largest eigenvalue $α_{\max}$. The result is a stiffness-mitigating, integration-free particle update for highly nonlinear measurement models. On a bearings-only tracking benchmark, it achieves the lowest error among the compared filters, at a per-update cost comparable to deterministic particle flow baselines and substantially lower than stochastic flows.

Rigidity conditions for a body considered as a discrete system of relativistic particles are proposed. They by themselves do not yet determine an evolution of the system, and some second-order equations must be added to them. Poincaré-covariant equations of motion compatible with these rigidity conditions are proposed and discussed. The resulting theory has the expected six dynamical degrees of freedom and therefore allows for more general motions than in Born's theory. Therefore, treating a relativistic body as a discrete system of particles could be a promising alternative to the standard approach based on Born's rigidity conditions.

Programming the static friction of mechanical interfaces is critical for soft robotics, haptics, and precision gripping. Static friction is governed by the real contact area, and standard rough surfaces exhibit a linear area-load scaling inherent to classical Archard and Greenwood-Williamson models, severely restricting their functional range. Here, we propose a framework for the inverse design of tribological metainterfaces engineered for programmable contact behaviors. By utilizing general axisymmetric asperities, we unlock nonlinear macroscopic responses unattainable by standard Hertzian contacts. To solve the inverse problem, we embed a fully differentiable contact mechanics engine within a neural network and a quadratic optimizer. We leverage regularized physical gradients to automatically discover non-standard topographies that reproduce complex target friction laws, with only a few asperities in unit cells. The predicted designs are strictly validated against high-fidelity Boundary Element Method (BEM) simulations. This framework bridges data-driven optimization and rigorous physics, offering a scale-invariant pathway for discovering functional tribological surfaces.

An important question in binary black hole mergers is to connect properties of the remnant black hole to those of the two initial black holes. These properties include not only the final mass and spin of the remnant, but also higher multipoles and answers to other questions such as, for a given initial configuration, which quasi-normal modes of the final black hole are excited, and what are the amplitudes of these modes? Such questions have thus far been primarily addressed through a study of the emitted gravitational wave signal. In this paper we consider a different alternative, namely using quasi-local black hole horizons themselves to establish the link between the initial and final states. Recent work has elucidated the behavior of black hole horizons in a merger. Cusps forming in such otherwise smoothly evolving horizons have been shown to play a central role in connecting the two initially separate black holes with the final remnant. In the present work, we will discuss from a numerical perspective how such cusps form in detail for the head-on collision of two non-spinning black holes. We show how the mass and higher mass multipole moments behave at the cusp and suggest a phenomenological model.

We show that the entanglement entropy (EE) of one-dimensional (1d) non-interacting fermions with $U(1)$ symmetry in the presence of a disordered or quasi-periodic potential in which the occupation number is being monitored by homodyne or projective protocols is always in an area-law phase so no measurement induced phase transition (MIPT) occurs. The reason for the previously claimed MIPT in these systems was a finite size effect related to the fact that the maximum lattice size $L \sim 500$ was of the order of the correlation length. By increasing the system size up to $L \leq 18000$, employing Graphics Processing Unit (GPU), and performing a careful finite size scaling analysis, we find that the critical monitoring strength is consistent with zero so no MIPT occurs. For the disordered case, these numerical results are fully supported by an analytical calculation based on mapping the problem onto a nonlinear sigma model (NLSM) that confirms the absence of the MIPT for any monitoring or disorder strength. The effect of disorder is captured by a change of symmetry, from BDI to AIII, which results in an enhanced correlation length in the weak disorder limit and, by an effective monitoring strength that increase linearly with disorder.

In this paper, we consider an autonomous semi-dynamical system driven by semilinear time-nonlocal evolution equations, these type equations are used to describe the Rayleigh-Stokes problem for a non-Newtonain fluid to a generalized second grade fluid. We first investigate the global well-posedness of solutions consisting of global Lipschitz condition by a weighted space $\mathcal C$. Utilizing the topology convergence on compact subsets of $\mathcal C$, we construct a semi-dynamical system that satisfies the semi-group structure. It also is shown that this semi-dynamical system has an attracting set when the vector field function satisfies a dissipativity condition and a local Lipschitz condition. With the asymptotic compactness, we also establish the existence of generalized attractors in $\mathcal C_α$ of subspace of $\mathcal C$ the weighted norm.

We describe a nonlinear kagome lattice with nonlinear dynamics described by Klein-Gordon interactions with a scalar unknown at each node, such as might occur in a nonlinear electrical lattice. We show that the dispersion relation has three bands - a flat band and two other surfaces which may meet in Dirac points or be separated by a gap. By using multiple scales asymptotic methods, we find a variety of reductions to nonlinear Schrodinger (NLS) systems, some of which are similar to those obtained previously, and have the Townes soliton as a solution. We find a novel system of coupled NLS equations, by forming an asymptotic expansion for small amplitude weakly nonlinear waves around the point where the flat band meets the upper surface of the dispersion relation. We analyse this 2+1 dimensional system using Lie symmetries, and find further reductions to more complicated solitary wave solutions. Numerical simulations of the wave are also presented.

Extremal functions for the $n$th coefficient in the Krzyż conjecture are atomic singular inner functions with at most $n$ atoms. This paper gives a lower bound on the number of atoms $N$ of the form $N\geq cn$, marking progress toward proving the expected $N=n$. Furthermore, we prove new formulas for extremal functions using variational techniques. Using these results and several other methods, we establish new conditions on extremal functions which are equivalent to the Krzyż conjecture being true. We also characterize the possible analytic invariants of extremal functions.

We present the AI weather and climate model intercomparison project (AIMIP), phase 1. Drawing from the rich tradition of intercomparisons in climate model development, we specify a common experiment, output data format, and training constraints (namely, training against historical reanalysis data) for AIMIP Phase 1 models. We aim to identify differences in modeling frameworks and AI architectural choices that influence model behavior, and build trust in AI weather and climate models through open data and evaluation. AIMIP Phase 1 models must simulate the atmosphere given specified historical sea surface temperatures over 1979-2024. We evaluate the models' performance using five major evaluation criteria: biases, trends, response to El Niño-related sea surface temperature anomalies, temporal variability, and out-of-sample generalization tests. We find that the AI models are able to simulate the historical climate and response to forcing as well as a conventional physically-based model, but some AI models underestimate historical warming trends, and their predictions diverge in the out-of-sample generalization tests. We describe the AIMIP Phase 1 dataset that is publicly available for additional evaluations.

We introduce the pro-tensor network, a categorification of the tensor network, as a fully rigorous yet graphically transparent framework for studying the collection of many many-body theories, which we dub many-many-body theory. We provide a comprehensive toolbox for the graphical calculations using pro-tensor networks. As applications, we recover the Levin-Wen model as a "uniform" pro-tensor network and generalize a result of Kitaev and Kong by characterizing particles as modules over promonads. One can also interpret the string-net pro-tensor network as the space of symmetric tensor networks, thus our framework also applies to the study of generalized symmetry and topological holography. Notably, our generalization dispenses with the assumptions of semisimplicity, finiteness, and rigidity, potentially facilitating the exploration of many-body physics beyond these constraints.

This work introduces a novel, nonparametric pixel-based framework for the Bayesian inference and imaging of transverse momentum dependent (TMD) parton distributions. The methodology is built upon a fully differentiable framework that integrates TMD evolution with the Collins-Soper-Sterman formalism, enabling the simultaneous extraction of partonic distributions and the nonperturbative evolution kernel. To achieve efficient and exact sampling of the high-dimensional posterior, we leverage generative AI through a hybrid normalizing flow-driven Metropolis-Hastings approach. The framework is validated through multi-scale closure tests of increasing complexity, ranging from basic functional models to convoluted structure functions. Using singular value decomposition (SVD), we rigorously characterize the uncertainty of the reconstructed distributions and reveal the existence of null TMDs, which are functional components in the null space of the kernel that remain unconstrained by observables. The new framework provides the first integration of pixel-based discretization, generative AI, and SVD within a Bayesian context to solve the TMD inverse problem. This synergy between machine learning and multi-scale data removes inherent degeneracies and enables unbiased 3D partonic imaging.

Electro-optic modulation is central to classical optical communications and emerging quantum technologies. High-confinement optomechanical crystal modulators enable microwave-optical transduction through strong optomechanical interactions and offer a promising interface between superconducting qubits and optical fibers. However, their performance is limited by thermal noise from optical absorption. Release-free optomechanical crystals provide improved thermal anchoring but have not yet been integrated into a microwave-optical transducer. Here, we demonstrate a release-free electro-optomechanical transducer combining strong optomechanical interactions in silicon with the efficient piezoelectricity of lithium niobate via micro-transfer printing. We observe electro- and optomechanical coupling rates compatible with quantum-level operation when co-integrated with a superconducting microwave circuit. This advance moves release-free electro-optomechanical devices toward practical microwave-optical interfaces.

We study finite probability theory through a category of finite probability schemes and probability-preserving maps, called \emph{bundles}. A bundle simultaneously records a quotient of a sample space, an algebra of random variables, and the family of conditional schemes over the quotient. The two natural linear functors associated with a bundle give a compact construction of conditional expectation and explain its projection properties. Within this framework we recover the laws of total expectation, variance, and covariance, the weak law of large numbers, and the variance decomposition behind simple linear regression. Fiber products then encode conditional independence and discrete-time Markov chains.

We give a polynomial-time algorithm to sample from the Gibbs measure of the Sherrington-Kirkpatrick model with negligible total-variation distance (TVD) error up to inverse temperature $β< 1/2$. Prior work obtained TVD error guarantees only up to $β\approx 0.295$, while results covering the entire replica-symmetric regime $β< 1$ gave guarantees only in Wasserstein distance. Our approach demonstrates that the same potential Hessian ascent previously developed for optimization also functions as a sampling algorithm by implementing algorithmic stochastic localization at high temperature. By estimating the covariance of the tilted Gibbs distribution via Gaussian integration by parts, overlap concentration, and precise cavity estimates, we show that a Hessian-ascent process achieves an $O(1)$ Wasserstein error guarantee for finite-time localization, improving on the previous $o(n)$. A careful comparison of stochastic localization with the Hessian ascent process and a free probability argument controlling the diagonal sub-algebra of the Hessian improves this to $O(1)$ in KL divergence. We then use Jarzynski's equality with rejection sampling, along with entropy contraction on the time-$T$ localized distribution, to refine the error to $o(1)$ in TVD up to a constant time $T$ and to complete the sampling with the polarized walk.

Based on a renormalization group (RG) analysis, we study the bosonization formulas in spin-$S$ Kitaev-Gamma and Kitaev-Heisenberg-Gamma chains in the $(K<0,Γ>0,J>0)$ parameter region, where $S$ is a half-odd integer. We find that the effects associated with the breaking of emergent continuous symmetries in bosonization formulas scale as $1/S$ in the large-$S$ limit, which is in qualitative agreement with DMRG numerical results for Kitaev-Gamma chains. In Kitaev-Heisenberg-Gamma chains, symmetry analysis reveals ten independent bosonization coefficients, five of which are predicted by the RG analysis to have no dependence on the Heisenberg coupling up to linear order. Our work may offer valuable input for determining magnetic ordering tendencies in two-dimensional Kitaev spin models within a quasi-one-dimensional approach.

We establish a novel connection between the minimal nilpotent orbit $\mathbb{O}_n$ in $\mathfrak{sl}_n$ and the minimal nilpotent orbit closure $\overline{\mathbf{O}}_n$ in $\mathfrak{so}_{2n+2}$, which differs from the shared-orbit paradigm of Brylinski and Kostant, where no direct type-A--type-D relation appears. More precisely, we show that the affine closure of the cotangent bundle $\overline{T^*\mathbb{O}_n}^{\mathrm{aff}}$ is isomorphic to a $\mathbb{C}^*$-Hamiltonian reduction of $\overline{\mathbf{O}}_n$. This provides a quasi-classical analogue of a quantum result of Levasseur and Stafford. A detailed study of the geometry of this Hamiltonian reduction reveals that $\overline{T^*\mathbb{O}_n}^{\mathrm{aff}}$ has no symplectic resolution.

In A Study of Spinoza's Ethics (1984, §17), Jonathan Bennett argues that the demonstration of Proposition V of Spinoza's Ethica contains identifiable invalid moves and that, even granted those moves, "cannot yield more than the conclusion that two substances could not have all their attributes in common" -- while Spinoza concludes that they cannot share any. Bennett doubts that any valid reconstruction is available from Spinoza's stated resources without importing further commitments. Michael Della Rocca (Spinoza, 2008, ch. 2) responds that the proposition can be derived if the Principle of Sufficient Reason (PSR) is committed substantively. The debate has remained at the level of prose argument for forty years. This paper provides the first machine-checked evidence in the debate. We formalise Ethica Pars I in Lean 4, encoding Bennett's reading of Spinoza's stated axioms as a typeclass and Della Rocca's substantive PSR as an extension class. The derivation attempt yields a partial result -- substances sharing all attributes are identical -- but cannot reach the full "sharing-any-attribute -> identity" content of Proposition V, mechanically tracking Bennett's own all-attributes ceiling. A four-element counter-model satisfying both axiom sets while falsifying Proposition V's content establishes the irreducibility against this specific augmentation. A second counter-model establishes the analogous result for axiom A15, a load-bearing universality clause for Spinoza's Proposition XIV. Bennett's diagnosis receives its first kernel-checked counter-model against the Della-Rocca PSR-substance reconstruction (the non-derivability claim itself a meta-logical consequence of kernel consistency); stronger PSR variants and the broader narrative claim against the full Section I + II + A1--A7 register remain open as future mechanical projects.

Quantum key distribution (QKD) theoretically offers information-theoretic security. The prevailing approach is the prepare-and-measure BB84 protocol, which implements QKD using conventional laser rather than single-photon source via the decoy-state method. However, side-channel attacks targeting sources severely threaten system security. Despite extensive efforts, including fully passive scheme, this vulnerability persists even with perfect single-photon source. Here, we propose a source-independent (SI) QKD protocol that resolves all known and unknown source-side attacks without pre-sending entanglement source. Aligning with advances in quantum light sources, our protocol simultaneously doubles the transmission distance while remaining robustness against imperfection of source. Theoretical analysis shows that non-classical light source provides practical security advantages unattainable with conventional laser.

Simulating quantum dynamics is one of the central applications of quantum computing. For Hamiltonians written as a sum of many terms, deterministic Trotter--Suzuki product formulas can require applying a large number of term-wise evolutions at each time step, leading to high circuit costs for large or dense systems. Randomized methods such as qDRIFT offer an alternative: each step samples only one Hamiltonian term, giving a circuit depth with no explicit dependence on the number of terms. However, when qDRIFT is used for observable estimation, high precision requires many independent random circuit realizations, resulting in a total gate complexity that scales as $\mathcal{O}(\varepsilon^{-3})$. We introduce a multilevel Monte Carlo framework for qDRIFT that reduces this sampling overhead. The method constructs a hierarchy of qDRIFT estimators with increasing circuit depths and couples adjacent levels by sharing their random Hamiltonian-term samples. This coupling makes the variance of the level differences decay with depth, allowing most samples to be taken on cheaper, coarse circuits and only a few on expensive, fine circuits. We prove that the resulting MLMC-qDRIFT estimator reduces the total gate complexity for fixed-precision observable estimation from the standard qDRIFT scaling $\mathcal{O}(\varepsilon^{-3})$ to $\mathcal{O}(\varepsilon^{-2}\log^2(1/\varepsilon))$, while preserving qDRIFT's lack of explicit dependence on the number of Hamiltonian terms. Numerical experiments for spin-chain dynamics confirm the predicted variance decay and demonstrate the practical gate-count savings of the multilevel construction.

Linking the composition of gas giant planets to their formation paths has long been a goal in exoplanet science. Especially, cold gas giants with temperatures below $\sim$500K have been out of reach for detailed atmospheric characterization. With JWST, however, we can reach high signal-to-noise (S/N) spectra for such cool worlds and can can measure not only their main trace gas abundances, but even their isotopic content unlocking new possibilities in linking them to their formation paths. In this study, we present the spectrum of one of the coldest planetary-mass companions COCONUTS-2b ($\mathrm{T_{eff}}\approx$480K, separation of $\sim$6400 au from its M dwarf host star) obtained with the Mid-InfraRed Instrument Medium Resolution Spectrometer (MIRI/MRS). Combining the MIRI and archival Gemini/FLAMINGOS-2 data sets, we aim to characterize the chemical composition and physical structure of its frigid atmosphere, setting the stage to uncover insights on the formation of COCONUTS-2b. For the first time on a MIRI/MRS data set, we use the full spectral resolution of MIRI/MRS and perform atmospheric retrievals to unlock the search for faint absorption features by rare molecules and isotopologues. The latter are identified using a leave-one-out analysis and Bayes factor comparison. We robustly detect three isotopologues, namely $^{15}$NH$_3$, H$_2^{18}$O and H$_2^{17}$O in the atmosphere of COCONUTS-2b. We find the first clear evidence of oxygen isotopes in water in a cold companion. This data set demonstrates the capability of MIRI/MRS to characterize such cold planetary-mass companion's atmospheres with respect to their compositional and isotopic content. In the future, the constrained elemental and isotope ratios provide a unique avenue in comparing with the host star's abundances and eventually in tracing formation scenarios.

We study Chebyshev quotients that arise in the representation theory of Lie algebras, specifically within the theory of Demazure flags for fusion products of $\mathfrak{sl}_2[t]$-modules. Using a recent formula that expresses numerical Demazure multiplicities as coefficients of such quotients, we prove a general eventual non-negativity theorem for the same rational functions that compute these multiplicities: each quotient either terminates or has strictly positive coefficients for sufficiently large degrees, which we in turn interpret in terms of matchings and bounded walks. In several natural infinite families, these are unsigned bounded Dyck path models, giving both a structural explanation for the observed positivity phenomenon and concrete combinatorial models for key families of Demazure multiplicities. The theorems in this paper were autonomously produced and formalized in Lean/Mathlib by AxiomProver from natural-language statements.

A subset of a topological space is constructible if it is a finite Boolean combination of closed sets. We prove that every NTP$_2$ expansion of $(\mathbb{R},<,+)$ by constructible sets defines only constructible sets, and that definable functions are generically piecewise continuous. The result also holds for all NTP$_2$ expansions of $(\mathbb{Q}_p,+,\cdot)$, and all NTP$_2$ definably complete expansions of ordered groups. In the latter case, the structure is generically locally o-minimal, has definable choice, and carries a well-behaved notion of naive topological dimension. For NIP uniform topological structures, constructibility of definable sets is preserved in the Shelah expansion. We classify strong expansions of $(\mathbb{R},<,+)$ by constructible sets, and obtain results on NTP$_2$ d-minimal structures.

Classical multivariate statistical methods such as covariance estimation and principal component analysis are well understood mathematically, yet their application at extreme data scales remains challenging. When the number of observations reaches billions, performance is limited by data movement, input-output bottlenecks, and numerical stability rather than arithmetic complexity. This work presents a case study of scaling classical multivariate statistics on a single multi-GPU node. Using C++ and CUDA, a GPU-accelerated workflow was developed to compute sufficient statistics in a single pass over a 10-billion-row dataset. Column sums and cross-product matrices are used to enable downstream computation of means, covariance, correlation, and principal component analysis without revisiting the raw data. The results highlight the importance of data representation, validation using known invariants, and careful numerical treatment when applying established statistical methods at large scale.

The radial acceleration relation (RAR) is a tight empirical correlation between the observed radial acceleration (a_tot) and the baryonic radial acceleration (a_bar) measured across galaxy radii: these two accelerations start to deviate significantly from each other below a characteristic acceleration scale, a0. So far, observational studies of the RAR have predominantly focused on galaxies in the local Universe, leaving its evolution with cosmic time largely unexplored. Using high signal-to-noise data from the MUSE Hubble Ultra Deep Field survey, we investigate the RAR with a sample of 79 star-forming galaxies (complete above M* >10^8.8 Msun) at intermediate redshifts (0.33 <z <1.44). We estimate the observed intrinsic acceleration and the baryonic acceleration from a disk-halo decomposition that incorporates stellar, gas, and dark matter components, with corrections for pressure support, using 3D forward modelling. We find a RAR in our intermediate-z sample offset from the local relation, with a higher characteristic acceleration scale, a0(z~1) = 2.38+/-0.1* 10^-10 m/s^2, and a larger intrinsic scatter (~0.17 dex). Dividing the sample into redshift bins and refitting the RAR in each bin, we find a characteristic acceleration scale that systematically increases with z. Parametrizing the z-dependence as a0(z)= a0(0) + a1 * z, we obtain a1 = 1.59+/-0.1 *10^-10 m/s^2, providing evidence for a z-evolution. We find similar results using various dark matter halo profiles as well as the Modified Newtonian Dynamics framework in our 3D forward modelling. Our results show that the RAR persists at intermediate redshift, with statistically significant redshift evolution of the characteristic acceleration, pointing to a possible evolution of the baryon-missing mass connection over cosmic time.

We study rational functions over finite fields under PGL-equivalence. We say that $f, g \in \Bbb F_q(X)$ are \emph{equivalent} if there exist $ψ, ϕ\in \Bbb F_q(X)$ of degree one such that $g = ψ\circ f \circ ϕ$. Most properties of rational functions over finite fields as they appear in theory and applications are preserved under this equivalence. In a recent work, Mattarei and Pizzato classified rational functions of degree three over finite fields in even characteristic. In the present paper, we classify all rational functions of degree three over finite fields in odd characteristic. Our approach is based on careful analyses of the value frequencies and the ramification points of the degree three rational functions. The completion of our classification also relies on an explicit formula for the number of equivalence classes of degree three rational functions over finite fields recently obtained by the first author.

Quantum key distribution (QKD) has reached a commercially viable stage, with several companies offering hardware systems designed for operational deployment. Evaluating the performance of commercial QKD devices under real-world deployment conditions is essential for users seeking to understand the practical limitations and operational reliability of these systems. In this paper, we present a long-term performance analysis of ID Quantique's Clavis XGR deployed within the Hermes Quantum Network, in Brazil. Our study provides a detailed characterization of key operational metrics, such as secret key rate, quantum bit error rate (QBER), visibility, and detection counts, mapping their behavior over extended periods of continuous operation. We analyze the system's stability across two distinct optical links: a 40.3 km indoor spooled fiber and a 3.5 km outdoor deployed underground fiber. Monitored under both unregulated tropical ambient fluctuations and actively controlled thermal stress, our results demonstrate excellent overall baseline resilience, with the system maintaining visibility above 97% and QBER below 1% on average. These findings provide practical insights into the expected behavior and thermal bottlenecks of commercial QKD systems in field deployments, particularly in tropical climates, helping to inform realistic expectations for operational quantum-safe infrastructures.

Interacting fermions exhibit a rich landscape of surface defects and associated critical phenomena. We investigate novel surface critical behavior in the three-dimensional Gross-Neveu-Yukawa model. For a class of defect renormalization group flows, we obtain exact infrared solutions and show how fermionic anomalies are encoded in the resulting surface dynamics. We further uncover emergent topological and geometric structures in the defect coupling space, and comment on their relation to a defect analogue of the CFT distance conjecture.

We conduct a photometric study of star clusters (or knots) in the collisional ring galaxy (CRG) Arp 147 to trace the star formation history across its empty ring. Using HST F450W, F606W and F814W images, we find that Arp 147 hosts 211 knots and six kpc-size clumps, nearly 60 per cent of which have ages below 10 Myr, and two thirds have masses above $\rm 10^{5}\,M_{\odot}$. The cluster mass function (CMF) of knots with ages between $10 - 200$ Myr deviates from a power-law and follows a Schechter function with a characteristic truncation mass of ${\rm M}_{c} = 6.2 \times 10^{5} \, {\rm M}_{\odot}$. This shape of the CMF is more prominent for a subsample of knots in the eastern region of the ring. Over the same age interval, we derive a low rate of disruption ($δ\sim 0.25$) from the cluster age function and a cluster formation efficiency (CFE) of $\sim$ 3 per cent. In contrast, the CFE in the $1 - 10$ Myr age range is nearly 40 per cent. We note the lack of high-resolution UV and H$α$ observations to help break age-extinction degeneracy which affects the derived ages for dusty young clusters and old ones with low reddening. Nevertheless, this study has shown, at least to a first-order approximation, that collision-triggered starburst events happening across the CRG offer an ideal environment for a second generation of young blue knots to form in abundance. It also suggests that the drop-through collision between the two galaxies can fuel at least mild cluster disruption over time.

Variability is a well known phenomenon in low-mass young stellar objects, but in recent years the monitoring of methanol masers and infrared continuum emission has permitted the detection of both burst-like episodes and periodic variations also in high-mass (proto)stars. Multi-epoch studies on large samples of these objects have become possible thanks to the NEOWISE database, which surveyed the sky in the mid-IR for about a decade. Our goal is to analyse the mid-IR emission from the well studied massive protostar IRAS20126+4104 and confirm the hypothesis that such emission is periodic, as proposed in previous studies. We take advantage of the NEOWISE, ALLWISE, and Spitzer databases to obtain 24 images of the 3.4 $μ$m emission from IRAS20126+4104 spanning 19 years, with $\sim$6 months sampling over a decade. With these data we create a light curve for each lobe of the bipolar nebulosity/outflow associated with the protostar. Our results confirm that the IR emission from IRAS20126+4104 varies regularly with a period of $\sim$6.8 yr. The period is the same for both lobes, but their emissions are anticorrelated with a phase difference of $\sim$2.5 yr. The variation is consistent with that found in previous studies for the 6 GHz CH$_3$OH masers and the near-IR emission from the lobes. After discussing four possible ``clocks'' that could determine the observed periodicity, we rule out all but a model involving rotation of the star with a spot obscuring $\sim$20% of the stellar surface. The long rotation period implies that the 12 $M_\odot$ protostar is bloated, with a radius of $\sim$200 $R_\odot$.

We introduce a two state vector formalism of quantum mechanics by generalizing Bell hidden variable model to higher dimensions and by attributing a physical significance, a state evolving backward in time, to the hidden variable. A simple deterministic and time symmetric rule for measurement outcomes allows us to obtain the Born rule. It turns out that probabilistic outcomes can be derived from a deterministic assignment and averaging over future states that propagate backward to the present. The assignment rule provides an alternative statement and demonstration of the Pusey, Barrett, Rudolph theorem.