The discovery of topological phases has introduced a new dimension to materials science. Three-dimensional (3D) topological insulators (TIs) are a remarkable class of matter that is insulating in the bulk while hosting conductive topological surface states (TSSs) with unique charge and spin properties. High-order harmonic generation (HHG) has emerged as a powerful tool to probe condensed matter systems by providing insights into their electronic structure and dynamic behavior. Here, we investigate HHG in the prototype 3D-TI Bi$_2$Se$_3$. We demonstrate that the contributions of bulk and surface states to the harmonic emission can be controlled by tuning the thickness of thin film samples. An ultrathin (6 nm) film substantially enhances HHG from the surface states, while the bulk states dominate HHG in a thicker (50 nm) film. By applying a quasi-static terahertz perturbing field, we disentangle the bulk and surface responses and reveal the significant impact of the surface states' shift vector and Berry curvature on HHG. Our study provides effective methods for isolating the optical responses of TSSs from those of the bulk, which opens the door to resolving an ongoing debate regarding whether it is possible to reliably extract topological signatures in HHG.

The theta cycle of a modular form modulo a prime $p\geq 5$ is well understood. By contrast, the theta cycle modulo a power of $p$ is still mysterious and experimentally erratic. Here we completely determine the theta cycle of a weight $k < p$ modular form modulo $p^2$ on the initial segment of length $p$ and we prove exact values or nontrivial bounds for the weight filtrations on $p-2$ further segments of length $p - k + 1$. In particular, asymptotically as $p \to \infty$ we establish 50% of the theta cycle exactly, and we provide nontrivial bounds for 100% of it. We determine the first two low points exactly and $\left\lfloor \frac{p - k + 1}{2} \right\rfloor$ further low points at regular positions. Moreover, we detect low points at exceptional positions which solve a quadratic equation modulo $p$, and which disturb the otherwise regular structure in the segments that we exhibit.

We present a data-driven approach for accelerating the discovery of high-performance CoSb$_3$-based skutterudites by curating a comprehensive dataset of compositions with various filler elements from over 300 research articles. Leveraging large language models (LLMs), we extract and embed compositional representations, which are then used to train a regression head for predicting thermoelectric figure of merit. Compared to traditional deep neural networks relying on elemental descriptors such as atomic radii, our LLM-based model achieves significantly lower mean-squared error losses. We further employ the trained model to propose novel filler compositions with promising thermoelectric properties. Finally, we support these predicted candidates through density functional theory and molecular dynamics calculations to assess their electrical and thermal conductivity. This data-driven approach demonstrates the potential of combining natural language processing, machine learning, and quantum simulations for thermoelectric materials design.

In this work we propose a single rounding algorithm for the fractional solutions of the standard LP relaxation for $k$-clustering. As a starting point, we obtain an iterative rounding $(\frac{3^p + 1}{2})$-Lagrangian Multiplier-Perserving (LMP) approximation for the $k$-clustering problem with the cost function being the $p$-th power of the distance. Such an algorithm outputs a random solution that opens $k$ facilities \emph{in expectation}, whose cost in expectation is at most $\frac{3^p + 1}{2}$ times the optimum cost. Thus, we recover the $2$-LMP approximation for $k$-median by Jain et al.~[JACM'03], which played a central role in deriving the current best $2$ approximation for $k$-median. Unlike the result of Jain et al., our algorithm is based on LP rounding, and it can be easily adapted to the $L_p^p$-cost setting. For the Euclidean $k$-means problem, the LMP factor we obtain is $\frac{11}{3}$, which is better than the $5$ approximation given by this framework for general metrics. Then, we show how to convert the LMP-approximation algorithms to a true-approximation, with only a $(1+\varepsilon)$ factor loss in the approximation ratio. We obtain a ($\frac{3^p + 1}{2}+\varepsilon$)-approximation algorithm for $k$-clustering with cost function being the $p$-th power of the distance, for $p \geq 1$. This reproduces the best known ($2+\varepsilon$)-approximation for $k$-median by Cohen-Addad et al. [STOC'25], and improves the approximation factor for metric $k$-means from 5.83 by Charikar at al. [FOCS'25] to $5+\varepsilon$ in our framework. Moreover, the same algorithm, but with a specialized analysis, attains ($4+\varepsilon$)-approximation for Euclidean $k$-means matching the recent result by Charikar et al. [STOC'26].

Dual control addresses the trade-off between exploitation and exploration, where control inputs both regulate the system and generate informative data for estimation and identification. For certain problem classes, control and estimation can be designed independently without loss of optimality, a property known as the separation principle. However, in stochastic control problems with model uncertainty and constraints, this principle generally breaks down, and introduces the need for dual control. In this paper, we propose an information-weighted dual model predictive control (MPC) formulation and introduce metrics that quantify the dependence of the MPC policy on the uncertainty. We focus on parametric uncertainty in linear systems with Gaussian noise, though the metrics can be applied more broadly. Numerical results show that the dependence of the MPC policy on the posterior covariance is largest under high uncertainty and vanishes as the posterior covariance contracts, providing empirical evidence of the dual effect in closed loop. Moreover, the dual controller improves regulation performance and model accuracy compared to certainty-equivalent MPC.

For a graph $G$, the general reduced second Zagreb index is defined as $$GRM_λ(G) = \sum_{uv \in E} (deg(u) + λ) (deg(v) + λ),$$ where $λ$ is an arbitrary real number and $deg (v)$ is the degree of the vertex $v$. In this paper, we extend and correct the equality results from [N. Dehgardia, S. Klav\v zar, {\it Improved lower bounds on the general reduced second Zagreb index of trees}, preprint (2023)] regarding the minimal value of $GRM_λ$ for $λ\geq -1$ among trees with $n$ vertices and a maximal degree $Δ$. Furthermore, we complement these results with two distinct approaches to determine the minimum value of the general reduced second Zagreb index for molecular trees with $Δ= 3$ and $Δ= 4$ in $λ= -2$, and characterize the extremal trees.

Understanding ligand properties is essential for computational high-throughput screening of transition metal complexes. However, ligand properties such as net charge and other information such as their application area are often absent or inconsistently recorded in crystallographic datasets. Here, we construct a ligand dataset from 126,985 mononuclear transition metal complexes curated from the Cambridge Structural Database. Using an iterative charge-balancing workflow that combines complex charges, metal oxidation states, and consensus across crystallographic observations, we confidently assign net charges to 66,810 ligands among 94,581 identified unique ligand structures to curate the Boston Open-Shell Ligand (BOS-Lig) dataset. The workflow assigns ligand charges in homoleptic complexes first and then iteratively propagates these assignments across heteroleptic environments, allowing charges to be inferred even when direct charge information is unavailable. We analyze cases where simple heuristics such as the octet rule would have failed and introduce a purity metric to identify when our charge assignments may be incorrect. Each ligand is also classified in terms of its metal coordinating atoms and whether there are multiple variants (i.e., hemilability). We then link complexes to their associated journal abstracts and apply a topic-modeling workflow to link 25,146 ligands with functional application areas spanning reactivity, redox chemistry, biological chemistry, and photophysical chemistry. Together, we provide an experimentally grounded dataset of ligand chemical space that connects charge and functional application as a foundation for computational screening and data-driven ligand design.

Synchronization in one dimension displays generic scale invariance with universal properties previously observed in surface kinetic roughening and the wider context of the Kardar-Parisi-Zhang (KPZ) universality class. This has been established for phase oscillators and also for some limit-cycle oscillators, both in the presence of columnar (quenched) disorder and of time-dependent noise, by extensive numerical simulations, and has been analytically motivated by continuum approximations in the strong oscillator coupling limit. The robustness and the precise boundaries in parameter space for such critical behavior remain unclear, however, which may preclude further developments, including the extension of these results to higher dimensions and the experimental observation of nonequilibrium criticality in synchronizing (e.g.~electronic or chemical) oscillators. We here present complete numerical phase diagrams of one-dimensional synchronization, including saturation times and values, but, most importantly, also dynamical features giving insight into the gradual emergence of synchronous dynamics, based on systems of phase oscillators with either type of randomness. In the absence of synchronization, the dynamics evolves as expected for random deposition (for time-dependent noise) or linear growth (for columnar disorder), while a crossover from Edwards-Wilkinson to Kardar-Parisi-Zhang behavior (with the corresponding type of randomness) is observed as the randomness strength, or the nonoddity of the coupling among oscillators, is increased in the synchronous region -- their combined effect being partially captured by the so-called KPZ coupling. The distortion of scaling due to phase slips near the desynchronization boundary, a feature that is likely to play a role in experimental contexts, is also discussed.

We investigate Lyman-alpha forest flux statistics in mixed fuzzy dark matter (FDM) and cold dark matter (CDM) cosmologies using the Fluctuating Gunn-Peterson Approximation (FGPA) applied to hybrid Schrödinger-Poisson and N-body simulations. We evolve the dark matter distribution from z = 120 to z = 2 for an axion mass ( m_22 = 0.01) and FDM fraction (f_A = 0.1), and compare two realizations with identical initial conditions: one evolved with a particle-only approximation and one with full wave--mechanical dynamics. We find that, despite near-degeneracy in the nonlinear matter power spectrum, the corresponding Ly α flux power spectra differ at the 10 percent level on intermediate scales. This discrepancy arises from a strong suppression of small-scale velocity power in the Schrödinger--Poisson evolution, which is not captured by N-body treatments with matched initial transfer functions. As a result, the flux statistics cannot be fully characterized by the matter power spectrum alone, but depend sensitively on the dynamical evolution of the velocity field. These results demonstrate that wave-mechanical effects in FDM leave distinct kinematic imprints in Ly α observables beyond those associated with initial-condition suppression. While our analysis is based on an idealized FGPA framework, it isolates a mechanism by which mixed dark matter models can break degeneracies present in standard structure-based probes, motivating further investigation with full hydrodynamical simulations.

This paper investigates the design of chirp-layered superposition coding for LoRa, where an additional waveform is linearly superposed on a standard LoRa transmission with minimal impact on the LoRa demodulation process. We first show that any non-zero superposed signal perturbs the output of the standard dechirp-and-DFT demodulator, and then characterize the class of superposed waveforms that minimize this degradation under a given power budget. In particular, we show that a high spreading factor (high-SF) LoRa waveform superposed on a low-SF signal (e.g., SF12 on SF7) can be designed so that its impact on the standard LoRa demodulation remains small. As a result, within each low-SF symbol interval, the high-SF segment can be treated as a quasi-narrowband carrier that conveys an additional BPSK stream. We derive analytical error-rate expressions for both the low-SF LoRa layer and the superposed high-SF layer, and validate them through Monte Carlo simulations. The proposed chirp-layered superposition coding scheme improves the spectral efficiency of LoRa-based links and uses a relatively simple transceiver architecture.

Given a positive integer $n$, an $n$-ladder is a lower finite lattice whose elements have at most $n$ lower covers. In 1984, Ditor proved that every $n$-ladder has cardinality at most $\aleph_{n-1}$ and asked whether this bound is sharp, i.e., whether for each $n$ there is an $n$-ladder of cardinality $\aleph_{n-1}$. We isolate the notion of maximal $n$-ladder and use it to study Ditor's problem and related questions. We show that $\text{Add}(ω, ω_ω)$ forces every maximal $n$-ladder to have cardinality $\aleph_{n-1}$, and hence forces a positive answer to Ditor's question for every $n$. In particular, it is consistent that there are no maximal $3$-ladders of cardinality $\aleph_1$. However, we show that the existence of such a ladder follows from $\mathfrak{d}=\aleph_1$. Under $\clubsuit$, we construct a maximal $3$-ladder of breadth $2$. Finally, we prove that, consistently (under $\diamondsuit$), there exists a maximal $3$-ladder that is destructible by forcing with a Suslin tree.

In this paper, we establish global $C^{1, α}$ regularity for viscosity solutions to a class of singular and degenerate fully nonlinear elliptic equations subject to oblique boundary conditions. Our work extends the findings in \cite{BKO25} to a broader class of equations, notably encompassing the singular case.

Bound states are dissipation-resilient states that may emerge when quantum systems are strongly coupled to reservoirs with band gaps. We analyze an exactly solvable bosonic model for bound state existence and reproduce these results by a weak-coupling treatment of a supersystem composed of the original system and multiple reaction coordinates, which are individually representing small energy intervals of the reservoir spectral function. Within the perturbative supersystem treatment, the bound state stability results from its energy being inside the band gap. We discuss cases of multiple band gaps and also show that already in presence of weak interactions the bound state's lifetime is finite -- but can be increased by raising the system-reservoir coupling strength.

We develop a framework for studying and quantifying the risk of cascading failures in time-delay consensus networks, motivated by a team of agents attempting temporal rendezvous under stochastic disturbances and communication delays. To assess how failures at one or multiple agents amplify the risk of deviation across the network, we employ the Average Value-at-Risk as a systemic measure of cascading uncertainty. Closed-form expressions reveal explicit dependencies of the risk of cascading failure on the Laplacian spectrum, communication delay, and noise statistics. We further establish fundamental lower bounds that characterize the best-achievable network performance under time-delay constraints. These bounds serve as feasibility certificates for assessing whether a desired safety or performance goal can be achieved without exhaustive search across all possible topologies. In addition, we develop an efficient single-step update law that enables scalable propagation of conditional risk as new failures are detected. Analytical and numerical studies demonstrate significant computational savings and confirm the tightness of the theoretical limits across diverse network configurations.

The generating series of descendent invariants of stable pairs on 3-folds is conjectured to be rational and to satisfy a $q\leftrightarrow q^{-1}$ symmetry. We prove this conjecture for Fano 3-folds. We utilize the same path of stability conditions that Toda used in his proof of the Calabi--Yau version of the conjecture, relating stable pairs and $L$ invariants, and work of the two authors that allows an extension of Joyce's descendent wall-crossing formula to non-standard hearts of $D^b(X)$. We use Ehrhart theory to deal with the combinatorics coming out of the wall-crossing formula. Furthermore, we specialize the wall-crossing formula to primary insertions and prove a strong rationality result predicted by the Pandharipande--Thomas/Gopakumar--Vafa correspondence.

Detection of parity violation on cosmological scales would have profound implications for fundamental physics. Motivated in part by recent measurements of parity-odd four-point correlation functions in BOSS and DESI luminous red galaxy samples, which probe parity violation in the scalar sector, we present the first measurement of parity-odd kurto spectra in spectroscopic galaxy survey data. We analyse two composite-field spectra, $\mathcal{P}_{2\times2}$ (vector--pseudo-vector) and $\mathcal{P}_{3\times1}$ (scalar--pseudo-scalar). Compared with parity-odd four-point correlation function analyses, the kurto-spectrum formalism performs physically motivated compression on the trispectrum into a substantially lower-dimensional data vector, allowing direct estimation of covariance matrices from mock catalogues and reducing sensitivity to covariance-modelling systematics. Using null-hypothesis $χ^2$ tests and cross-patch consistency checks, we find no evidence for a cosmological parity-violating signal in either survey. We examine the impact of the adopted mock catalogues and find that the high-fidelity mocks provide a better match to the data of both surveys than the approximate mocks. The DESI DR1 measurements exhibit a scatter smaller than that of BOSS DR12 by about a factor of four, consistent with the improved statistical precision expected from the higher tracer number density. Future DESI data releases, with larger volume and number density, together with larger suites of high-fidelity mocks, can enable significantly sharper tests of parity violation using kurto spectra.

To optimize diffusion MRI acquisition protocols for IMPULSED model at clinical 3T scanner using Bayesian experimental design, enabling accurate cellular-scale parameter estimation under realistic scan time and scanner hardware constraints. Expected Information Gain (EIG) was used as the optimization objective to maximize the information content of acquired measurements for IMPULSED model fitting. Bayesian optimization with Gaussian process surrogates efficiently searched the high-dimensional acquisition parameter space, including pulse types (PGSE, OGSEn1, and OGSEn2), diffusion times, and b-values. Optimized protocols were systematically evaluated against a heuristically designed baseline protocol through simulation studies assessing classification accuracy and parameter estimation performance across SNR levels of 5-40. Robustness to optimization assumptions was examined by varying prior distributions and assumed SNR. In-vivo validation was performed using canine tumor data acquired at 3T. The optimized protocol eliminated OGSEn2 acquisitions, concentrated measurements at high b-values, employing concurrently optimized diffusion timing. Compared to the baseline protocol, the optimized design achieved superior classification accuracy for distinguishing cell populations and reduced parameter estimation error across biologically relevant parameter ranges at various SNRs. Performance advantages were consistent across diverse optimization scenarios, demonstrating robustness to prior knowledge and noise assumptions. In-vivo parameter maps showed substantially improved quality and smoothness. Bayesian optimization substantially improves IMPULSED acquisition design for clinical 3T scanners. This principled, algorithm-agnostic framework enables accurate diffusion MRI cytometry under clinical constraints, with potential applications to tumor characterization and treatment monitoring.

Spectral hypergraph theory studies the structural properties of a hypergraph that can be inferred from the eigenvalues and the eigenvectors of either matrices or tensors associated with it. In this paper we study the spectral indistinguishability in the hypergraph setting. We present a general switching method to construct uniform $E$-cospectral hypergraphs (hypergraphs with the same $E$-spectrum), and discuss some of its multiple applications. Our method not only provides a framework to unify the existing methods for obtaining $E$-cospectral hypergraphs via switching, but also generalizes most of the existing switching tools, yielding multiple new constructions. Finally, we compare common methods of computing $E$-characteristic polynomials, and in particular show that one standard method, while useful for generic tensors, is uninformative for almost all hypergraphs.

This paper extends the study of fringe trees in random plane trees with a given degree statistic. While previous work established the asymptotic normality of the count of fringe trees isomorphic to a fixed tree, we investigate the case where the target tree grows with the size of the random tree. We consider three primary subtree counts: the number of fringe trees isomorphic to a specific growing tree, the number of fringe trees sharing a given growing degree statistic, and the number of fringe trees of a specific growing size. To establish our results, we employ and compare four distinct probabilistic frameworks: the method of moments with the Gao-Wormald theorem, Stein's method with coupling (to provide explicit error bounds in total variation distance), the Cai-Devroye method, and Stein's method with exchangeable pairs. Our findings provide conditions for Poisson and normal convergence for these subtree counts. Additionally, we provide a local limit theorem for sums of values obtained via sampling without replacement that may be of independent interest. Finally, our results and methods are also applied to conditioned critical Galton-Watson trees.

This note reports the following observation: the finite-volume expectation value of the spin operator (the one-point function) between the $\mathbb{Z}_2$-even and odd ground states in the critical periodic Ising chain, when continued as a complex-analytic function of the system length $N$ through the Borel resummation of its large-$N$ expansion, has a natural boundary of analyticity along the negative real axis. The singular behavior near the negative real axis, after an exponential map, is the same as that of a Lambert-type series for the odd-divisor-squared sum near the unit circle $|z|=1$. The same divisor sum also governs the strengths of the Borel discontinuities of the one-point function's factorially-divergent large-$N$ asymptotics. We also report the all-order large-$N$ asymptotics of the leg function for the finite-volume spin-operator form factor, and the similarities to certain known quantities in the literature.

Menstrual health education (MHE) in Pakistan is constrained by cultural taboos and inadequate formal curricula, leaving women with few trusted resources to lean on. In response to these challenges, we introduce a WhatsApp-based chatbot powered by a large language model (LLM) and Retrieval Augmented Generation (RAG), co-designed with Pakistani college women. Workshops (N=30) revealed key design requirements -- support for Roman Urdu, use of subsidized platforms, and an expert -- curated knowledge base. We then deployed the chatbot with 13 participants for two weeks (403 messages and interviews). Women used it to challenge cultural taboos, legitimize health concerns often dismissed as normal, and build reproductive health knowledge through iterative questioning. Yet, interactions also exposed tensions: reliance on cultural explanatory models, questions of trust and validation, and gendered persona of the chatbot itself. We contribute empirical insights, a stigma-aware design framework for culturally sensitive conversational AI, and a methodological lens foregrounding expert validation in intimate health domains.

Hybrid quantum neural networks are increasingly explored for classification, yet it remains unclear how their performance and quantum behavior scale with circuit depth and qubit count. We present a controlled scaling study of hybrid quantum-classical classifiers along two axes: (1) increasing the number of quantum layers L at fixed qubits Q, and (2) increasing the number of qubits Q at fixed depth L. Across multiple datasets, we evaluate predictive performance using Accuracy, PR-AUC, Precision, Recall, and F1, and track quantum-specific metrics (QCE, EEE, QGN) to characterize how quantum properties evolve under scaling. Our results summarize scaling trends, saturation regimes, and dataset-dependent sensitivity, and further analyze how quantum metrics relate to predictive performance. This study provides practical guidance for selecting (Q,L) in hybrid QNN classifiers and establishes a consistent evaluation protocol for scaling analysis.

The orbital inclination of an eclipsing binary is generally determined through light curve analysis. Binary parameters in the light curve analysis are typically constrained through the use of optimization and sampling techniques. We propose a new simple method, based on the derivatives of light curves, for estimating the orbital inclinations of overcontact systems. Our sample consists of 89670 synthetic light curves for overcontact binaries, covering a parameter space typical of overcontact systems. We classified the sample light curves on the basis of a recently proposed classification scheme: DP, SPp, SPb, SPf, and SPs types. For each type, we found that the orbital inclination is closely associated either with the time interval between local extrema in the derivatives of light curves or with the depth of the local minimum at phase 0.5 in the second derivative. Using regression analysis of the identified associations, we developed empirical formulae to estimate the orbital inclinations for each type of light curve. We also provide the associated uncertainties for the estimated inclinations. Application of the proposed method to real overcontact binary data demonstrated that our method can reasonably estimate both the inclinations and their uncertainties.

The pseudogap phenomenon is a hallmark of strongly interacting Fermi systems, from high-temperature superconductors to ultracold atomic gases, yet its precise origin remains debated. Here we calculate the spectral function and rf spectra of ultracold atomic gases across the BCS-BEC crossover to quantitatively investigate the pairing mechanism of the pseudogap. We advance our pairing fluctuation theory by incorporating particle-hole fluctuations, which renormalize the effective interaction in the particle-particle channel. To achieve quantitative accuracy, we employ a full numerical convolution for the pair susceptibility and self-energy, moving beyond previous analytic pseudogap approximations. This convolution approach automatically captures two critical effects: (i) the full spectral broadening of fermions due to finite pair lifetime, and (ii) the previously neglected pair-hole scattering effect, which manifests as a substantial Hartree energy. We calculate the spectral function, and use rf spectral intensity maps and energy distribution curves to determine the quasiparticle dispersion. From these, we extract the pseudogap $Δ$, Hartree energy, and chemical potential, mapping their evolution across the crossover. Our results show that the pseudogap emerges continuously as the system moves from the BCS regime toward BEC. Furthermore, the pair spectral function reveals that pairs become diffusive at energies above 2$Δ$, indicating that the pair lifetime is governed by virtual binding and unbinding processes. Our calculations achieve quantitative agreement with recent experiments across the BCS-BEC crossover, including at unitarity, providing strong support for a pairing-based origin of the pseudogap as described by our pairing fluctuation theory.

Let $(V, ϕ)$ be a holomorphic Lie algebroid over an irreducible smooth complex projective variety $X$ of dimension at least three, and let $E$ be a holomorphic vector bundle on $X$. We establish a necessary and sufficient condition for the existence of a holomorphic $(V, ϕ)$--connection on $E$.

We report a 3.3 $σ$ measurement of coherent elastic neutrino-nucleus scattering from solar $^8$B neutrinos using a 6.77 t$\times$yr exposure from the XENONnT experiment, inferring a solar $^8$B neutrino flux of $(5_{-2}^{+3})\times 10^6\,\mathrm{cm}^{-2}\mathrm{s}^{-1}$, consistent with previous measurements. In the presence of the $^8$B "neutrino fog", we find no evidence for light dark matter, and observe diminishing returns in sensitivity with increasing exposure. A 93% increase in exposure from the previous search improves the median sensitivity to 5 GeV/$c^2$ weakly interacting massive particles-nucleon cross section by 10%. The dataset was also used to measure the weak mixing angle at $\sim$ 0.02 GeV/$c$ momentum transfer and constrain physics beyond the Standard Model.

HCI work has explored the effective integration of AI/ML tools across "application domains" from healthcare to finance to transportation. We add to this literature with an analysis of AI/ML tools in meteorology, a domain that already uses "big data" and massive physics-based models. Drawing from 12 interviews with forecasters and meteorologists with varied connections to AI/ML weather modeling, we trace tensions in AI/ML weather application arising from what we call "regimes of scale," different ways that AI/ML and meteorological regimes make observations, data, and models scale. Rather than seeing AI/ML as a domain-agnostic tool, we argue that AI/ML methods were born from specific platform and internet infrastructures, and so they can struggle to integrate with very different (in this case meteorological) ways of organizing data pipelines.

Consider a branching process $\{Z_n\}$ in a varying environment. Let $\{W_n\}$ be the natural martingale $Z_n/{\bf E}Z_n$. It converges to some random variable $W$ as $n\to\infty$. An important problem is to show that ${\bf P}(W>0)$ equals the survival probability, so that $Z_n$ is either $0$ or of the order ${\bf E}Z_n$. We find a new kind of sufficient conditions, applicable to branching processes in a random environment. An important property of our estimates is that we don't necessary assume that ${\bf E}X_{i,1}^2$ are finite for every $i$.

Current theories of high-temperature superconductivity in flat-band systems predict a linear dependence of the transition temperature on the attractive interaction, $T_c(U) = c|U|$. However, neither the value of $c$ nor the full nonlinear $T_c(U)$ curve -- with a maximum at large $|U|$ -- is known beyond mean-field and quantum geometry estimates. Using a controlled diagrammatic Monte Carlo technique, we trace the onset of superfluid response in the Lieb lattice with attractive Hubbard interaction. Focusing on the half-filled flat-band case, where the ordering mechanism differs fundamentally from both BCS and preformed Cooper pair scenarios, we find that the pairing response diverges linearly with decreasing temperature over a broad range of $U$, leading to a sharp crossover to long-range correlations at a characteristic temperature $T_*$, which provides a controlled upper bound on $T_c$. The highest $T_*$ occurs when all three bands touch at a single momentum point, potentially corresponding to high $T_c$ values.

A search for a Higgs boson produced in association with a W or Z boson and decaying via a soft unclustered energy pattern (SUEP) is presented. The analysis is based on proton-proton collision data corresponding to an integrated luminosity of 138 fb$^{-1}$ collected between 2016 and 2018 at the LHC. Final states with a leptonic W or Z boson decay associated with a high multiplicity of low-momentum charged particles are explored for the first time. The results show no significant excess over the standard model background expectation. Limits are set on the production cross section of a Higgs boson that decays to a SUEP, for a range of parameters of the SUEP model. Material is provided to facilitate further interpretation of the results.